Preface

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About *Contemporary Mathematics*

*Contemporary Mathematics* is designed to meet the requirements for a liberal arts mathematics course. The textbook covers a range of topics that are typically found in a liberal arts course as well as some topics to connect mathematics to the world around us. The text provides stand-alone sections with a focus on showing relevance in the features as well as the examples, exercises, and exposition.

Pedagogical Foundation

About the Authors

Additional Resources

Sets

Introduction

Basic Set Concepts

Sets and Ways to Represent Them

Think back to your kitchen organization. If the drawer is the set, then the forks and knives are elements in the set. Sets can be described in a number of different ways: by roster, by set-builder notation, by interval notation, by graphing on a number line, and by Venn diagrams. Sets are typically designated with capital letters. The simplest way to represent a set with only a few members is the **roster** (or *listing*) **method**, in which the elements in a set are listed, enclosed by curly braces and separated by commas. For example, if $F$ represents our set of flatware, we can represent $F$ by using the following set notation with the roster method:

Computing the Cardinal Value of a Set

Almost all the sets most people work with outside of pure mathematics are finite sets. For these sets, the cardinal value or cardinality of the set is the number of elements in the set. For finite set $A$, the cardinality is denoted symbolically as $n\left(A\right)$. For example, a set that contains four elements has a cardinality of 4.

Equal versus Equivalent Sets

When speaking or writing we tend to use equal and equivalent interchangeably, but there is an important distinction between their meanings. Consider a new Ford Escape Hybrid and a new Toyota Rav4 Hybrid. Both cars are hybrid electric sport utility vehicles; in that sense, they are equivalent. They will both get you from place to place in a relatively fuel-efficient way. In this example we are comparing the single member set {Toyota Rav4 Hybrid} to the single member set {Ford Escape Hybrid}. Since these two sets have the same number of elements, they are also equivalent mathematically, meaning they have the same cardinality. But they are not equal, because the two cars have different looks and features, and probably even handle differently. Each manufacturer will emphasize the features unique to their vehicle to persuade you to buy it; if the SUVs were truly equal, there would be no reason to choose one over the other.

Subsets

Exponential Notation

So far, we have figured out how many subsets exist in a finite set by listing them. Recall that in [link], when we listed all the subsets of the three-element set $L=\left\{\mathrm{newspaper, magazine, book}\right\}$ we saw that there are eight subsets. In Your Turn 1.11, we discovered that there are four subsets of the two-element subset, $S=\left\{\mathrm{heads, tails}\right\}$. A one-element set has two subsets, the empty set and itself. The only subset of the empty set is the empty set itself. But how can we easily figure out the number of subsets in a very large finite set? It turns out that the number of subsets can be found by raising 2 to the number of elements in the set, using exponential notation to represent repeated multiplication. For example, the number of subsets of the set $L=\left\{\mathrm{newspaper, magazine, book}\right\}$ is equal to ${2}^{3}=2\cdot 2\cdot 2=8$. Exponential notation is used to represent repeated multiplication, ${b}^{n}=b\cdot b\cdot b\cdot \dots \cdot b$, where $b$ appears as a factor $n$ times.

Equivalent Subsets

In the early 17th century, the famous astronomer Galileo Galilei found that the set of natural numbers and the subset of the natural numbers consisting of the set of square numbers, ${n}^{2}$, are equivalent. Upon making this discovery, he conjectured that the concepts of less than, greater than, and equal to did not apply to infinite sets.

Understanding Venn Diagrams

Interpreting Venn Diagrams

Venn diagrams are the graphical tools or pictures that we use to visualize and understand relationships between sets. Venn diagrams are named after the mathematician John Venn, who first popularized their use in the 1880s. When we use a Venn diagram to visualize the relationships between sets, the entire set of data under consideration is drawn as a rectangle, and subsets of this set are drawn as circles completely contained within the rectangle. The entire set of data under consideration is known as the universal set.

Creating Venn Diagrams

The main purpose of a Venn diagram is to help you visualize the relationship between sets. As such, it is necessary to be able to draw Venn diagrams from a written or symbolic description of the relationship between sets.

The Complement of a Set

Recall that if set $A$ is a proper subset of set $U$, the universal set (written symbolically as $A\subset U$), then there is at least one element in set $U$ that is not in set $A$. The set of all the elements in the universal set $U$ that are not in the subset $A$ is called the complement of set $A$, $A\text{'}$. In set builder notation this is written symbolically as: $A\text{'}=\left\{x\in U|x\notin A\right\}.$ The symbol $\in$ is used to represent the phrase, “is a member of,” and the symbol $\notin$ is used to represent the phrase, “is not a member of.” In the Venn diagram below, the complement of set $A$ is the region that lies outside the circle and inside the rectangle. The universal set $U$ includes all of the elements in set $A$ and all of the elements in the complement of set $A$, and nothing else.

Set Operations with Two Sets

The Intersection of Two Sets

The members that the two sets share in common are included in the intersection of two sets. To be in the intersection of two sets, an element must be in both the first set and the second set. In this way, the intersection of two sets is a logical AND statement. Symbolically, $A$ intersection $B$ is written as: $A\cap B$. $A$ intersection $B$ is written in set builder notation as: $A\cap B=\left\{x|x\in A{\rule{0.28em}{0ex}}\text{and}{\rule{0.28em}{0ex}}x\in B\right\}$.

Set Operations with Three Sets

Venn Diagrams with Three Sets

Below is a Venn diagram with two intersecting sets, which breaks the universal set up into four distinct regions.

Creating Venn Diagrams with Three Sets

In general, when creating Venn diagrams from data involving three subsets of a universal set, the strategy is to work from the inside out. Start with the intersection of the three sets, then address the regions that involve the intersection of two sets. Next, complete the regions that involve a single set, and finally address the region in the universal set that does not intersect with any of the three sets. This method can be extended to any number of sets. The key is to start with the region involving the most overlap, working your way from the center out.

Applying Set Operations to Three Sets

Set operations are applied between two sets at a time. Parentheses indicate which operation should be performed first. As with numbers, the inner most parentheses are applied first. Next, find the complement of any sets, then perform any union or intersections that remain.

Proving Equality of Sets Using Venn Diagrams

To prove set equality using Venn diagrams, the strategy is to draw a Venn diagram to represent each side of the equality, then look at the resulting diagrams to see if the regions under consideration are identical.

Logic

Introduction

Statements and Quantifiers

Identifying Logical Statements

An example of logical statement with a false truth value is, “All roses are red.” It is a logical statement because it has the form of a complete sentence and makes a claim that can be determined to be either true or false. It is a false statement because not all roses are red: some roses are red, but there are also roses that are pink, yellow, and white. Requests, questions, or directives may be complete sentences, but they are not logical statements because they cannot be determined to be true or false. For example, suppose someone said to you, “Please, sit down over there.” This request does not make a claim and it cannot be identified as true or false; therefore, it is not a logical statement.

Representing Statements in Symbolic Form

When analyzing logical arguments that are made of multiple logical statements, symbolic form is used to reduce the amount of writing involved. Symbolic form also helps visualize the relationship between the statements in a more concise way in order to determine the strength or validity of an argument. Each logical statement is represented symbolically as a single lowercase letter, usually starting with the letter $p$.

Negating Statements

Consider the false statement introduced earlier, “All roses are red.” If someone said to you, “All roses are red,” you might respond with, “Some roses are not red.” You could then strengthen your argument by providing additional statements, such as, “There are also white roses, yellow roses, and pink roses, to name a few.”

Negating Logical Statements Symbolically

The symbol for negation, or not, in logic is the tilde, ~. So, not $p$ is represented as $~p$. To negate a statement symbolically, remove or add a tilde. The negation of not (not $p$) is $p$. Symbolically, this equation is $~\left(~p\right)=p.$

Translating Negations Between Words and Symbols

In order to analyze logical arguments, it is important to be able to translate between the symbolic and written forms of logical statements.

Expressing Statements with Quantifiers of All, Some, or None

A quantifier is a term that expresses a numerical relationship between two sets or categories. For example, all squares are also rectangles, but only some rectangles are squares, and no squares are circles. In this example, *all, some,* and *none* are quantifiers. In a logical argument, the logical statements made to support the argument are called premises, and the judgment made based on the premises is called the conclusion. Logical arguments that begin with specific premises and attempt to draw more general conclusions are called inductive arguments.

Negating Statements Containing Quantifiers

Recall that the negation of a statement will have the opposite truth value of the original statement. There are four basic forms that logical statements with quantifiers take on.

Compound Statements

Common Logical Connectives

Understanding the following logical connectives, along with their properties, symbols, and names, will be key to applying the topics presented in this chapter. The chapter will discuss each connective introduced here in more detail.

Translating Compound Statements to Symbolic Form

To translate a compound statement into symbolic form, we take the following steps.

Translating Compound Statements in Symbolic Form with Parentheses into Words

When parentheses are written in a logical argument, they group a compound statement together just like when calculating numerical or algebraic expressions. Any statement in parentheses should be treated as a single component of the expression. If multiple parentheses are present, work with the inner most parentheses first.

The Dominance of Connectives

The order of operations for working with algebraic and arithmetic expressions provides a set of rules that allow consistent results. For example, if you were presented with the problem $1+3×2$, and you were not familiar with the order of operation, you might assume that you calculate the problem from left to right. If you did so, you would add 1 and 3 to get 4, and then multiply this answer by 2 to get 8, resulting in an incorrect answer. Try inputting this expression into a scientific calculator. If you do, the calculator should return a value of 7, not 8.

Constructing Truth Tables

Interpret and Apply Negations, Conjunctions, and Disjunctions

The negation of a statement will have the opposite truth value of the original statement. When $p$ is true, $~p$ is false, and when $p$ is false, $~p$ is true.

Construct Truth Tables to Analyze All Possible Outcomes

Recall from Statements and Questions that the negation of a statement will always have the opposite truth value of the original statement; if a statement $p$ is false, then its negation $~p$ is true, and if a statement $p$ is true, then its negation $~p$ is false. To create a truth table for the negation of statement $p$, add a column with a heading of $~p$, and for each row, input the truth value with the opposite value of the value listed in the column for $p$, as depicted in the table below.

Determine the Validity of a Truth Table for a Compound Statement

A logical statement is valid if it is always true regardless of the truth values of its component parts. To test the validity of a compound statement, construct a truth table to analyze all possible outcomes. If the last column, representing the complete statement, contains only true values, the statement is valid.

Truth Tables for the Conditional and Biconditional

Use and Apply the Conditional to Construct a Truth Table

A conditional is a logical statement of the form if $p$, then $q$. The conditional statement in logic is a promise or contract. The only time the conditional, $p\to q,$ is false is when the contract or promise is broken.

Use and Apply the Biconditional to Construct a Truth Table

The biconditional, $p↔q$, is a two way contract; it is equivalent to the statement $\left(p\to q\right)\wedge \left(q\to p\right).$ A biconditional statement, $p↔q,$ is true whenever the truth value of the hypothesis matches the truth value of the conclusion, otherwise it is false.

Equivalent Statements

Determine Logical Equivalence

Two statements, $p$ and $q$, are logically equivalent when $p↔q$ is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology. To determine whether two statements $p$ and $q$ are logically equivalent, construct a truth table for $p↔q$ and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and $p$ is logically equivalent to $q$; otherwise, $p$ is not logically equivalent to $q$.

Compose the Converse, Inverse, and Contrapositive of a Conditional Statement

The converse, inverse, and contrapositive are variations of the conditional statement, $p\to q.$

* The converse is if $q$ then $p$, and it is formed by interchanging the hypothesis and the conclusion. The converse is logically equivalent to the inverse.
* The inverse is if $~p$ then $~q$, and it is formed by negating both the hypothesis and the conclusion. The inverse is logically equivalent to the converse.
* The contrapositive is if $~q$ then $~p$, and it is formed by interchanging and negating both the hypothesis and the conclusion. The contrapositive is logically equivalent to the conditional.

De Morgan’s Laws

Negation of Conjunctions and Disjunctions

In Chapter 1, [link] used a Venn diagram to prove De Morgan’s Law for set complement over union. Because the complement of a set is analogous to negation and union is analogous to an *or* statement, there are equivalent versions of De Morgan’s Laws for logic.

Negation of a Conditional Statement

The negation of any statement has the opposite truth values of the original statement. The **negation of a conditional**, $~\left(p\to q\right)$, is the conjunction of $p$ and not $q$, $p\text{}\wedge ~q.$ Consider the truth table below for the negation of the conditional.

Evaluating De Morgan’s Laws with Truth Tables

In Chapter 1, you learned that you could prove the validity of De Morgan’s Laws using Venn diagrams. Truth tables can also be used to prove that two statements are logically equivalent. If two statements are logically equivalent, you can use the form of the statement that is clearer or more persuasive when constructing a logical argument.

Logical Arguments

Law of Detachment

The law of detachment is a valid form of a conditional argument that asserts that if both the conditional, $p\to q$, and the hypothesis, $p$, are true, then the conclusion $q$ must also be true. The law of detachment is also called affirming the hypothesis (or antecedent) and modus ponens. Symbolically, it has the form $\left(\left(p\to q\right)\wedge p\right)\to q$.

Law of Denying the Consequent

Another form of a valid conditional argument is called the law of denying the consequent, or modus tollens. Recall, that the conditional statement, $p\to q$, is logically equivalent to the contrapositive, $~q\to \text{}~p.$ So, if the conditional statement is true, then the contrapositive statement is also true. By the law of detachment, if $~q$ is also true, then it follows that $~p$ must also be true. Symbolically, it has the form $\left(\left(p\to q\right)\wedge ~q\right)\to \text{}~p$.

Chain Rule for Conditional Arguments

The chain rule for conditional arguments is another form of a valid conditional argument. It is also called hypothetical syllogism or the transitivity of implication. Recall that the conditional statement $p\to q$ can also be read as $p$ implies $q$. This is where the name *transitivity of implication* comes from. The transitive property for numbers states that, if $3<4$ and $4<5,$ then it follows that $3<5.$ The chain rule extends this property to conditional statements. If the premises of the argument consist of two conditional statements, with the form “$p\to q$” and “$q\to r,$” then it follows that $p\to r.$ Symbolically, it has the form $\left(\left(p\to q\right)\wedge \left(q\to r\right)\right)\to \left(p\to r\right)$.

Real Number Systems and Number Theory

Introduction

Prime and Composite Numbers

Applying Divisibility Rules

Prime and Composite Numbers

Sometimes, a natural number has only two unique divisors, 1 and itself. For instance, 7 and 19 are prime. In other words, there is no way to divide a prime number into groups with an equal number of things, unless there is only one group, or those groups have one item per group. Other natural numbers have more than two unique divisors, such as 4, or 26. These numbers are called composite. The number 1 is special; it is neither prime nor composite.

Finding the Prime Factorization of Composite Numbers

Before we can start with prime factorization, let’s remind ourselves what it means to factor a number. We factor a number by identifying two (or more) numbers that, when multiplied, result in the original number. For instance, 3 and 24, when multiplied, give 72. So, 72 can be factored into $3×24$. Notice that we could have factored the 72 differently, say as $72=6×12$, or $72=2×36$, or even as $72=3×4×6$.

Fundamental Theorem of Arithmetic

Every natural number, other than 1, can be expressed in exactly one way, apart from the arrangement, as a product of primes.

Finding the Greatest Common Divisor

Two numbers often have more than one divisor in common (all pairs of natural numbers have the common divisor 1). When listing the common divisors, it’s often the case that the largest is of interest. This divisor is called the greatest common divisor and is denoted **GCD**. It is also sometimes referred to as the greatest common factor (GCF).

Applications of the Greatest Common Divisor

The greatest common divisor has uses that are related to other mathematics (reducing fractions) but also in everyday applications. We’ll look at two such applications, which have very similar underlying structures. In each case, something must divide the groups or measurements equally.

Finding the Least Common Multiple

The flip side to a divisor of a number is a multiple of a number. For example, 5 is a divisor of 45 and so 45 is a multiple of 5. More generally, if the number $a$ divides the number $b$, then $b$ is a multiple of $a$.

Applications of the Least Common Multiple

Some applications of LCM involve events that repeat at fixed intervals, such as visits to a location. Other applications involve getting things to be of equal magnitude when using parts that are different sizes (see [link], for instance). In each case, we may be looking to determine when two processes “line up.”

The Integers

Defining and Identifying Integers

Extending the counting numbers to include negative numbers and zero forms the integers. Any other number that cannot be written as $\left\{\dots -3,-2,-1,0,1,2,3,\dots \right\}$ is not an integer.

Graphing Integers on a Number Line

Integers are often imagined as steps along a path. You start at 0, and going to the left is going backward, or in the negative direction, while going to the right is going forward, or in the positive direction. A number line ([link]) helps envision the integers. This also means that an integer gives magnitude (size) and direction (positive is to the right, negative is to the left). Graphing an integer on the number line means placing a solid dot at the integer on the number line.

Comparing Integers

When determining if one quantity or size is larger than another, we know it means there is more of whatever is being discussed. In terms of positive integers, we can envision that larger integers are further to the right on the number line. This idea applies to negative integers also. This means that $\mathbit{a}$ is greater than $\mathbit{b}$ when $a$ is to the right of $b$ on the number line. We write $a>b$. When $a$ is greater than $b$, we can also say that $\mathbit{b}$ is less than $\mathbit{a}$. On the number line, $b$ would be to the left of $a$. We write $b.

The Absolute Value of an Integer

When talking about graphing integers on the number line, one interpretation suggests it is like walking along a path. Negative is going to the left of 0, and positive going to the right. If you take 30 steps to the right, you are 30 steps away from 0. On the other hand, when you take 30 steps to the left, you are still 30 steps away from 0. So, in a way, even though one is negative and the other positive, these two numbers, 30 and −30, are equal since both are 30 steps away from 0. The absolute value of an integer $n$ is the distance from $n$ to 0, regardless of the direction. The notation for absolute value of the integer $n$ is $|n|$.

Adding and Subtracting Integers

You may recall having approached adding and subtracting integers using the number line from earlier in your academic life. Adding a positive integer results in moving to the right on the number line. Adding a negative integer results in moving to the left. Subtracting a positive integer results in a move to the left on the number line. But subtracting a negative integer results in a move to the right.

Multiplying and Dividing Integers

Similar to addition and subtraction, the signs of the integers impact the results when multiplying and dividing integers. The rules are fairly straightforward, but again rely on the direction on the number line. There are only two rules.

Order of Operations

Simplify Expressions Using Order of Operations

The order in which mathematical operations is performed is a convention that makes it easier for anyone to correctly calculate. They follow the acronym EMDAS:

Using the Order of Operations Involving Grouping Symbols

We have examined how to use the order of operations, denoted by EMDAS, to correctly calculate expressions. However, there may be expressions where a multiplication should happen before an exponent, or a subtraction before a division. To indicate an operation should be performed out of order, the operation is placed inside parentheses. When parentheses are present, the operations inside the parentheses are performed first. Adding the parentheses to our list, we now have PEMDAS, as shown below.

Rational Numbers

Defining and Identifying Numbers That Are Rational

A rational number (called rational since it is a ratio) is just a fraction where the numerator is an integer and the denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here that any integer is a rational number. An integer, $n$, written as a fraction of two integers is $\frac{n}{1}$.

Simplifying Rational Numbers and Expressing in Lowest Terms

A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the same value with different rational numbers. For instance, $\frac{4}{5}$ and $\frac{12}{15}$ are the same value. If we enter them into a calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two fractions are equal.

Adding and Subtracting Rational Numbers

Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or by finding a common denominator for the rational numbers being added or subtracted.

Converting Between Improper Fractions and Mixed Numbers

One way to visualize a fraction is as parts of a whole, as in $\frac{5}{12}$ of a pizza. But when the numerator is larger than the denominator, as in $\frac{23}{12}$, then the idea of parts of a whole seems not to make sense. Such a fraction is an **improper fraction.** That kind of fraction could be written as an integer plus a fraction, which is a **mixed number**. The fraction $\frac{23}{12}$ rewritten as a mixed number would be $1\frac{11}{12}$. Arithmetically, $1\frac{11}{12}$ is equivalent to $1+\frac{11}{12}$, which is read as “one and 11 twelfths.”

Converting Rational Numbers Between Decimal and Fraction Forms

Understanding what decimals represent is needed before addressing conversions between the fractional form of a number and its **decimal form**, or writing a number in **decimal notation**. The decimal number 4.557 is equal to $4\frac{557}{1,000}$. The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were three decimal digits in 4.557, we divided 557 by ${10}^{3}=1000$.

Multiplying and Dividing Rational Numbers

Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product. Symbolically, $\frac{a}{b}×\frac{c}{d}=\frac{a×c}{b×d}$. As always, rational numbers should be reduced to lowest terms.

Applying the Order of Operations to Simplify Expressions

The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations. The order of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym PEMDAS:

Applying the Density Property of Rational Numbers

Between any two rational numbers, there is another rational number. This is called the **density property** of the rational numbers.

Solving Problems Involving Rational Numbers

Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines.

Using Fractions to Convert Between Units

A common application of fractions is called **unit conversion**, or **converting units**, which is the process of changing from the units used in making a measurement to different units of measurement.

Defining and Applying Percent

A **percent** is a specific rational number and is literally per 100. $n$ percent, denoted $n$%, is the fraction $\frac{n}{100}$.

Solve Problems Using Percent

In the media, in research, and in casual conversation percentages are used frequently to express proportions. Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using percentages comes down to identifying which of the three components of a percentage you are given, the total, the percentage, or the percentage of the total. If you have two of those components, you can find the third using the methods outlined previously.

Irrational Numbers

Defining and Identifying Numbers That Are Irrational

We defined rational numbers in the last section as numbers that could be expressed as a fraction of two integers. **Irrational numbers** are numbers that cannot be expressed as a fraction of two integers. Recall that rational numbers could be identified as those whose decimal representations either terminated (ended) or had a repeating pattern at some point. So irrational numbers must be those whose decimal representations do not terminate or become a repeating pattern.

Simplifying Square Roots and Expressing Them in Lowest Terms

To **simplify a square root** means that we rewrite the square root as a rational number times the square root of a number that has no perfect square factors. The act of changing a square root into such a form is simplifying the square root.

Adding and Subtracting Irrational Numbers

Just like any other number we’ve worked with, irrational numbers can be added or subtracted. When working with a calculator, enter the operation and a decimal representation will be given. However, there are times when two irrational numbers may be added or subtracted without the calculator. This can happen only when the irrational parts of the irrational numbers are the same.

Multiplying and Dividing Irrational Numbers

Just like any other number that we’ve worked with, irrational numbers can be multiplied or divided. When working with a calculator, enter the operation and a decimal representation will be given. Sometimes, though, you may want to retain the form of the irrational number as a rational part times an irrational part. The process is similar to adding and subtracting irrational numbers when they are in this form. We do not need the irrational parts to match. Even though they need not match, they do need to be similar, such as both irrational parts are square roots, or both irrational parts are multiples of pi. Also, if the irrational parts are square roots, we may need to reduce the resulting square root to lowest terms.

Rationalizing Fractions with Irrational Denominators

Fractions often represent that some amount is being equally divided into some number of parts. But to conceptualize a fraction in that manner, the denominator needs to be an integer. An irrational number in the denominator interferes with that interpretation of a fraction. Fractions that have denominators that are just the square root of an integer can be altered into fractions with integer denominators using a process called **rationalizing the denominator**. The process relies on the following property of square roots: $\sqrt{a}×\sqrt{a}=a$ and the following property of fractions: $\frac{a}{b}=\frac{ac}{bc}$ for any non-zero number $c$.

Real Numbers

Defining and Identifying Real Numbers

Real numbers are the rational and irrational numbers combined. The real numbers represent the collection of all physical distances that exist, along with 0 and the negatives of those physical distances. For example, if you take a measure of three units, and divide that distance into eight (8) equal lengths, the distance you have formed is $\frac{3}{8}$ units. Also, if you draw a right triangle (a triangle with one angle equal to 90 degrees) with one side length of 1, and the other side length of 3, the long side of the triangle will have length $\sqrt{10}$ units, as shown in [link].

Identifying Subsets of Real Numbers

The real numbers were built out of pieces, including integers, rational numbers, and irrational numbers. As such, the real numbers have named subsets, as shown in the table below.

Recognizing Properties of Real Numbers

The real numbers behave in very regular ways. These behaviors are called the properties of the real numbers. Knowing these properties helps when evaluating formulas, working with equations, or performing algebra. Being familiar with these properties is helpful in all settings where numbers are used and manipulated. For example, when multiplying $4×13×25$, you could multiply the 4 and 25 first. If you know that product is 100, it makes the multiplication easier.

Clock Arithmetic

Adding, Subtracting, and Multiplying Using Clock Arithmetic

When we do arithmetic, numbers can become larger and larger. But when we work with time, specifically with clocks, the numbers cycle back on themselves. It will never be 49 o’clock. Once 12 o’clock is reached, we go back to 1 and repeat the numbers. If it's 11 AM and someone says, “See you in four hours,” you know that 11 AM plus 4 hours is 3 PM, not 15 AM (ignoring military time for now). Math worked on the clock, where numbers restart after passing 12, is called clock arithmetic.

Calculating Real-World Applications with Clock Arithmetic

Clock arithmetic processes can be applied to days of the week. Every 7 days the day of the week repeats, much like every 12 hours the time on the clock repeats. The only difference will be that we work with remainders after dividing by 7. In technical terms, this is referred to as modulo 7. More generally, let $n$ be a positive integer. Then $n$ modulo 7, written $n$ mod 7, is the remainder when $n$ is divided by 7. If that value is $x$, we may write $n=x$ (mod 7).

Exponents

Applying the Rules of Exponents to Simplify Expressions

Squaring a number is multiplying it by itself, and has that name because it is the area of a square with that side length. Cubing a number is finding the volume of a cube with that length of sides. That’s why we refer to ${5}^{2}$ as five squared, or ${10}^{3}$ as ten cubed. Exponents represent that multiplication.

Scientific Notation

Writing Numbers in Standard or Scientific Notation Form

When we say that a number is in scientific notation, we are specifying the form in which that number is written. That form begins with an integer with an absolute value between 1 and 9, then perhaps followed the decimal point and then some more digits. This is then multiplied by 10 raised to some power. When the number only has one non-zero digit, the scientific notation form is the digit multiplied by 10 raised to an exponent. When the number has more than one non-zero digit, the scientific notation form is a single digit, followed by a decimal, which is then followed by the remaining digits, which is then multiplied by 10 to a power.

Converting Numbers from Scientific Notation to Standard Form

In the previous section, converting a number from standard form to scientific notation was explored. Now, we explore converting from scientific notation back into standard form. Doing so involves moving the decimal according to the power of the 10. The decimal is moved a number of steps equal to the exponent of the 10. As demonstrated previously, when the exponent of the 10 is negative, the decimal is moved to the left and when the exponent of the 10 is positive, the decimal is moved to the right.

Adding and Subtracting Numbers in Scientific Notation

To add or subtract numbers in scientific notation, the numbers first need to have the same exponent for the 10s. It is possible to add the following since the powers of 10 match:
 $4.5×{10}^{4}+3.15×{10}^{4}=7.65×{10}^{4}$

Multiplying and Dividing Numbers in Scientific Notation

Multiplying and dividing numbers in scientific notation is somewhat easier than adding or subtracting, because the exponents of the 10s do not have to match. However, it is much more likely that the result will not be in scientific notation, and so that will have to be adjusted at the end. Generally, we multiply or divide the number parts of the two values, and then apply exponent rules to the 10 raised to the powers.

Dividing Numbers in Scientific Notation

To divide two numbers that are in scientific notation:

Using Scientific Notation in Computing Real-World Applications

As noted at the start of this section, scientific notation is useful when the standard representation of a number is awkward or impractical, which occurs when the numbers being used are extremely large or extremely small. For example, Venus is 67,667,000 miles from the sun. In scientific notation, this is $6.7667×{10}^{7}$. Planetary and galaxy distances is one set of numbers that is easier to express using scientific notation.

What Numbers Could Be Considered “Too Big” or “Too Small”?

One wonders when the numbers we represent become too large or small for consideration. Perhaps the following examples put limits on what is meaningful. The number of particles in the known universe has been estimated at $4×{10}^{80}$ particles. The smallest distance that has been measured is $1×{10}^{-18}{\rule{0.28em}{0ex}}\mathrm{m}$, though the theoretical smallest measurable value is $1×{10}^{-35}{\rule{0.28em}{0ex}}\mathrm{m}$. The distance across the universe is $4.4×{10}^{26}{\rule{0.28em}{0ex}}\mathrm{m}$. Considering what those numbers represent, the extreme largest and extreme smallest, they might be numbers that constrain what we should reasonably be expected to deal with.

Arithmetic Sequences

Identifying Arithmetic Sequences

A sequence of numbers is just that, a list of numbers in order. It can be a short list, such as the number of points earned on each assignment in a class, such as {10, 10, 8, 9, 10, 6, 10}. Or it can be a longer list, even infinitely long, such as the list of prime numbers. For example, here’s a sequence of numbers, specifically, the squares of the first 12 natural numbers.

Finding the Sum of a Finite Arithmetic Sequence

Sometimes we want to determine the sum of the numbers of a finite arithmetic sequence. The formula for this is fairly straightforward.

Using Arithmetic Sequences to Solve Real-World Applications

Applications of arithmetic sequences occur any time some quantity increases by a fixed amount at each step. For instance, suppose someone practices chess each week and increases the amount of time they study each week. The first week the person practices for 3 hours, and vows to practice 30 more minutes each week. Since the amount of time practicing increases by a fixed number each week, this would qualify as an arithmetic sequence.

Geometric Sequences

Identifying Geometric Sequences

We know what a sequence is, but what makes a sequence a geometric sequence? In an arithmetic sequence, each term is the previous term plus the constant difference. So, you add a (possibly negative) number at each step. In a geometric sequence, though, each term is the previous term multiplied by the same specified value, called the common ratio. In the sequence $\left\{3,6,12,24,48,96,192,384,728,1456\right\}$ the common ratio is 2. To see the difference between an arithmetic sequence and geometric sequence, examine these two sequences (Figures 3.52 and 3.53).

Finding the Sum of a Finite Geometric Sequence

As with arithmetic sequences, it is possible to add the terms of the geometric sequence. Like arithmetic sequences, the formula for the finite sum of the terms of a geometric sequence has a straightforward formula.

Using Geometric Sequences to Solve Real-World Applications

Geometric sequences have a multitude of applications, one of which is compound interest. Compound interest is something that happens to money deposited into an account, be it savings or an individual retirement account, or IRA. The interest on the account is calculated and added to the account at regular intervals. This means the interest that was earned later gains its own interest. This allows the money to grow faster. If that interest is added every month, we say it is compounded monthly. If the interest is added daily, then we say it is compounded daily. The amount of money that is deposited into the account is called the principal and is denoted $P$. The account earns money on that principal. The amount it earns is a percentage of the money in the account. The interest rate, expressed as a decimal, is denoted $r$.

Number Representation and Calculation

Introduction

Hindu-Arabic Positional System

Evaluating Exponential Expressions

Most modern numerical systems depend on place values, where the quantity represented depends not only on the digit, but also on where the digit is in the number. The place value is a power of some specific number, which means most numbering systems are actually exponential expressions. An exponential expression is any mathematical expression that includes exponents. So, evaluating such an expression means performing the calculation. In this chapter, we will be using exponents that are positive integer values. Before we do so, let’s remind ourselves about exponents and what they represent. Suppose you want to multiply a number. Let’s label that number $a$, by itself some number of times. Let’s label the number of times $b$. We denote that as ${a}^{b}$. We say $a$, or the **base**, raised to the $b$th power, or the **exponent**. For example, if we are multiplying 13 by itself eight times, we write ${13}^{8}$ and say 13 to the eighth power.

Converting Hindu-Arabic Numerals to Expanded Form

When you see the number 738, and you speak the number out loud, what do you say? You probably said “seven hundred thirty-eight” while wondering what point could possibly be made by asking this. What you didn’t say was “seven, and three, and eight.” A pre-K student might say that. Which should make you wonder, why?

Converting Numbers in Expanded Form to Hindu-Arabic Numerals

Converting from expanded form back into a Hindu-Arabic numeral is the reverse process of expanding a number, and is equivalent to evaluating the exponential expression.

Early Numeration Systems

Understand and Convert Babylonian Numerals to Hindu-Arabic Numerals

The Babylonians used a mix of an additive system of numbers and a positional system of numbers. An additive system is a number system where the value of repeated instances of a symbol is added the number of times the symbol appears. A positional system is a system of numbers that multiplies a “digit” by a number raised to a power, based on the position of the “digit.”

Understand and Convert Mayan Numerals to Hindu-Arabic Numerals

The Mayans employed a positional system just as we do and the Babylonians did, but they based their position values on powers of 20 and they had a dedicated symbol for zero. Similar to the Babylonians, the Mayans would repeat symbols to indicate certain values. A single dot was a 1, two dots were a 2, up to four dots. Then a five was a horizontal bar. The horizontal bars could be used three times, since the fourth horizontal bar would make a 20, which was a new position in the number. The 0 was a special picture, which appears like a turtle lying on its back. The shell would then be "empty," so maybe that’s why the symbol was 0. The complete list is provided in [link]. Another feature of Mayan numbers was that they were written vertically. The powers of 20 increased from bottom to top.

Understand and Convert Between Roman Numerals and Hindu-Arabic Numerals

The Mayan and Babylonian systems shared two features, one of which we are familiar with (place value) and one that we don’t use (repeated symbols). The  **Roman system of numbers**  used repeated symbols, but does not employ a place value. It also lacks a 0. The Roman system is built on the following symbols in [link].

Converting with Base Systems

Conversion of Another Base into Base 10 and Other Bases

We saw in Hindu-Arabic Positional System that our Hindu-Arabic system uses base 10, which is a system using place values of digits that depend on powers of 10 (or, are based on powers of 10). We’ve already worked with bases other than base 10: The Babylonian system was base 60, while the Mayan system was base 20.

Conversion of Base 10 into Another Base

Converting from base 10 into another base uses repeated division, recording the remainder at each step. Then, the number in the new base is the remainder starting from the last remainder found. To be accurate in what we’re saying, we need to remind ourselves of some terminology associated with division. When integers are divided, the one being divided is the dividend, and the one that is dividing the dividend is the divisor. The quotient is the largest natural number that can be multiplied by the divisor where the product is less than the dividend.

Converting from Hindu-Arabic Numbers to Mayan Numbers

To convert from a Hindu-Arabic number to a Mayan number involves two distinct processes. First, the number must be converted to base 20, using the process described and demonstrated previously. Next, that base 20 number has to be written using Mayan numerals. For reference, the Mayan numerals and their values are below.

Errors in Converting Between Bases

There are some common errors that are made when converting between bases. Often, it comes down to using an “illegal” symbol in the new base.

Addition and Subtraction in Base Systems

Addition in Bases Other Than Base 10

Now that we understand what it means for numbers to be expressed in a base other than 10, we can look at arithmetic using other bases, starting with addition. When you think back to when you first learned addition, it is very likely you learned the addition table. Once you knew the addition table, you moved on to addition of numbers with more than one digit. The same process holds for addition in other bases. We begin with an addition table, and then move on to adding numbers with two or more digits.

Subtraction in Bases Other Than Base 10

Subtraction in bases other than base 10 follow the same processes as base 10 subtraction, but, as with addition, using the addition table for the base.

Errors When Adding and Subtracting in Bases Other Than Base 10

Errors when computing in bases other than 10 often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base. For instance, if a 9 shows up when working in base 7, you know an error has happened because 9 is not a legal symbol in base 7.

Multiplication and Division in Base Systems

Multiplication in Bases Other Than 10

Multiplication is a way of representing repeated additions, regardless of what base is being used. However, different bases have different addition rules. In order to create the multiplication tables for a base other than 10, we need to rely on addition and the addition table for the base. So let’s look at multiplication in base 6.

Division in Bases Other Than 10

Just as with the other operations, division in a base other than 10, the process of division in a base other than 10 is the same as the process when working in base 10. For instance, $72÷9=8$ because, we know that $9×8=72$. So, for many division problems, we are simply looking to the multiplication table to identify the appropriate multiplication rule.

Errors in Multiplying and Dividing in Bases Other Than Base 10

The types of errors encountered when multiplying and dividing in bases other than base 10 are the same as when adding and subtracting. They often involve applying base 10 rules or symbols to an arithmetic problem in a base other than base 10. The first type of error is using a symbol that is not in the symbol set for the base.

Algebra

Introduction

Algebraic Expressions

Algebraic Expressions and Equations

Xavier and Yasenia have the same birthday, but they were born in different years. This year Xavier is 20 years old and Yasenia is 23, so Yasenia is three years older than Xavier. When Xavier was 15, Yasenia was 18. When Xavier will be 33, Yasenia will be 36. No matter what Xavier’s age is, Yasenia’s age will always be 3 years more.

Simplifying and Evaluating Algebraic Expressions

To simplify an expression means to do all the math possible. For example, to simplify $4•2+1$ we would first multiply $4•2$ to get 8 and then add 1 to get 9. We have introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. Consider $2+7•3$. Do you add first or multiply first? Do you get different answers?

Operations of Algebraic Expressions

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Examples of terms are 7, $y$, 5${x}^{2}$, 9$a$, and ${b}^{5}$. The constant that multiplies the variable is called the coefficient. Think of the coefficient as the number in front of the variable. Consider the algebraic expressions 5${x}^{2}$, which has a coefficient of 5, and 9$a$, which has a coefficient of 9. If there is no number listed in front of the variable, then the coefficient is 1 since $x=1•x$.

Linear Equations in One Variable with Applications

Linear Equations and Applications

Solving any equation is like discovering the answer to a puzzle. The purpose of solving an equation is to find the value or values of the variable that makes the equation a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle! There are many types of equations that we will learn to solve. In this section, we will focus on a linear equation, which is an equation in one variable that can be written as

Linear Equations with No Solutions or Infinitely Many Solutions

Every linear equation we have solved thus far has given us one numerical solution. Now we'll look at linear equations for which there are no solutions or infinitely many solutions.

Solving a Formula for a Given Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be able to manipulate formulas and solve for specific variables.

Linear Inequalities in One Variable with Applications

Graphing Inequalities on the Number Line

In Algebraic Expressions, we introduced equality and the $=$ symbol. In this section, we look at inequality and the symbols $<$, $>$, $\le$, and $\ge$. The table below summarizes the symbols and their meaning.

| Symbol | Meaning |
| --- | --- |
| $<$ | less than |
| $>$ | greater than |
| $\le$ | less than or equal to |
| $\ge$ | greater than or equal to |

Solving Linear Inequalities

A **linear inequality** is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms $ax+b or $ax+b>c,$ where $a$, $b$, and $c$ are all real numbers.

Solving Applications with Linear Inequalities

Many real-life situations require us to solve inequalities. The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.

Ratios and Proportions

Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as: $a$ to $b$ OR $a:b$ OR the fraction $a/b$. Which method you use often depends upon the situation. For the most part, we will want to write our ratios using the fraction notation. Note that, while all ratios are fractions, not all fractions are ratios. Ratios make part to part, part to whole, and whole to part comparisons. Fractions make part to whole comparisons only.

Using and Applying Proportional Relationships to Solve Problems

Using proportions to solve problems is a very useful method. It is usually used when you know three parts of the proportion, and one part is unknown. Proportions are often solved by setting up like ratios. If $\frac{a}{b}$ and $\frac{c}{d}$ are two ratios such that $\frac{a}{b}=\frac{c}{d},$ then the fractions are said to be **proportional**. Also, two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are proportional $\left(\frac{a}{b}=\frac{c}{d}\right)$ if and only if $a×d=b×c$.

Determining and Applying a Constant of Proportionality

In the last example, we were given that $2\frac{1}{4}$ cups of flour could make 60 cookies; we then calculated that $38\frac{1}{4}$ cups of flour would make 1,020 cookies, and 720 cookies could be made from 27 cups of flour. Each of those three ratios is written as a fraction below (with the fractions converted to decimals). What happens if you divide the numerator by the denominator in each?

Using Proportions to Solve Scaling Problems

Ratio and proportions are used to solve problems involving **scale**. One common place you see a scale is on a map (as represented in [link]). In this image, 1 inch is equal to 200 miles. This is the scale. This means that 1 inch on the map corresponds to 200 miles on the surface of Earth. Another place where scales are used is with models: model cars, trucks, airplanes, trains, and so on. A common ratio given for model cars is 1:24—that means that 1 inch in length on the model car is equal to 24 inches (2 feet) on an actual automobile. Although these are two common places that scale is used, it is used in a variety of other ways as well.

Graphing Linear Equations and Inequalities

Plotting Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the $x$$y$-plane or the “coordinate plane.”

Graphing Linear Equations in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation, you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form $Ax+By=C$. An equation of this form, where $A$ and $B$ are both not zero, is called a **linear equation in two variables**. Here is an example of a linear equation in two variables, $x$ and $y$.

$\begin{array}{l}\mathbit{A}x+\mathbit{B}y=\mathbit{C}\\ \mathbit{A}+\mathbf{4}y=\mathbf{8}\\ \mathbit{A}\mathbf{=}\mathbf{1},\mathbit{B}\mathbf{=}\mathbf{4},\mathbit{C}\mathbf{=}\mathbf{8}\end{array}$

Solving Applications Using Linear Equations in Two Variables

Many fields use linear equalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Graphing Linear Inequalities

Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables that can be written in one of the following forms: $Ax+By\ge C$, $Ax+By>C$, $Ax+By\le C$, and $Ax+By where $A$ and $B$ are not both zero. We will look at **linear inequalities in two variables,** which are very similar to linear equations in two variables.

Solving Applications Using Linear Inequalities in Two Variables

Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.

Quadratic Equations with Two Variables with Applications

Identify Polynomials, Monomials, Binomials and Trinomials

You have learned that a term is a constant, or the product of a constant and one or more variables. When it is of the form $a{x}^{m}$, where $a$ is a constant and ${x}^{m}$ is a positive whole number, it is called a monomial. Some examples of monomial are 8, $-2{x}^{2}$, $4{y}^{3}$, and $11z$.

Multiply Binomials

Recall multiplying algebraic expressions from Algebraic Expressions. In this section, we will continue that work and multiply binomials as well. We can use an area model to do multiplication.

Factoring Trinomials

We’ve just covered how to multiply binomials. Now you will need to “undo” this multiplication—to start with the product and end up with the factors. Let us review an example of multiplying binomials to refresh your memory.

Solving Quadratic Equations by Graphing

We have already solved and graphed linear equations in Graphing Linear Equations and Inequalities, equations of the form $Ax+By=C$. In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. The following are some examples of quadratic equations:

Solving Quadratic Equations by Factoring

Another way of solving quadratic equations is by factoring. We will use the Zero Product Property that says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

Solving Quadratic Equations Using the Square Root Property

We just solved some quadratic equations by factoring. Let us use factoring to solve the quadratic equation ${x}^{2}=9$.

Solving Quadratic Equations Using the Quadratic Formula

This last method we will look at for solving quadratic equations is the quadratic formula. This method works for all quadratic equations, even the quadratic equations we could not factor! To use the quadratic formula, we substitute the values of $a$, $b$, and $c$ into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.

Solving Real-World Applications Modeled by Quadratic Equations

There are problem solving strategies that will work well for applications that translate to quadratic equations. Here’s a problem-solving strategy to solve word problems:

Functions

Use Function Notation

It is very convenient to name a function; most often functions are named $f$, $g$, $h$, $F$, $G$, or $H$. In any function, for each $x$-value from the domain, we get a corresponding $y$-value in the range. In the function $f$, we write this range value $y$ as $f$($x$). This notation $f$($x$) is called function notation and is read "f of $x$" or "the value of f at $x$." In this case the parentheses do not indicate multiplication.

Determining If a Relation Is a Function with Different Representations

We can determine whether a relation is a function by identifying the input and the output values. If each input value leads to only one output value, classify the relation as a function. If any input value leads to two or more outputs, do not classify the relation as a function.

Applying the Vertical Line Test

We reviewed how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping, or an equation. We will now cover how to tell if a graph is that of a function.

Determining the Domain and Range of a Function

For the function $y=f\left(x\right),x$ is the independent variable as it can be any value in the domain, and $y$ is the dependent variable since its value depends on $x$. For the function $y=f\left(x\right)$, the values of $x$ make up the domain and the values of $y$ make up the range.

Graphing Functions

Graphing Functions Using Intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points. At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the $x$-axis and the $y$-axis. These points are called the intercepts of a line. Let us review the graphs of the lines in [link].

Computing Slope

When graphing linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. In mathematics, the measure of the steepness of a line is called the slope of the line. To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices of the triangle and one side is horizontal and one side is vertical. Next, we measure or calculate the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run.

Graphing Functions Using Slope and $y$-Intercept

We have graphed linear equations by plotting points and using intercepts. Once we see how an equation in slope-intercept form and its graph are related, we will have one more method we can use to graph lines. Review the graph of the equation $y=\frac{1}{2}x+3$ in [link] and find its slope and $y$-intercept.

Graphing Horizontal and Vertical Lines

Some linear equations have only one variable. They may have just $x$ without the $y$, or just $y$ without an $x$. This changes how we make a table of values to get the points to plot. Let us consider the equation $x=-3$. This equation has only one variable, $x$. The equation says that $x$ is always equal to $-3$, so its value does not depend on $y$. No matter what the value of $y$ is, the value of $x$ is always $-3$. To make a table of values, write $-3$ in for all the $x$-values. Then choose any values for $y$. Since $x$ does not depend on $y$, you can choose any numbers you like. But to fit the points on our coordinate graph, we will use 1, 2, and 3 for the $y$-coordinates in the table below.

Interpreting Graphs of Functions

An important yet often overlooked area in algebra involves interpreting graphs. Oftentimes in math classes, students are given mathematical functions and can make graphs to represent them. But the interpretation of graphs is a more applicable skill to the real world. Being able to “read” a graph—understanding its domain and range, what the intercepts mean, and what the slope (or curve) means— that's a real-world skill.

Modeling Applications Using Slope and $y$-Intercept

Many real-world applications are modeled by linear equations. We will review a few applications here so you can understand how equations written in slope-intercept form relate to real-world situations. Usually when a linear equation model uses real-world data, different letters are used for the variables instead of using only $x$ and $y$. The variable names often remind us of what quantities are being measured. Also, we often need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.

Systems of Linear Equations in Two Variables

Determining If an Ordered Pair Is a Solution to a System of Equations

When we solved linear equations in Linear Equations in One Variable with Applications and Linear Inequalities in One Variable with Applications, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a **system of linear equations**.

Solving Systems of Linear Equations Using Graphical Methods

We will use three methods to solve a system of linear equations. The first method we will use is graphing. The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what points the lines have in common, we will find the solution to the system.

Solving Systems of Linear Equations Using Substitution

We will now solve systems of linear equations by the substitution method. We will use the same system we used for graphing.

Solving Systems of Linear Equations Using Elimination

We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small, and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.

Identifying Systems with No Solution or Infinitely Many Solutions

In all the systems of linear equations so far, the lines intersected, and the solution was one point. In [link] and [link], we will look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.

Solving Applications of Systems of Linear Equations

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we will first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.

Systems of Linear Inequalities in Two Variables

Determining If an Ordered Pair Is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is similar to the definition of a system of linear equations. A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown here.

Solving Systems of Linear Inequalities Using Graphical Methods

The solution to a single linear inequality was the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. We will review graphs of linear inequalities and solve the linear inequality from its graph.

Graphing Systems of Linear Inequalities

We learned that the solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region by graphing, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Interpreting and Solving Applications of Linear Inequalities

When solving applications of systems of inequalities, first translate each condition into an inequality. Then graph the system, as we did above, to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so add inequalities to the system as additional requirements.

Linear Programming

Compose an Objective Function to Be Minimized or Maximized

An **objective function** is a linear function in two or more variables that describes the quantity that needs to be maximized or minimized.

Composing Inequalities Representing a System Application

For our two examples of profit and production, in an ideal world the profit a person makes and/or the number of products a company produces would have no restrictions. After all, who wouldn’t want to have an unrestricted profit? However in reality this is not the case; there are usually several variables that can restrict how much profit a person can make or how many products a company can produce. These restrictions are called constraints.

Applying Linear Programming to Solve Application Problems

There are four steps that need to be completed when solving a problem using linear programming. They are as follows:

Money Management

Introduction

Understanding Percents and Discounts

Define and Calculate Percent

The word percent comes from the Latin phrase per centum, which means “by the hundred.” So any percent is a number divided by 100. Changing a percent to a fraction is to write the percent in its fractional form. To write $n$% in its fractional form is to write the percent as the fraction $\frac{n}{100}$.

Convert Between Percent, Decimal, and Fractional Values

When any calculation with a percent is to be performed, the form of the percent must be changed, either to its fractional form or its **decimal form**. We can change a percent into decimal form by dividing the percent by 100 and representing the result as a decimal.

Calculate the Total, Percent, or Part

The word “of” is used to indicate multiplication using fractions, as in “one-fourth of 56.” To find “one-fourth of 56” we would multiply 56 by one-fourth. We can think of percents as fractions with a specific denominator—100. So, to calculate “25% of 52,” we multiply 52 by 25%. But, first we need to convert the percent to either fractional form (25/100) or decimal form. Using the decimal form of 25% we have 0.25 × 52, which equals 13.

Solve Application Problems Involving Percents

Percents are frequently used in finance, research, science experiments, and even casual conversation. Understanding these types of values helps when consuming media or discussing finances, for instance. Effectively working with and interpreting numbers and percents will help you become an informed consumer of this information.

Discounts, Markups, and Sales Tax

Calculating Discounts

Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off the original price. These are discounts, a reduction in the price of something. The price after the discount is sometimes referred to as the reduced price or the sale price.

Solve Application Problems Involving Discounts

In application problems, identify what is given and what is to be found, using the terms that have been learned, such as discount, original price, percent discount, and sale price. Once you have identified those, use the appropriate formula (or formulas) to find the solution(s).

Calculate Markups

When retailers purchase goods to sell, they pay a certain price, called the **cost**. The retailer then charges more than that amount for the goods. This increase is called the **markup**. This selling price, or **retail price**, is what the retailer charges the consumer in order to pay their own costs and make a profit. Markup, then is very similar to discount, except we add the markup, while we subtract the discount.

Solve Application Problems Involving Markups

As before when working with application problems, be sure to look for what is given and identify what you are to find. Once you have evaluated the problem, use the appropriate formula to find the solution(s). These application problems address markups.

Compute Sales Tax

Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government.

Solve Application Problems Involving Sales Tax

Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will use the following formula:

Simple Interest

Compute Simple Interest

Let’s get some terminology understood. Interest to be paid by a borrower is often expressed as an annual percentage rate, which is the percent of the principal that is paid as interest for each year the money is borrowed. This means that the more that is borrowed, the more that must be paid back. Sometimes, the interest to be paid back is simple interest, which means that the interest is calculated on the amount borrowed only.

Understand and Compute Future Value

Money can be invested for a specific amount of time and earn simple interest while invested. The terminology and calculations are the same as we’ve already seen. However, instead of the total to be paid back, the investor is interested in the total value of the investment after the interest is added. This is called the **future value** of the investment.

Paying Simple Interest Loans with Partial Payments

In every example above, there was one payment for the loan, or one withdrawal for the investment. However, for many loans (house, car, in-ground swimming pool), the loan will be paid back in two or more payments. Such a payment is called a **partial payment**, because they only pay off part of the loan.

Understand and Compute Present Value for Simple Interest Investments

When finding the future value of an investment, we know how much is deposited, but we have no idea how much that money will be worth in the future. If we set a goal for the future, it would be useful to know how much to deposit now so an account reaches the goal. The amount that needs to be deposited now to hit a goal in the future is called the **present value**.

Compound Interest

Understand and Compute Compound Interest

As we saw in Simple Interest, an account that pays simple interest only pays based on the original principal and the term of the loan. Accounts offering compound interest pay interest at regular intervals. After each interval, the interest is added to the original principal. Later, interest is calculated on the original principal plus the interest that has been added previously.

Determine the Difference in Interest Between Simple and Compound Calculations

It is natural to ask, does compound interest make much of a difference? To find out, we revisit Abena’s CD.

Understand and Compute Future Value

Imagine investing for 30 years and compounding the interest every month. Using the method above, there would be 360 periods to calculate interest for. This is not a reasonable approach. Fortunately, there is a formula for finding the future value of an investment that earns compound interest.

Understand and Compute Present Value

When investing, there is often a goal to reach, such as “after 20 years, I’d like the account to be worth \$100,000.” The question to be answered in this case is “How much money must be invested now to reach the goal?” As with simple interest, this is referred to as the present value.

Compute and Interpret Effective Annual Yield

As we’ve seen, quarterly compounding pays interest 4 times a year or every 3 months; monthly compounding pays 12 times a year; daily compounding pays interest every day, and so on. Effective annual yield allows direct comparisons between simple interest and compound interest by converting compound interest to its equivalent simple interest rate. We can even directly compare different compound interest situations. This gives information that can be used to identify the best investment from a yield perspective.

Making a Personal Budget

Creating a Budget

You should view creating a **budget** as a financial tool that will help you achieve your long-term goals. A budget is an estimation of income and expenses over some period of time. You will be able to track your progress, which will help you to prepare for the future by making smart investment decisions.

Methods of Savings

Distinguish Various Basic Forms of Savings Plans

There are at least three types of savings accounts. Traditional savings accounts, certificates of deposit (CDs), and money market accounts are three main savings account vehicles.

Compute Payment to Reach a Financial Goal

The formula used to get the future value of an ordinary annuity is useful, finding out what the final amount in the account will be. However, that isn’t how planning works. To plan, we need to know how much to put into the ordinary annuity each compounding period in order to reach a goal. Fortunately, that formula exists.

Investments

Distinguish Between Basic Forms of Investments

Bonds, stocks, and mutual funds tend to offer higher returns, but to varying degrees, come with higher risks. Stocks and mutual funds also vary in how much they earn. Their predicted rates of return on investment are not guaranteed, but educated guesses based on market trends and historical performance.

Return on Investment

As in Methods of Savings, the formula for return on investment is $\text{ROI}=\frac{FV-P}{P}$. As indicated before, this formula does not take into account how long the investment took to reach its current value. It depends only on the initial value, $P$, and the value at the end of the investment, $FV$.

As in Methods of Savings, determining the payment necessary to reach a financial goal uses the payment formula for an ordinary annuity, $pmt=\frac{FV×\left(r/n\right)}{{\left(1+r/n\right)}^{n×t}-1}$. If dealing with mutual funds or stocks, an assumed annual interest rate, compounded, will be used. This value is often determined through research and informed speculation.

Retirement Savings Plans

We close this section by investigating the three main forms of retirement savings accounts: traditional individual retirement accounts (IRAs), Roth IRAs, and 401(k) accounts. Each has distinct characteristics that are suited to different investors’ needs.

The Basics of Loans

Reasons for Loans

Even if you want a new car because you need one, or if you need a new computer since your current one no longer runs as fast or smoothly as you would like, or you need a new chimney because the one on your house is crumbling, it’s likely you do not have that cost in cash. Those are very large purchases. How do you buy that if you don ’t have the cash?

 You borrow the money.

The Terminology of Loans

There are many words and acronyms that get used in relation to loans. A few are below.

Calculating Loan Payments

Loan payments are made up of two components. One component is the interest that accrued during the payment period. The other component is part of the principal. This should remind you of partial payments from Simple Interest.

Reading Amortization Tables

An **amortization table** or amortization schedule is a table that provides the details of the periodic payments for a loan where the payments are applied to both the principal and the interest. The principal of the loan is paid down over the life of the loan. Typically, the payments each period are equal. Importantly, one of the columns will show how much of each payment is used for interest, another column shows how much is applied to the outstanding principal, and another column shows the remaining principal or balance [link].

Cost of Finance

There are often costs associated with a loan beyond the interest being paid. The **cost of finance** of a loan is the sum of all costs, fees, interest, and other charges paid over the life of the loan.

Understanding Student Loans

Obtaining a Student Loan

All college students are eligible to apply for a loan regardless of their financial situation or credit rating. Federal student loans do not require a co-signer or a credit check. Most students do not have a credit history when they begin college, and the federal government is aware of this. However, private loans will generally require a co-signer as well as a credit check. The co-signer will assume responsibility for paying off the loan if the student cannot make the payments.

Types and Features of Student Loans

Once a tuition statement is received, and all the non-loan awards are analyzed that are applicable to the costs of college (such as scholarships and grants), there still may be quite of bit of an expense to attend college. This difference between what college will cost (including tuition, room and board, books, computers) and the non-loan awards received is the **college funding gap.**

Limits on Student Loans

As mentioned earlier, there are limits to how much a student can borrow, per year and in total. The following table shows a general breakdown of the amounts the federal government and private lenders will lend. Amounts are based on level of need and whether the student is a dependent or an independent student. Independent students include those who are at least 24 years old, married, a professional, a graduate student, a veteran, a member of the armed forces, an emancipated minor, or an orphan. The amounts shown are as of this writing in 2022.

Student Loan Interest Rates

Student loans are first and foremost loans. Students will pay them back and will pay interest. In the fall of 2022, the federal student loan interest rate was 4.99%. Private student loans rates ranged between 3.22% and 13.95%. Finding the lowest interest rate you can helps with the payments, and especially helps if the loan is not federally subsidized. Remember, if the loan is not federally subsidized, the student is on the hook for the interest that is accumulating with the loan.

Interest Accrual

The interest on student loans begins as soon as the loan is disbursed (paid to the borrower). When the loan is federally subsidized, the government pays that interest for the student. This means the loan for a subsidized loan of \$3,000 is still a loan for \$3,000 when the student graduates. However, if the loan is not federally subsidized, the student is responsible for the interest that accrues on the loan. The \$3,000 loan from year 1 of college is now a loan for more due to that added interest. The interest on that loan grew while the student was in college. The formula for growth of the loan’s balance is the same as compound interest formula from Compound Interest, $A=P{\left(1+\frac{r}{n}\right)}^{nt}$.

The Standard Repayment Plan

There are various repayment plans available. The one most likely to apply to a student loan is the **standard repayment** **plan,** which is available to everyone. Borrowers pay a fixed amount monthly so the loan is paid in full within 10 years. Consolidated loans, discussed later in this section, also qualify for the standard repayment plan, and may allow the payoff period to range from 10 to 30 years. Direct subsidized and unsubsidized loans, PLUS loans, and federal Stafford loans are eligible.

Student Loan Consolidation

When a student graduates, they may have multiple different student loans. Keeping track of them and paying them off separately can be a burden. Instead, these loans can be consolidated into a single loan. If they are federal loans the combination is called **federal consolidation.** Combining private loans is often referred to as **refinancing.** Refinancing, or **private consolidation,** can be used to combine both private and federal student loans. Be aware that consolidated federal loans may still be subject to the rules and protections that govern subsidized loans. Refinancing loans, private or federal, are no longer subject to those rules and guidelines. Check out this Experian article about consolidation and refinancing for more deatil.

Other Repayment Plans

There are various other repayment plans available to students. Plans other than the standard repayment plan typically require the student to meet certain criteria. The following plans are independent of student income, but may make early payments easier.

Student Loan Default and Consequences

The first day a payment is late, the account becomes **delinquent**. After 90 days, this delinquency is reported to the credit bureaus, and goes into **default***.* This is serious, as now a credit score is affected, meaning that it will be harder to buy a car, a home, get a credit card, or a cell phone. Even renting an apartment may be a task not easily overcome. The default rate for students who do not complete their degree is three times higher than for students who do.

Credit Cards

Types of Credit Cards

There are basically three types of credit cards: bank-issued credit cards, store-issued credit cards, and travel/entertainment credit cards. We will look at all three and explain the good and the bad qualities of each.

Bank-Issued Credit Cards

Perhaps the most widely used credit card type is the **bank-issued credit card**, like Visa or MasterCard (and even American Express and Discover cards). These types of cards are an example of revolving credit, meaning that additional credit is extended before the previous balance is paid—but only up to the assigned credit limit. Bank-issued cards are considered the most convenient, as they can be used to purchase anything, including apparel, furniture, groceries, fuel for automobiles, meals, hotel bills, and so on, just as if paying with cash. The interest rates on bank-issued credit cards are usually lower than those for other credit cards we’ll discuss, and the credit limits are generally higher. Currently, bank-issued cards have an average 20.09% APR.

Store-Issued Credit Cards

**Store-issued credit cards** are issued by retailers. One can hardly walk into a store these days without being offered a discount on purchases if one applies for the store credit card. These cards can only be used in that store or family of stores that issues the card. However, if a store credit card is associated with Visa, MasterCard, or American Express, then the card might be used the same way that the bank-issued cards are used. This is called cobranding. The logo of the bank-issued card will be present on the store card. Many stores offer both types. Like other credit cards, they may come with an annual fee.

Travel/Entertainment Cards, or Charge Cards

This is the third type of credit card. The **travel and entertainment cards,** also known as **charge cards,** first and foremost offer very high limits or unlimited credit, but they must be paid in full every month. They generally charge high annual fees and impose expensive penalties should a payment be late. On the other hand, they typically have longer grace periods and offer many and various kinds of rewards.

Credit Card Statements

Cardholders usually receive monthly statements and have 21 days to pay the minimum amount due. The statements itemize and summarize activity on the credit card for that statement’s billing period. The billing period for a credit card is generally a month long, but typically does not start and end on the first and last days of the month. The statement will include the current balance, interest rate, the minimum payment due, and the due date. Be aware, different companies produce statements that are laid out differently. The information will be clearly labeled though.

Compute Interest, Balance Due, and Minimum Payment Due for a Credit Card

Computing all of these values depends on understanding and computing the average daily balance on a credit card. Once that is known, the interest, balance due, and minimum payment can be found.

Average Daily Balance

Most credit card companies compute interest using the **average daily balance** method.

Calculating the Interest for a Credit Card

The interest charged for a credit card is based on the daily interest rate of the card, the number of days in the billing cycle, and the average daily balance on the card.

Calculating the Balance of a Credit Card

The **balance,** or sometimes balance due, on a credit card is the previous balance, plus all expenses, minus all payments and credits, plus the interest on the card. As stated before, if the card was paid off, there is no interest to be paid.

Minimum Payment Due

The **minimum payment** due is the smallest required amount to be paid on a credit card to avoid late fees and penalties, such as an increased interest rate. The calculations for this may differ from card to card. They also depend in the balance of the credit card. General guidelines for minimum payment due are:

Buying or Leasing a Car

The Basics of Car Purchasing

The biggest questions you will answer before purchasing a car are, what do you want and what do you need?

Purchase Payments and Interest

Whether or not you buy a new car or a used car, if you finance the purchase, you are taking out a loan. The interest rates available for used cards are frequently higher than those for new cars. These loan payments work exactly the same way as other loans do as far as payments are concerned. The payment function comes from The Basics of Loans. The difference between financing a new car or a used car is that financing a new car typically comes with a lower interest rate and a longer term that financing a used car.

The Basics of Leasing a Car

Leasing a car is an alternative to purchasing a car. It is still a loan, and acts like one in many respects. They typically last either 24 months or 36 months, though other terms are available. Leases also come with mileage limits, frequently 10,000, 12,000, or 15,000 miles per year. When the lease is over, the car is returned to the dealer. At that time there may be fees that have to be paid, such as for damage to the car or for extra miles driven over the limit.

Lease Payments

Lease payments are similar to regular loan payments, but have some other details. Calculating a lease payment involves knowing the following values:

Comparing Purchasing and Leasing

When deciding to buy or lease a car, the differences between the two options should be carefully evaluated. The following is a list of points of comparison between the two.

Car Insurance

Car insurance is meant to cover costs associated with accidents involving cars. Most states (all except New Hampshire and Virginia) require some insurance. Without insurance, the state may not let you get a license for your car or register your car. Your state’s requirements can be hard to follow. Fortunately, insurance companies and brokers will make sure your insurance is sufficient for your state and will warn you if you try to not meet the requirements. Of course, they may offer more than what is sufficient, so it is your responsibility to determine how much coverage you want, as long as the minimum insurance requirements are met. The cost of insurance should be accounted for when evaluating the affordability of buying or leasing a car.

Maintaining a Car

Cars are not a buy it and forget it item. They require upkeep, which adds to the cost of owning the car. Tires, brakes, and wipers need replacing. Oli changes, inspections, so many things other than gasoline. Below is a list of some maintenance requirements for cars, along with cost and roughly how often they should happen.

Renting and Homeownership

Advantages and Disadvantages of Renting

When renting, you will likely sign a lease, which is a contract between a renter and a landlord. A landlord is the person or company that owns property that is rented. The lease will detail your responsibilities, restrictions on activities, deposits, fees, maintenance, repairs, and rent during the term of the lease. It also defines what your landlord can, and cannot, do with the property while you occupy the property.

Advantages of Buying a Home

The advantages to buying a home mirror the disadvantages of renting, and the disadvantages of home ownership mirror the advantages of renting.

Mortgages

Some people will purchase a home or condo with cash, but the majority of people will apply for a mortgage. A mortgage is a long-term loan and the property itself is the security. The bank decides the minimum down payment (with your input), the payment schedule, the duration of the loan, whether the loan can be assumed by another party, and the penalty for late payments. The title of the home belongs to the bank.

Monthly Mortgage Payments

The formula to calculate your monthly payments of principal and interest uses APR as the annual interest rate.

Reading and Interpreting Amortization Tables

Amortization tables were addressed in The Basics of Loans. They are most frequently encountered when analyzing mortgages.

Escrow Payments

The last few examples have looked at mortgage payments, which cover the principal and interest. However, when you take out a mortgage, the payment is sometimes much higher than that. This is because your mortgage company also has you pay into an **escrow account**, which is a savings account maintained by the mortgage company.

Income Tax

Gross, Adjusted Gross, and Taxable Income

Your income drives how much you pay in taxes. The more you earn, the more you are likely to pay. But your income alone is not the full story. When you add all the money you earned from your job, freelance work, interest from savings, and other sources, you have your gross income. If you are an employee, your income from your job will be reported on a W-2, which is sent to you by your employer. Income from freelance work will be reported on a 1099-MISC form, and is sent by the company that paid you. Income from interest is reported on a 1099-INT form and comes from the entity that paid the interest.

Tax Credits

Another piece of the tax puzzle is tax credits. This is money subtracted from the tax you owe.

Computing FICA Taxes

FICA stands for the Federal Insurance Contributions Act of 1935. FICA taxes are used solely to fund Social Security and Medicare and are separate from federal income tax. It amounts to 7.65% of your gross pay, which is withheld from your paycheck automatically. Your employer is required to match the 7.65% amount. Of the 7.65%, 6.2% goes to Social Security (SSI), and 1.45% goes to Medicare.

Calculating Your Income Tax

Your income tax bill and your income tax rate are based on your taxable income. The tax system in the United States is progressive, meaning that the tax rates are marginal so the higher your taxable income the higher the tax rate you will pay. Taxable income is broken into brackets, or ranges of income. Each bracket has a different tax rate. The tax brackets and rates for single filers as of 2022 are given below:

Probability

Introduction

The Multiplication Rule for Counting

Multiplication as a Combinatorial Short Cut

One of the first combinatorial short cuts to counting students learn in school has to do with areas of rectangles. If we have a set of objects to be counted that can be physically arranged into a rectangular shape, then we can use multiplication to do the counting for us. Consider this set of objects ([link]):

Permutations

Applying the Multiplication Rule for Counting to Permutations

In the case of the swimming finals, one possible permutation of length 3 would be the list of medal winners (first, second, and third place finishers). A permutation of length 8 would be the full order of finish (first place through eighth place). Let’s use the Multiplication Rule for Counting to figure out how many of each of these permutations there are.

Factorials

The pattern we see in [link] occurs commonly enough that we have a name for it: factorial.
For any positive whole number $n$, we define the factorial of $n$ (denoted $n!$ and read "$n$ factorial") to be the product of every whole number less than or equal to $n$. We also define 0! to be equal to one. We will use factorials in a couple of different contexts, so let's get some practice doing computations with them.

Permutations

As we’ve seen, factorials can pop up when we’re computing permutations. In fact, there is a formula that we can use to make that connection explicit. Let’s define some notation first. If we have a collection of $n$ objects and we wish to create an ordered list of $r$ of the objects (where $1\le r\le n$), we’ll call the number of those permutations ${}_{n}{P}_{r}$ (read “the number of permutations of $n$ objects taken $r$ at a time”). We formalize the formula we'll use to compute permutations below.

Combinations

Combinations: When Order Doesn’t Matter

In situations in which the order of a list of objects doesn’t matter, the lists are no longer permutations. Instead, we call them combinations.

Counting Combinations

Permutations and combinations are certainly related, because they both involve choosing a subset of a large group. Let’s explore that connection, so that we can figure out how to use what we know about permutations to help us count combinations. We’ll take a basic example. How many ways can we select 3 letters from the group A, B, C, D, and E? If order matters, that number is ${}_{5}{P}_{3}=60$. That’s small enough that we can list them all out in the table below.

Tree Diagrams, Tables, and Outcomes

Single Stage Experiments

When we are talking about combinatorics or probability, the word “experiment” has a slightly different meaning than it does in the sciences. Experiments can range from very simple (“flip a coin”) to very complex (“count the number of uranium atoms that undergo nuclear fission in a sample of a given size over the course of an hour”). Experiments have unknown outcomes that generally rely on something random, so that if the experiment is repeated (or replicated) the outcome might be different. No matter what the experiment, though, analysis of the experiment typically begins with identifying its sample space.

Multistage Experiments

Some experiments have more complicated sample spaces because they occur in stages. These stages can occur in succession (like drawing cards one at a time) or simultaneously (rolling 2 dice). Sample spaces get more complicated as the complexity of the experiment increases, so it’s important to choose a systematic method for identifying all of the possible outcomes. The first method we’ll discuss is the table.

Basic Concepts of Probability

Introducing Probability

Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; for example, in the *Monopoly* example at the beginning of this section, we were interested in the roll of 2 dice that might fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event. In other words, an event is a subset of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter.

Three Ways to Assign Probabilities

The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we do about those in-between cases, for events that might or might not occur? There are three methods to assign probabilities that we can choose from. We’ll discuss them here, in order of reliability.

New Probabilities from Old: Complements

One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will come later.

Probability with Permutations and Combinations

Using Permutations to Compute Probabilities

Recall that we can use permutations to count how many ways there are to put a number of items from a list in order. If we’re looking at an experiment whose sample space looks like an ordered list, then permutations can help us to find the right probabilities.

Combinations to Computer Probabilities

If the sample space of our experiment is one in which order doesn’t matter, then we can use combinations to find the number of outcomes in that sample space.

What Are the Odds?

Computing Odds

The ratio of the number of equally likely outcomes in an event $E$ to the number of equally likely outcomes *not* in the event $E\prime$ is called the odds for (or odds in favor of) the event. The opposite ratio (the number of outcomes not in the event to the number in the event $E\prime$ to the number in the event $E$ is called the odds against the event.

Odds as a Ratio of Probabilities

We can also think of odds as a ratio of probabilities. Consider again the instant-win game from the section opener, with 500,000 winning tickets out of 2,000,000 total tickets. If a player buys one ticket, the probability of winning is $\frac{500,000}{2,000,000}=\frac{1}{4}$, and the probability of losing is $1-\frac{1}{4}=\frac{3}{4}$. Notice that the ratio of the probability of winning to the probability of losing is $\frac{1}{4}:\frac{3}{4}=1:3$, which matches the odds in favor of winning.

The Addition Rule for Probability

Mutual Exclusivity

Before we get to the key techniques of this section, we must first introduce some new terminology. Let’s say you’re drawing a card from a standard deck. We’ll consider 3 events: $H$ is the event “the card is a $♡$,” $T$ is the event “the card is a 10,” and $S$ is the event “the card is a $♠$.” If the card drawn is $\text{J}♠$, then $H$ and $T$ didn’t occur, but $S$ did. If the card drawn is instead $10♠$, then $H$ didn’t occur, but both $T$ and $S$ did.

The Addition Rule for Mutually Exclusive Events

If two events are mutually exclusive, then we can use addition to find the probability that one or the other event occurs.

Finding Probabilities When Events Aren’t Mutually Exclusive

Let’s return to the example we used to explore the Addition Rule: We’re about to draw a *Scrabble* tile from a bag containing A, A, B, E, E, E, R, S, S, U. Consider these events: $J$ is “draw a vowel” and $K$ is “draw a letter that comes after L in the alphabet.” Since there are 6 vowels, $P\left(J\right)=\frac{6}{10}$. There are 4 tiles with letters that come after L alphabetically, so $P\left(K\right)=\frac{4}{10}$. What is $P\left(J\text{or}K\right)$? If we blindly apply the Addition Rule, we get $\frac{6}{10}+\frac{4}{10}=1$, which would mean that the compound event $J$ or $K$ is certain. However, it’s possible to draw a B, in which case neither $J$ nor $K$ happens. Where’s the error?

Conditional Probability and the Multiplication Rule

Conditional Probabilities

When we analyze experiments with multiple stages, we often update the probabilities of the possible final outcomes or the later stages of the experiment based on the results of one or more of the initial stages. These updated probabilities are called conditional probabilities.

Compound Events Using “And” and the Multiplication Rule

For multistage experiments, the outcomes of the experiment as a whole are often stated in terms of the outcomes of the individual stages. Commonly, those statements are joined with “and.” For example, in the sock drawer example just above, one outcome might be “the left sock is black and the right sock is blue.” As with “or” compound events, these probabilities can be computed with basic arithmetic.

The Binomial Distribution

Binomial Experiments

The tools of this section apply to multistage experiments that satisfy some pretty specific criteria. Before we move on to the analysis, we need to introduce and explain those criteria so that we can recognize experiments that fall into this category. Experiments that satisfy each of these criteria are called binomial experiments. A binomial experiment is an experiment with a fixed number of repeated independent binomial trials, where each trial has the same probability of success.

The Binomial Formula

If we flip a coin 100 times, you might expect the number of heads to be around 50, but you probably wouldn’t be surprised if the actual number of heads was 47 or 52. What is the probability that the number of heads is exactly 50? Or falls between 45 and 55? It seems unlikely that we would get more than 70 heads. Exactly how unlikely is that?

The Binomial Distribution

If we are interested in the probability of more than just a single outcome in a binomial experiment, it’s helpful to think of the Binomial Formula as a function, whose input is the number of successes and whose output is the probability of observing that many successes. Generally, for a small number of trials, we’ll give that function in table form, with a complete list of the possible outcomes in one column and the probability in the other.

Expected Value

Expected Value

Many experiments have numbers associated with their outcomes. Some are easy to define; if you roll 2 dice, the sum of the numbers showing is a good example. In some card games, cards have different point values associated with them; for example, in some forms of the game rummy, aces are worth 15 points; 10s, jacks, queens, and kings are worth 10; and all other cards are worth 5. The outcomes of casino and lottery games are all associated with an amount of money won or lost. These outcome values are used to find the expected value of an experiment: the mean of the values associated with the outcomes that we would observe over a large number of repetitions of the experiment. (See Conditional Probability and the Multiplication Rule for more on means.)

Interpreting Expected Values

As we noted, the expected value of an experiment is the mean of the values we would observe if we repeated the experiment a large number of times. (This interpretation is due to an important theorem in the theory of probability called the Law of Large Numbers.) Let’s use that to interpret the results of the previous example.

Using Expected Value

Now that we know how to find and interpret expected values, we can turn our attention to using them. Suppose someone offers to play a game with you. If you roll a die and get a 6, you get \$10. However, if you get a 5 or below, you lose \$1. Is this a game you’d want to play? Let’s look at the expected value: The probability of winning is $\frac{1}{6}$ and the probability of losing is $\frac{5}{6}$, so the expected value is $10×\frac{1}{6}+\left(-1\right)×\frac{5}{6}=\frac{5}{6}\approx 0.83$. That means, on average, you’ll come out ahead by about 83 cents every time you play this game. It’s a great deal! On the other hand, if the winnings for rolling a 6 drop to \$3, the expected value becomes $3×\frac{1}{6}+\left(-1\right)×\frac{5}{6}=-\frac{1}{3}\approx -0.33$, meaning you should expect to lose about 33 cents on average for every time you play. Playing that game is not a good idea! In general, this is how casinos and lottery corporations make money: Every game has a negative expected value for the player.

Statistics

Introduction

Gathering and Organizing Data

Sampling and Gathering Data

The *Digest's* failure highlights the need for what is now considered the most important criterion for sampling: randomness. This randomness can be achieved in several ways. Here we cover some of the most common.

Organizing Data

Once data have been collected, we turn our attention to analysis. Before we analyze, though, it’s useful to reorganize the data into a format that makes the analysis easier. For example, if our data were collected using a paper survey, our *raw data* are all broken down by respondent (represented by an individual response sheet). To perform an analysis on all the responses to an individual question, we need to first group all the responses to each question together. The way we organize the data depends on the type of data we’ve collected.

Visualizing Data

Visualizing Categorical Data

If the data we’re visualizing is categorical, then we want a quick way to represent graphically the relative numbers of units that fall in each category. When we created the frequency distributions in the last section, all we did was count the number of units in each category and record that number (this was the *frequency* of that category). Frequencies are nice when we’re organizing and summarizing data; they’re easy to compute, and they’re always whole numbers. But they can be difficult to understand for an outsider who’s being introduced to your data.

Visualizing Quantitative Data

There are several good ways to visualize quantitative data. In this section, we’ll talk about two types: stem-and-leaf plots and histograms.

Misleading Graphs

Graphical representations of data can be manipulated in ways that intentionally mislead the reader. There are two primary ways this can be done: by manipulating the scales on the axes and by manipulating or misrepresenting areas of bars. Let’s look at some examples of these.

Mean, Median and Mode

The Mode

In our discussion of average heights, the first possible definition we offered was the height that more people share than any other. This is the mode, or the value that appears most often. If there are two modes, the data are bimodal.

The Median

Let's revisit our example of trying to identify the height of the “average” person. If we lined everyone up in order by height and found the person right in the middle, that person’s height is called the median, or the value that is greater than no more than half and less than no more than half of the values.

The Mean

Recall our example of ways we could identify the “average” height of an individual. The last method we discussed was also the most complicated. It involved a game where the player guesses a height, then figures out how far off that guess is from every single person’s height. Those differences get squared and added together to get a score. Our next measure of centrality gives the lowest possible score: No other guess would beat it in the game. Given a dataset containing *n* total values, the mean of the dataset is the sum of all the data values, divided by *n*.

Which Is Better: Mode, Median, or Mean?

If the mode, median, and mean all purport to measure the same thing (centrality), why do we need all three? The answer is complicated, as each measure has its own strengths and weaknesses. The mode is simple to compute, but there may be more than one. Further, if no data value appears more than once, the mode is entirely unhelpful. As for the mean and median, the main difference between these two measures is how each is affected by extreme values.

Range and Standard Deviation

The Range

Our first measure of dispersion is the **range**, or the difference between the maximum and minimum values in the set. It’s the measure we used in the standardized test example above.

The Standard Deviation

The standard deviation is a measure of dispersion that can be interpreted as approximately the average distance of every data value from the mean. (This distance from the mean is the “deviation” in “standard deviation.”)

Percentiles

Finding Percentiles

There are some other terms that are related to "percentile" with meanings you may infer from their roots. Remember that the word *percent* means “per hundred.” This reflects that percentiles divide our data into 100 pieces. The word quartile has a root that means “four.” So, if a data value is at the first quantile of a dataset, that means that if you break the data into four parts (because of the *quart-*), this data value comes after the first of those four parts. In other words, it’s greater than 25% of the data, placing it at the 25th percentile. Quintile has a root meaning “five,” so a data value at the third quintile is greater than three-fifths of the data in the set. That would put it at the 60th percentile. The general term for these is quantiles (the root *quant*– means “number”).

The Normal Distribution

Moving Toward Normality

Let’s take a look at a histogram for the dataset in our section opener:

The Normal Distribution

In the coin flipping example above, the distribution of the number of heads for 10,000 trials was close to perfectly symmetric and bell-shaped:

![A combinational graph titled, results of 100 flips. The combinational graph includes a histogram and a normal distribution curve. The horizontal axis representing the number of heads ranges from 30 to 70, in increments of 10. The vertical axis representing frequency ranges from 0 to 800, in increments of 200. The histogram shows bars of varying heights. A normal distribution curve is drawn across the heights of the bars and it infers the following data. The curve begins at (30, 0), has a peak value at (48, 780), and ends at (70, 0).](../../media/CS_Figure_08_06_068.jpg)

Properties of Normal Distributions: The 68-95-99.7 Rule

The most important property of normal distributions is tied to its standard deviation. If a dataset is perfectly normally distributed, then 68% of the data values will fall within one standard deviation of the mean. For example, suppose we have a set of data that follows the normal distribution with mean 400 and standard deviation 100. This means 68% of the data would fall between the values of 300 (one standard deviation *below* the mean: $400-100=300$) and 500 (one standard deviation *above* the mean: $400+100=500$). Looking at the histogram below, the shaded area represents 68% of the total area under the graph and above the axis:

Standardized Scores

When we want to apply the 68-95-99.7 Rule, we must first figure out how many standard deviations above or below the mean our data fall. This calculation is common enough that it has its own name: the standardized score. Values above the mean have positive standardized scores, while those below the mean have negative standardized scores. Since it's common to use the letter $z$ to represent a standard score, this value is also often referred to as a $\mathbit{z}$**-score**.

Using Google Sheets to Find Normal Percentiles

The 68-95-99.7 Rule is great when we’re dealing with whole-number $z$-scores. However, if the $z$-score is not a whole number, the Rule isn’t going to help us. Luckily, we can use technology to help us out. We’ll talk here about the built-in functions in Google Sheets, but other tools work similarly.

Applications of the Normal Distribution

College Entrance Exams

There are two good ways to compare two data values from different groups: using $z$-scores and using percentiles. The two methods will always give consistent results (meaning that we won’t find, for example, that the first value is better using $z$-scores but the second value is better using percentiles), so use whichever method is more comfortable for you.

Coin flipping

In the opening of The Normal Distribution, we saw that the number of heads we get when we flip a coin 100 times is distributed normally. It can be shown that if $n$ is the number of flips, then the mean of that distribution is $\frac{n}{2}$ and the standard deviation is $\frac{\sqrt{n}}{2}$ (as long as $n\ge 20$). So, for 100 flips, the mean of the distribution is 50 and the standard deviation is 5.
 In that opening example, one of our early runs gave us 70 heads in 100 flips, which we noted seemed unusual. Using the normal distribution, we can identify exactly how unusual that really is. Using Google Sheets, the formula “=NORM.DIST(70, 50, 5, TRUE)” gives us 0.999968, which is the 99.997th percentile! How is that useful? Suppose you need to test whether a coin is fair, and so you flip it 100 times. While we might be suspicious if we get 70 heads out of the 100 flips, we now have a numerical measure for how unusual that is: If the coin were fair, we would expect to see 70 heads (or more) only $100-99.9968=0.0032%$ of the time. That’s really unlikely! Analysis like this is related to *hypothesis testing*, an important application of statistics in the sciences and social sciences.

Analyzing Data That Are Normally Distributed

Whenever we’re working with a dataset that has a distribution that looks symmetric and bell-shaped, we can use techniques associated with the normal distribution to analyze the data.

Scatter Plots, Correlation, and Regression Lines

Relationships Between Quantitative Datasets

Before we can evaluate a relationship between two datasets, we must first decide if we feel that one might depend on the other. In our exam example, it is appropriate to say that the score on the final depends on the score on the midterm, rather than the other way around: if the midterm depended on the final, then we’d need to know the final score *first*, which doesn’t make sense.

Linear Regression

The final step in our analysis of the relationship between two datasets is to find and use the equation of the regression line. For a given set of explanatory and response data, the regression line (also called the **least-squares line** or line of best fit) is the line that does the best job of approximating the data.

Extrapolation

A very common misuse of regression techniques involves extrapolation, which involves making a prediction about something that doesn't belong in the dataset.

Correlation Does Not Imply Causation

One of the most common fallacies about statistics has to do with the relationship between two datasets. In the dataset “Public”, we find that the correlation coefficient between the 75th percentile math SAT score and the 75th percentile verbal SAT score is 0.92, which is really strong. The slope of the regression line that predicts the verbal score from the math score is 0.729, which we might interpret as follows: “If the 75th percentile math SAT score goes up by 10 points, we’d expect the corresponding verbal SAT score to go up by just over 7 points.”

Metric Measurement

Introduction

The Metric System

Units of Measurement in the Metric System

Units of measurement provide common standards so that regardless of where or when an object or substance is measured, the results are consistent. When measuring distance, the units of measure might be feet, meters, or miles. Weight might be expressed in terms of pounds or grams. Volume or capacity might be measured in gallons, or liters. Understanding how metric units of measure relate to each other is essential to understanding the metric system:

Metric Prefixes

Unlike the U.S. Customary System of Measurement in which 12 inches is equal to 1 foot and 3 feet are equal to 1 yard, the metric system is structured so that the units within the system get larger or smaller by a power of 10. For example, a centimeter is ${10}^{-2}$, or 100 times smaller than a meter, while the kilometer is 103, or 1,000 times larger than a meter.

Converting Metric Units of Measure

Imagine you order a textbook online and the shipping detail indicates the weight of the book is 1 kg. By attaching the letter “k” to the base unit of gram (g), the unit used to express the measure is ${10}^{3}$or 1,000 times greater than a gram. One kilogram is equivalent to 1,000 grams.

Measuring Area

Reasonable Values for Area

Because area is determined by multiplying two lengths, the magnitude of difference between different square units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a square meter $\left({\text{m}}^{\text{2}}\right)$ is $100×100,\mathrm{or}10,000$ times greater in area than a square centimeter $\left({\text{cm}}^{\text{2}}\right)$. The relationships between benchmark metric area units are shown in the following table.

Converting Units of Measures for Area

Just like converting units of measure for distance, you can convert units of measure for area. However, the conversion factor, or the number used to multiply or divide to convert from one area unit to another, is not the same as the conversion factor for metric distance units. Recall that the conversion factor for area is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are shown in [link].

Solving Application Problems Involving Area

While it may seem that solving area problems is as simple as multiplying two numbers, often determining area requires more complex calculations. For example, when measuring the area of surfaces, you may need to account for portions of the surface that are not relevant to your calculation.

Measuring Volume

Reasonable Values for Volume

Because volume is determined by multiplying three lengths, the magnitude of difference between different cubic units is exponential. In other words, while a meter is 100 times greater in length than a centimeter, a cubic meter, m3, is $100×100×100,\mathrm{or}1,000,000$ times greater in area than a cubic centimeter, cm3. This relationship between benchmark metric volume units is shown in the following table.

Converting Like Units of Measures for Volume

Just like converting units of measure for distance, you can convert units of measure for volume. However, the conversion factor, the number used to multiply or divide to convert from one volume unit to another, is different from the conversion factor for metric distance units. Recall that the conversion factor for volume is exponentially relative to the conversion factor for distance. The most frequently used conversion factors are illustrated in [link].

Understanding Other Metric Units of Volume

When was the last time you purchased a bottle of soda? Was the volume of the bottle expressed in cubic centimeters or liters? The liter (L) is a metric unit of capacity often used to express the volume of liquids. A liter is equal in volume to 1 cubic decimeter. A milliliter is equal in volume to 1 cubic centimeter. So, when a doctor orders 10 cc (cubic centimeters) of saline to be administered to a patient, they are referring to 10 mL of saline.

Solving Application Problems Involving Volume

Knowing the volume of an object lets you know just how much that object can hold. When making a bowl of punch you might want to know the total amount of liquid a punch bowl can hold. Knowing how many liters of gasoline a car’s tank can hold helps determine how many miles a car can drive on a full tank. Regardless of the application, understanding volume is essential to many every day and professional tasks.

Measuring Weight

Reasonable Values for Weight

To have an essential understanding of metric weight, you must be able to identify reasonable values for weight. When testing for reasonableness, you should assess both the unit and the unit value. Only by examining both can you determine whether the given weight is reasonable for the situation.

Converting Like Units of Measures for Weight

Just like converting units of measure for distance, you can convert units of measure for weight. The most frequently used conversion factors for metric weight are illustrated in [link].

Solving Application Problems Involving Weight

From children’s safety to properly cooking a pie, knowing how to solve problems involving weight is vital to everyday life. Let’s review some ways that knowing how to work with metric weight can facilitate important decisions and delicious eating.

Measuring Temperature

Converting Between Fahrenheit and Celsius Temperatures

Understanding how to convert between Fahrenheit and Celsius temperatures is an essential skill in understanding metric temperatures. You likely know that below 32 °F means freezing temperatures and perhaps that the same holds true for 0 °C. While it may be difficult to recall that water boils at 212 °F, knowing that it boils at 100 °C is a fairly easy thing to remember.

Reasonable Values for Temperature

While knowing the exact temperature is important in most cases, sometimes an approximation will do. When trying to assess the reasonableness of values for temperature, there is a quicker way to convert temperatures for an approximation using mental math. These simpler formulas are listed in [link].

Solving Application Problems Involving Temperature

Whether traveling abroad or working in a clinical laboratory, knowing how to solve problems involving temperature is an important skill to have. Many food labels express sizes in both ounces and grams. Most rulers and tape measures are two-sided with one side marked in inches and feet and the other in centimeters and meters. And while many thermometers have both Fahrenheit and Celsius scales, it really isn’t practical to pull out a thermometer when cooking a recipe that uses metric units. Let’s review at few instances where knowing how to fluently use the Celsius scale helps solve problems.

Geometry

Introduction

Points, Lines, and Planes

Points and Lines

The first definition Euclid wrote was that of a point. He defined a point as “that which has no part.” It was later expanded to “an indivisible location which has no width, length, or breadth.” Here are the first two of the five postulates, as they are applicable to this first topic:

Parallel Lines

**Parallel lines** are lines that lie in the same plane and move in the same direction, but never intersect. To indicate that the line ${l}_{1}$ and the line ${l}_{2}$ are parallel we often use the symbol ${l}_{1}\parallel {l}_{2}.$ The distance $d$ between parallel lines remains constant as the lines extend infinitely in both directions. See [link].

Perpendicular Lines

Two lines that intersect at a ${90}^{\circ }$ angle are perpendicular lines and are symbolized by $\perp$. If ${l}_{1}$ and ${l}_{2}$ are perpendicular, we write ${l}_{1}\perp {l}_{2}.$ When two lines form a right angle, a ${90}^{\circ }$ angle, we symbolize it with a little square $\square .$ See [link].

Defining Union and Intersection of Sets

Union and intersection of sets is a topic from set theory that is often associated with points and lines. So, it seems appropriate to introduce a mini-version of set theory here. First, a **set** is a collection of objects joined by some common criteria. We usually name sets with capital letters. For example, the set of odd integers between 0 and 10 looks like this: $A=\left\{1,3,5,7,9\right\}.$ When it involves sets of lines, line segments, or points, we are usually referring to the **union** or **intersection** of set.

Planes

A **plane,** as defined by Euclid, is a “surface which lies evenly with the straight lines on itself.” A plane is a two-dimensional surface with infinite length and width, and no thickness. We also identify a plane by three noncollinear points, or points that do not lie on the same line. Think of a piece of paper, but one that has infinite length, infinite width, and no thickness. However, not all planes must extend infinitely. Sometimes a plane has a limited area.

Angles

Classifying Angles

Angles are measured in radians or degrees. For example, an angle that measures $\pi$ radians, or 3.14159 radians, is equal to the angle measuring ${180}^{\circ }.$ An angle measuring $\frac{\pi }{2}$ radians, or 1.570796 radians, measures ${90}^{\circ }.$ To translate degrees to radians, we multiply the angle measure in degrees by $\frac{\pi }{180}.$ For example, to write ${45}^{\circ }$ in radians, we have

${45}^{\circ }\left(\frac{\pi }{180}\right)=\frac{\pi }{4}=0.785398{\rule{0.28em}{0ex}}\text{radians.}$

Notation

Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols $\angle$ or $\measuredangle$ before the points. When we are referring to the measure of the angle, we use the symbol $m\measuredangle$. See [link].

Adjacent Angles

Two angles with the same starting point or vertex and one common side are called adjacent angles. In [link], angle $\angle DBC$ is adjacent to $\angle CBA$. Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle.

Supplementary Angles

Two angles are **supplementary** if the sum of their measures equals ${180}^{\circ }.$ In [link], we are given that $m\measuredangle FBE={35}^{\circ },$ so what is $m\measuredangle ABE?$ These are supplementary angles. Therefore, because $m\measuredangle ABF={180}^{\circ }$, and as ${180}^{\circ }-{35}^{\circ }={145}^{\circ },$ we have $m\measuredangle ABE={145}^{\circ }.$

Complementary Angles

Two angles are **complementary** if the sum of their measures equals ${90}^{\circ }.$ In [link], we have $m\measuredangle ABC={30}^{\circ },$ and $m\measuredangle ABD={90}^{\circ }.$ What is the $m\measuredangle CBD?$ These are complementary angles. Therefore, because ${90}^{\circ }-{30}^{\circ }={60}^{\circ },$ the $\measuredangle CBD={60}^{°}.$

Vertical Angles

When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example, [link] shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.

Transversals

When two parallel lines are crossed by a straight line or **transversal**, eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. See [link]. Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles.

Alternate Interior Angles

Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example, $\measuredangle 3$ and $\measuredangle 6$ are alternate interior angles and have equal measure; $\measuredangle 4$ and $\measuredangle 5$ are alternate interior angles and have equal measure as well. See [link].

Alternate Exterior Angles

Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in [link], $\measuredangle 2$ and $\measuredangle 7$ are alternate exterior angles and have equal measures; $\measuredangle 1$ and $\measuredangle 8$ are alternate exterior angles and have equal measures as well.

Corresponding Angles

Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In [link], $\measuredangle 1$ and $\measuredangle 5$ are corresponding angles and have equal measures; $\measuredangle 3$ and $\measuredangle 7$ are corresponding angles and have equal measures; $\measuredangle 2$ and $\measuredangle 6$ are corresponding angles and have equal measures; $\measuredangle 4$ and $\measuredangle 8$ are corresponding angles and have equal measures as well.

Triangles

Identifying Triangles

Joining any three noncollinear points with line segments produces a triangle. For example, given points $A$, $B$, and $C$, connected by the line segments $\overline{AB},\overline{BC,}$ and $\overline{AC},$ we have a triangle, as shown in [link].

Congruence

If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent.

The Congruence Theorems

The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol $\cong$ to define congruence. For example, $\text{Δ}ABC\cong \text{Δ}DEF$.

Similarity

If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction $\frac{a}{b}$ is called a ratio, but two fractions equal to each other is called a proportion, such as $\frac{a}{b}=\frac{c}{d}.$

Polygons, Perimeter, and Circumference

Identifying Polygons

A polygon is a closed, two-dimensional shape classified by the number of straight-line sides. See [link] for some examples. We show only up to eight-sided polygons, but there are many, many more.

Perimeter

Perimeter refers to the outside measurements of some area or region given in linear units. For example, to find out how much fencing you would need to enclose your backyard, you will need the perimeter. The general definition of **perimeter** is the sum of the lengths of the sides of an enclosed region. For some geometric shapes, such as rectangles and circles, we have formulas. For other shapes, it is a matter of just adding up the side lengths.

Sum of Interior and Exterior Angles

To find the sum of the measurements of interior angles of a regular polygon, we have the following formula.

Circles and Circumference

The perimeter of a circle is called the **circumference**. To find the circumference, we use the formula $C=\pi d,$ where $d$ is the diameter, the distance across the center, or $C=2\pi r,$ where $r$ is the radius.

Tessellations

Tessellation Properties and Transformations

A regular tessellation means that the pattern is made up of congruent regular polygons, same size and shape, including *some type of movement*; that is, some type of transformation or symmetry. Here we consider the rigid motions of translations, rotations, reflections, or glide reflections. A plane of tessellations has the following properties:

* Patterns are repeated and fill the plane.
* There are no gaps or overlaps. Shapes must fit together perfectly. (It was Escher who determined that a proper tessellation could have no gaps and no overlaps.)
* Shapes are combined using a transformation.
* All the shapes are joined at a vertex. In other words, if you were to draw a circle around a vertex, it would include a corner of each shape touching at that vertex.
* For a tessellation of regular congruent polygons, the sum of the measures of the interior angles that meet at a vertex equals ${360}^{\circ }.$

Interior Angles

The sum of the interior angles of a tessellation is ${360}^{\circ }$. In [link], the tessellation is made of six triangles formed into the shape of a hexagon. Each angle inside a triangle equals ${60}^{\circ }$, and the six vertices meet the sum of those interior angles, $6\left({60}^{°}\right)={360}^{°}$.

Tessellating Shapes

We might think that all regular polygons will tessellate the plane by themselves. We have seen that squares do and hexagons do. The pattern of squares in [link] is a translation of the shape horizontally and vertically. The hexagonal pattern in [link], is translated horizontally, and then on the diagonal, either to the right or to the left. This particular pattern can also be formed by rotations. Both tessellations are made up of congruent shapes and each shape fits in perfectly as the pattern repeats.

Area

Area of Triangles

The formula for the area of a triangle is given as follows.

Area of Quadrilaterals

A quadrilateral is a four-sided polygon with four vertices. Some quadrilaterals have either one or two sets of parallel sides. The set of quadrilaterals include the square, the rectangle, the parallelogram, the trapezoid, and the rhombus. The most common quadrilaterals are the square and the rectangle.

Rectangle

Similarly, the area for a rectangle is found by multiplying length times width. The rectangle in [link] has width equal to 5 in and length equal to 12 in. The area is $A=5\left(12\right)=60{\rule{0.28em}{0ex}}{\text{in}}^{2}.$

Parallelogram

The area of a parallelogram can be found using the formula for the area of a triangle. Notice in [link], if we cut a diagonal across the parallelogram from one vertex to the opposite vertex, we have two triangles. If we multiply the area of a triangle by 2, we have the area of a parallelogram:

$\begin{array}{ccc}\hfill A& \hfill =\hfill & 2\left(\frac{1}{2}bh\right)\hfill \\ \hfill A& \hfill =\hfill & bh\hfill \end{array}$

Trapezoid

Another quadrilateral is the trapezoid. A trapezoid has one set of parallel sides or bases. The formula for the area of a trapezoid with parallel bases $a$ and $b$ and height $h$ is given here.

Area of Polygons

To find the area of a regular polygon, we need to learn about a few more elements. First, the apothem $a$ of a regular polygon is a line segment that starts at the center and is perpendicular to a side. The radius $r$ of a regular polygon is also a line segment that starts at the center but extends to a vertex. See [link].

Area of Circles

Just as the circumference of a circle includes the number $\pi ,$ so does the formula for the area of a circle. Recall that $\pi$ is a non-terminating, non-repeating decimal number: $\pi =3.14159\dots$. It represents the ratio of the circumference to the diameter, so it is a critical number in the calculation of circumference and area.

Area within Area

Suppose you want to install a round hot tub on your backyard patio. How would you calculate the space needed for the hot tub? Or, let’s say that you want to purchase a new dining room table, but you are not sure if you have enough space for it. These are common issues people face every day. So, let’s take a look at how we solve these problems.

Volume and Surface Area

Three-Dimensional Objects

In geometry, three-dimensional objects are called geometric solids. Surface area refers to the flat surfaces that surround the solid and is measured in square units. Volume refers to the space inside the solid and is measured in cubic units. Imagine that you have a square flat surface with width and length. Adding the third dimension adds depth or height, depending on your viewpoint, and now you have a box. One way to view this concept is in the Cartesian coordinate three-dimensional space. The $x$-axis and the $y$-axis are, as you would expect, two dimensions and suitable for plotting two-dimensional graphs and shapes. Adding the $z$-axis, which shoots through the origin perpendicular to the $xy$-plane, and we have a third dimension. See [link].

Right Prisms

A right prism is a particular type of three-dimensional object. It has a polygon-shaped base and a congruent, regular polygon-shaped top, which are connected by the height of its lateral sides, as shown in [link]. The lateral sides form a right angle with the base and the top. There are rectangular prisms, hexagonal prisms, octagonal prisms, triangular prisms, and so on.

Right Cylinders

There are similarities between a prism and a cylinder. While a prism has parallel congruent polygons as the top and the base, a right cylinder is a three-dimensional object with congruent circles as the top and the base. The lateral sides of a right prism make a ${90}^{\circ }$ angle with the polygonal base, and the side of a cylinder, which unwraps as a rectangle, makes a ${90}^{\circ }$ angle with the circular base.

Applications of Surface Area and Volume

The following are just a small handful of the types of applications in which surface area and volume are critical factors. Give this a little thought and you will realize many more practical uses for these procedures.

Optimization

Problems that involve optimization are ones that look for the best solution to a situation under some given conditions. Generally, one looks to calculus to solve these problems. However, many geometric applications can be solved with the tools learned in this section. Suppose you want to make some throw pillows for your sofa, but you have a limited amount of fabric. You want to make the largest pillows you can from the fabric you have, so you would need to figure out the dimensions of the pillows that will fit these criteria. Another situation might be that you want to fence off an area in your backyard for a garden. You have a limited amount of fencing available, but you would like the garden to be as large as possible. How would you determine the shape and size of the garden? Perhaps you are looking for maximum volume or minimum surface area. Minimum cost is also a popular application of optimization. Let’s explore a few examples.

Right Triangle Trigonometry

Pythagorean Theorem

The Pythagorean Theorem is used to find unknown sides of right triangles. The theorem states that the sum of the squares of the two legs of a right triangle equals the square of the hypotenuse (the longest side of the right triangle).

Distance

The applications of the Pythagorean Theorem are countless, but one especially useful application is that of distance. In fact, the distance formula stems directly from the theorem. It works like this:

${30}^{\circ }\text{-}{60}^{\circ }\text{-}{90}^{\circ }$ Triangles

In geometry, as in all fields of mathematics, there are always special rules for special circumstances. An example is the *perfect square* rule in algebra. When expanding an expression like ${\left(2x+5y\right)}^{2},$ we do not have to expand it the long way:

$\begin{array}{ccc}\hfill {\left(2x+5y\right)}^{2}& \hfill =\hfill & \left(2x+5y\right)\left(2x+5y\right)\hfill \\ \hfill & \hfill =\hfill & {\left(2x\right)}^{2}+10xy+10xy+{\left(5y\right)}^{2}\hfill \\ \hfill & \hfill =\hfill & 4{x}^{2}+20xy+25{y}^{2}\hfill \end{array}$

$4{5}^{\circ }\text{-}4{5}^{\circ }\text{-}9{0}^{\circ }$ Triangles

The ${45}^{\circ }\text{-}{45}^{\circ }\text{-}{90}^{\circ }$ triangle is another *special* triangle such that with the measure of one side we can find the measures of all the sides. The two angles adjacent to the ${90}^{\circ }$ angle are equal, and each measures ${45}^{\circ }.$ If two angles are equal, so are their opposite sides. The ratio among sides is $1:1:\sqrt{2},$ or $x:x:x\sqrt{2},$ as shown in [link].

Trigonometry Functions

Trigonometry developed around 200 BC from a need to determine distances and to calculate the measures of angles in the fields of astronomy and surveying. Trigonometry is about the relationships (or ratios) of angle measurements to side lengths in primarily right triangles. However, trigonometry is useful in calculating missing side lengths and angles in other triangles and many applications.

Angle of Elevation and Angle of Depression

Other problems that involve trigonometric functions include calculating the angle of elevation and the angle of depression. These are very common applications in everyday life. The angle of elevation is the angle formed by a horizontal line and the line of sight from an observer to some object at a higher level. The angle of depression is the angle formed by a horizontal line and the line of sight from an observer to an object at a lower level.

Voting and Apportionment

Introduction

Voting Methods

Majority versus Plurality Voting

When an election involves only two options, a simple majority is a reasonable way to determine a winner. A majority is a number equaling more than half, or greater than 50 percent of the total.

Runoff Voting

Has your family ever debated what to have for dinner? Suppose your family is deciding on a restaurant and exactly half of you want to have pizza but the other half want hamburgers. How do you decide when the result is a tie? You need a tiebreaker!

Steps to Determine Winner by Plurality or Majority Election with Runoff

To determine the winner by plurality or when a majority election with runoff occurs, we take these three steps:

Ranked-Choice Voting

In [link] and Your Turn 11.4, you were given a list that ordered each voter’s preferences. This ordering is called a preference ranking. A ballot in which a voter is required to give an ordering of their preferences is a ranked ballot, and any voting system in which a voter uses a ranked ballot is referred to as ranked voting.

Steps to Determine Winner by Ranked-Choice Voting

To determine the winner when ranked-choice voting occurs, we take these three steps:

Borda Count Voting

Ranked-choice voting is one type of ranked voting that simulates multiple runoffs based on ranked ballots. Another type of ranked voting is the **Borda count method**, which uses ranked ballots that award candidates points corresponding to the number of candidates ranked lower on each ballot.

Pairwise Comparison and Condorcet Voting

We have discussed two kinds of ranked voting methods so far: ranked-choice and Borda count. A third type of ranked voting is the **pairwise comparison method**, in which the candidates receive a point for each candidate they would beat in a one-on-one election and half a point for each candidate they would tie. If one candidate earns more points than the others, then that candidate wins. This method is one of several **Condorcet voting methods**, which are methods in which candidates are ranked and then compared pairwise to each other, a candidate having to beat all others in order to win. These methods vary in the way candidates are scored, and there is not always a clear winner. A candidate who wins each possible pairing is known as a **Condorcet candidate**. These terms are named after the Marquis de Condorcet, a French philosopher and mathematician who preferred the pairwise comparison method to the Hare method and made public arguments in its favor.

Steps to Determine a Winner by Pairwise Comparison Method Using a Matrix

To determine the winner when the pairwise comparison method is used, we take these three steps:

Three Key Questions

Before you decide if you want to use the pairwise comparison method for Imaginarian elections, let’s consider three questions that might affect your decision.

Approval Voting

The last type of voting system you will consider for your budding democracy is an approval voting system. In this system, each voter may approve any number of candidates without rank or preference for one over another (among the approved candidates), and the candidate approved by the most voters wins.
 This voting system has aspects in common with plurality voting and Condorcet voting methods, but it has characteristics that distinguish it from both. An approval voting ballot lists the candidates and provides the option to approve or not approve each candidate.

Compare and Contrast Voting Methods to Identify Flaws

Wow! We have covered a lot of options for the voting methods. Now, you need to decide which one is best for Imaginaria. Imaginarians might consider characteristics of certain voting systems desirable and others undesirable. In some cases, voters may consider these undesirable traits to be flaws in a voting system that are significant enough to motivate them to reject that system. If you are feeling a bit overwhelmed by this decision, maybe it would help to read about the experiences of others who have faced similar questions.

Fairness in Voting Methods

The Majority Criterion

One of the most fundamental concepts in voting is the idea that most voters should be in favor of a candidate for a candidate to win, and that a candidate should not win without majority support. This concept is known as the majority criterion.

Head-to-Head Criterion

Another fairness criterion you must consider as you select a voting method for Imaginaria is the Condorcet criterion, also known as the head-to-head criterion. An election method satisfies the Condorcet criterion provided that the Condorcet candidate wins the election whenever a Condorcet candidate exists. A Condorcet method is any voting method that satisfies the Condorcet criterion.

Monotonicity Criterion

The citizens of Imaginaria might be surprised to learn that it is possible for a voter to cause a candidate to lose by ranking that candidate higher on their ballot. Is that fair? Most voters would say, “Absolutely not!!” This is an example of a violation of the fairness criterion called the **monotonicity criterion**, which is satisfied when no candidate is harmed by up-ranking nor helped by down-ranking, provided all other votes remain the same.

Irrelevant Alternatives Criterion

We have covered a lot about voting fairness, but there is one more fairness criterion that you and the other Imaginarians should know. Consider this well-known anecdote that is sometimes attributed to the American philosopher Sidney Morgenbesser:

Standard Divisors, Standard Quotas, and the Apportionment Problem

The Apportionment Problem

In the new democracy of Imaginaria, there are four states: Fictionville, Pretendstead, Illusionham, and Mythbury. Each state will have representatives in the Imaginarian Legislature. You might now have an agreement on which voting method your citizens will use to elect representatives. However, before that process can even begin, you must decided on how many representatives each state will receive. This decision will present its own challenges.

The Standard Divisor

There are two houses of congress in the United States: the Senate and the House of Representatives. Each state has two senators, but the number of representatives depends on the population of the state. The number of representative seats in the U.S. House of Representatives is currently set by law to be 435. In order to distribute the seats fairly to each state, the ratio of the population of the U.S. to the number of representative seats must be calculated. The ratio of the total population to the house size is called the standard divisor, and it is the number of members of the total population represented by one seat.

The Standard Quota

Once the standard divisor for the Imaginarian legislature is calculated, the next task is to determine the number of seats that each state should receive, which is referred to as the state’s standard quota. Unless all the states have the same population, each state will receive a different number of seats because the quantities will be proportionate to the state populations. To determine those amounts, we will use an idea we learned earlier. Recall that, when the number of units of item $A$ is proportionate to the number of units of item $B$, we have: $\text{units of}{\rule{0.28em}{0ex}}A=\frac{\text{units of}{\rule{0.28em}{0ex}}B}{\text{ratio of}{\rule{0.28em}{0ex}}B\text{'s to}{\rule{0.28em}{0ex}}A\text{'s}}$

Apportionment Methods

A Closer Look at the Apportionment Problem

In Standard Divisors, Standard Quotas, and the Apportionment Problem we calculated the standard divisor and the standard quotas in various apportionment scenarios. The results of those calculations routinely led to fractions and decimals of units. However, the seats in the House of Representatives, laptops in a classroom, or a variety of other resources, are indivisible, meaning they cannot be divided up into fractional parts. This leaves a decision to be made. For example, if the standard quota for the number of laptops to be distributed to a classroom is 12.44 units, how do we deal with the fractional part of 0.44? It is unclear if the classroom should receive 12 units, 13 units, or some other value. Let’s try traditional rounding to the nearest whole number value.

Hamilton's Method of Apportionment

One of the problems encountered when standard quotas are transformed into whole numbers using traditional rounding is that it is possible for the sum of the values to be greater than the number of seats available. A reasonable way to avoid this is to always round down, even when the first decimal place is five or greater. For example, a standard quota of 12.33 and a standard quota of 12.99 would both round down to 12. This is called the lower quota.

Steps for Hamilton’s Method of Apportionment

There are five steps we follow when applying Hamilton’s Method of apportionment:

The Quota Rule

A characteristic of an apportionment that is considered favorable is when the final quota values all either result from rounding down or rounding up from the standard quotas. The value that results from rounding down is called the lower quota, and the value that results from rounding up is called the upper quota.

Jefferson’s Method of Apportionment

Another approach to dealing with the fractional parts of the standard quotas is to modify the standard divisor so that the total of the resulting modified lower quotas is the necessary number of seats. This is the approach used by Jefferson.

Steps for Jefferson’s Method of Apportionment

We take four steps to apply Jefferson’s Method of apportionment:

Adams’s Method of Apportionment

Adams’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did, Adams used the upper quotas.

Webster’s Method of Apportionment

Webster’s method of apportionment is another method of apportionment that is based on a modified divisor. However, instead of basing the changes on the sum of the lower quotas, as Jefferson did or the sum of the upper quotas as Adams did, Webster used traditional rounding.

Comparing Apportionment Methods

Recall that the four apportionment methods discussed in this chapter differ in two main ways:

Flaws in Apportionment Methods

As we have seen, different apportionment methods can have the same results in some scenarios but different results in others. Citizens of states which receive fewer seats with a particular apportionment method will view the apportionment method as flawed and argue in favor of a different method. This inevitably creates debates regarding the use of one method over another. Methods that favor larger states are likely to be challenged by smaller states, methods that favor smaller states are likely to be challenged by larger states, and methods that violate the quota rule are likely to be challenged by states of any size depending on the circumstances.

Fairness in Apportionment Methods

Apportionment Paradoxes

The citizens of Imaginaria will want the apportionment method to be as fair as possible. There are certain characteristics that they would reasonably expect from a fair apportionment.

The Alabama Paradox

At the time of the 1880 U.S. Census, the Hamilton method of apportionment had replaced the Jefferson method. The number of seats in the House of Representatives was not fixed. To achieve the fairest apportionment possible, the house sizes were chosen so that the Hamilton and Webster methods would result in the same apportionment. The chief clerk of the Census Bureau calculated the apportionments for house sizes between 275 and 350. There was a surprising result that became known as the Alabama paradox, which is said to occur when an increase in house size reduces a state’s quota. Alabama would receive eight seats with a house size of 299, but only receive seven seats if the house size increased to 300. (Michael J. Caulfield (Gannon University), "Apportioning Representatives in the United States Congress - Paradoxes of Apportionment," *Convergence* (November 2010), DOI:10.4169/loci003163)

The Population Paradox

It is important for the founders of Imaginaria to keep in mind that the populations of states change as time passes. Some populations grow and some shrink. Some populations increase by a large amount while others increase by a small amount. These changes may necessitate a reapportionment of seats, or the recalculation of state quotas due to a change in population. It would be reasonable for Imaginarians to expect that the state with a population that has grown more than others will gain a seat before the other states. Once again, this is not always the case with the Hamilton method of apportionment. The population growth rate of a state is the ratio of the change in the population to the original size of the population, often expressed as a percentage. This value is positive if the population is increasing and negative if the population is decreasing. The population paradox occurs when a state with an increasing population loses a seat while a state with a decreasing population either retains or gains seats. More generally, the population paradox occurs when a state with a higher population growth rate loses seats while a state with a lower population growth rate retains or gains seats.

The New-States Paradox

As a founder of Imaginaria, you might also consider the possibility that Imaginaria could annex nearby lands and increase the number of states. This occurred several times in the United States such as when Oklahoma became a state in 1907. The House size was increased from 386 to 391 to accommodate Oklahoma’s quota of five seats. When the seats were reapportioned using Hamilton’s method, New York lost a seat to Maine despite the fact that their populations had not changed. This is an example of the new-state paradox, which occurs when the addition of a new state is accompanied by an increase in seats to maintain the standard ratio of population to seats, but one of the existing states loses a seat in the resulting reapportionment.

The Search for the Perfect Apportionment Method

The ideal apportionment method would simultaneously satisfy the following four fairness criteria.

Graph Theory

Introduction

Graph Basics

Parts of a Graph

In a graph, the objects are represented with dots and their connections are represented with lines like those in [link]. [link] displays a simple graph labeled *G* and a multigraph labeled *H*. The dots are called vertices; an individual dot is a **vertex**, which is one object of a set of objects, some of which may be connected. We often label vertices with letters. For example, Graph *G* has vertices *a, b, c*, and *d*, and Multigraph *H* has vertices, *e, f, g*, and *h*. Each line segment or connection joining two vertices is referred to as an edge. *H* is considered a multigraph because it has a double edge between *f* and *h*, and a double edge between *h* and *g*. Another reason *H* is called a multigraph is that it has a loop connecting vertex *e* to itself; a loop is an edge that joins a vertex to itself. Loops and double edges are not allowed in a simple graph.

Analyzing Geographical Maps with Graphs

When graphs are used to model and analyze real-world applications, the number of edges that meet at a particular vertex is important. For example, a graph may represent the direct flight connections for a particular airport as in [link]. Representing the connections with a graph rather than a map shifts the focus away from the relative positions and toward which airports are connected. In [link], the vertices are the airports, and the edges are the direct flight paths. The number of flight connections between a particular airport and other South Florida airports is the number of edges meeting at a particular vertex. For example, Key West has direct flights to three of the five airports on the graph. In graph theory terms, we would say that vertex FYW has degree 3. The degree of a vertex is the number of edges that connect to that vertex.

Graphs of Social Interactions

Geographical maps are just one of many real-world scenarios which graphs can depict. Any scenario in which objects are connected to each other can be represented with a graph, and the connections don’t have to be physical. Just think about all the connections you have to people around the world through social media! Who is in your network of Twitter followers? Whose Snapchat network are you connected to?

Graph Structures

Importance of the Degrees of Vertices

One reason scientists study these networks is to determine how successful the communication within a network continues to be when it experiences failures in nodes and connections. Graphs can be used to study the resilience of these networks. (Mikey Taylor, "Graph Theory & Machine Learning in Neuroscience," Medium.com, June 24, 2020)

Relating the Number of Edges to the Degrees of Vertices

In the applications of graph theory to neuroscience and sociology in [link] and YOUR TURN 12.6, there was a correlation between the degrees of vertices and the resilience of a network. Researchers in many fields have also observed a direct relationship between the number of edges in a graph and the degrees of the vertices. To begin to understand this relationship, consider a graph with five vertices and zero edges as in [link].

Completeness

Suppose that there were five strangers in a room, *A*, *B*, *C*, *D*, and *E*, and each one would be introduced to each of the others. How many introductions are necessary? One way to begin to answer this question is to draw a graph with each vertex representing an individual in the room and each edge representing an introduction as in [link].

Subgraphs

Sometimes a graph is a part of a larger graph. For example, the graph of South Florida Airports from [link] is part of a larger graph that includes Orlando International Airport in Central Florida, which is shown in [link]

Identifying and Naming Cycles

When you think of a cycle in everyday life, you probably think of something that begins and ends the same way. For example, the water cycle ([link]) begins with water in a lake or ocean, which evaporates into water vapor, condenses into clouds, and then returns to earth as rain or some other form of precipitation that settles into lakes or oceans. A cycle in graph theory is similar in that it begins and ends in the same way: It is a series of connected edges that begin and end at the same vertex but otherwise never repeat any vertices.

Cyclic Subgraphs and Cliques

When cycles appear as subgraphs within a larger graph, they are called **cyclic subgraphs**. Cyclic subgraphs are named by listing their vertices sequentially. The vertex where you begin is not important. Graph *K* in [link] has two triangle cycles (*g*, *h*, *j*) and (*h*, *i*, *j*), and one quadrilateral cycle (*g*, *h*, *i*, *j*).

Three-Cycles in Complete Graphs

Just as complete graphs have a predictable number of edges, complete graphs have a predictable number of cyclic subgraphs. Let’s look at the three-cycles within complete graphs with up to six vertices, which are shown in [link].

Comparing Graphs

When Are Two Graphs Really the Same Graph?

In arithmetic, when two numbers have the same value, we say they are equal, like ½ = 0.5. Although ½ and 0.5 look different, they have the same value, because they are assigned the same position on the real number line. When do we say that two graphs are equal?

When Are Two Graphs Really Different?

Verifying that two graphs are isomorphic can be a challenging process, especially for larger graphs. It makes sense to check for any obvious ways in which the graphs might differ so that we don’t spend time trying to verify that graphs are isomorphic when they are not. If two graphs have any of the differences shown in [link], then they cannot be isomorphic.

Recognizing Isomorphic Graphs

Isomorphic graphs that represent the same pattern of connections can look very different despite having the same underlying structure. The edges can be stretched and twisted. The graph can be rotated or flipped. For example, in [link], each of the diagrams represents the same pattern of connections.

Identifying and Naming Isomorphisms

When two graphs are isomorphic, meaning they have the same structure, there is a correspondence between their vertices, which can be named by listing corresponding pairs of vertices. This list of corresponding pairs of vertices in such a way that any adjacent pair in one graph corresponds to an adjacent pair in the other graph is called an **isomorphism**. Consider the isomorphic graphs in [link]. In [link], we could replace the labels Graph *F* with the labels from Graph *A* and have an identical graph, as in [link].

Complementary Graphs

Suppose that you are a camp counselor at Camp Woebegone and you are holding a camp Olympics with four events. The campers have signed up for the events. You drew a graph in [link] to help you visualize which events have campers in common.

Navigating Graphs

Walks

Suppose [link] is a maze you want to solve. You want to get from the start to the end.

Paths and Trails

A walk is the most basic way of navigating a graph because it has no restrictions except staying on the graph. When there are restrictions on which vertices or edges we can visit, we will call the walk by a different name. For example, if we want to find a walk that avoids travelling the same edge twice, we will say we want to find a trail (or directed trail). If we want to find a walk that avoids visiting the same vertex twice, we will say, we want to find a path (or directed path).

Circuits

In many applications of graph theory, such as creating efficient delivery routes, beginning and ending at the same location is a requirement. When a walk, path, or trail end at the same location or vertex they began, we call it closed. Otherwise, we call it open (does not begin and end at the same location or vertex). Some examples of closed walks, closed trails, and closed paths are given in in the following table.

Graph Colorings

In this section so far, we have looked at how to navigate graphs by proceeding from one vertex to another in a sequence that does not skip any vertices, but in some applications we may want to skip vertices. Remember the camp Olympics at Camp Woebegone in Comparing Graphs? You were planning a camp Olympics with four events. The campers signed up for the events. You drew a graph to help you visualize which events have campers in common. The vertices of Graph *E* in [link] represent the events and adjacent vertices indicate that there are campers who are participating in both.

Creating Colorings to Solve Problems

Let’s see how these facts can help us color the graph in [link].

The Four Color Problem

The idea of coloring graphs to solve problems was inspired by one of the most famous problems in mathematics, the “four color problem.” The idea was that, no matter how complicated a map might be, only four colors were needed to color the map so that no two regions that shared a boundary would be the same color. For many years, everyone suspected this to be true, because no one could create a map that needed more than four colors, but they couldn’t prove it was true in general. Finally, graphs were used to solve the problem!

Euler Circuits

Connectedness

Before we can talk about finding the best delivery route, we have to make sure there is a route at all. For example, suppose that you were tasked with visiting every airport on the graph in [link] by plane. Could you accomplish that task, only taking direct flight paths between airports listed on this graph? In other words, are all the airports connected by paths? Or are some of the airports disconnected from the others?

Origin of Euler Circuits

The city of Konigsberg, modern day Kaliningrad, Russia, has waterways that divide up the city. In the 1700s, the city had seven bridges over the various waterways. The map of those bridges is shown in [link]. The question as to whether it was possible to find a route that crossed each bridge exactly once and return to the starting point was known as the Konigsberg Bridge Problem. In 1735, one of the most influential mathematicians of all time, Leonard Euler, solved the problem using an area of mathematics that he created himself, graph theory!

Chinese Postman Problem

At Camp Woebegone, campers travel the waterways in canoes. As part of the Camp Woebegone Olympics, you will hold a canoeing race. You have placed a checkpoint on each of the 11 different streams. The competition requires each team to travel each stream, pass through the checkpoints in any order, and return to the starting line, as shown in the [link].

Identifying Euler Circuits

Solving the Chinese postman problem requires finding a shortest circuit through any graph or multigraph that visits every edge. In the case of Eulerian graphs, this means finding an Euler circuit. The method we will use is to find any circuit in the graph, then find a second circuit starting at a vertex from the first circuit that uses only edges that were not in the first circuit, then find a third circuit starting at a vertex from either of the first two circuits that uses only edges that were not in the first two circuits, and then continue this process until all edges have been used. In the end, you will be able to link all the circuits together into one large Euler circuit.

Eulerization

The Chinese postman problem doesn’t only apply to Eulerian graphs. Recall the postal delivery person who needed to deliver mail to every block of every street in a subdivision. We used the map in [link] to create the graph in [link].

Euler Trails

Euler Trails

If we need a trail that visits every edge in a graph, this would be called an Euler trail. Since trails are walks that do not repeat edges, an Euler trail visits every edge exactly once.

The Five Rooms Puzzle

Just as Euler determined that only graphs with vertices of even degree have Euler circuits, he also realized that the only vertices of odd degree in a graph with an Euler trail are the starting and ending vertices. For example, in [link], Graph *H* has exactly two vertices of odd degree, vertex *g* and vertex *e*. Notice the Euler trail we saw in Excercise 3 of [link] began at vertex *g* and ended at vertex *e*.

Bridges and Local Bridges

Now that we know which graphs have Euler trails, let’s work on a method to find them. The method we will use involves identifying bridges in our graphs. A bridge is an edge which, if removed, increases the number of components in a graph. Bridges are often referred to as cut-edges. In [link], there are several examples of bridges. Notice that an edge that is not part of a cycle is always a bridge, and an edge that is part of a cycle is never a bridge.

Finding an Euler Trail with Fleury’s Algorithm

Now that we are familiar with bridges, we can use a technique called Fleury’s algorithm, which is a series of steps, or algorithm, used to find an Euler trail in any graph that has exactly two vertices of odd degree.

Hamilton Cycles

Hamilton’s Puzzle

Before we look at the solution to Hamilton's puzzle, let’s review some vocabulary we used in [link]. It will be helpful to remember that directed cycle is a type of circuit that doesn’t repeat any edges or vertices.

Hamilton Cycles vs. Euler Circuits

Let’s practice naming and identifying Hamilton cycles, as well as distinguishing them from Euler circuits. It is important to remember that Euler circuits visit all edges without repetition, while Hamilton cycles visit all vertices without repetition. Hamilton cycles are named by their vertices just like all circuits. An example is given in [link].

Factorials

In many areas of mathematics, we must calculate products like $7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ or $11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$, products that involve multiplying all the counting numbers from a particular number down to 1. Imagine that the product happened to be all the numbers from 100 down to 1. That’s a lot of writing! Instead of writing all of that out, mathematicians came up with a shorthand notation. For example, instead of $7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$, we write $7!$, which is read “7 factorial.” In other words, the product of all the counting numbers from $n$ down to 1 is called $n$ factorial and it is written $n!$

Counting Hamilton Cycles in Complete Graphs

Now, let’s get back to answering the question of how many Hamilton cycles are in a complete graph. In [link], we have drawn all the four cycles in a complete graph with four vertices. Remember, cycles can be named starting with any vertex in the cycle, but we will name them starting with vertex $a$.

Weighted Graphs

Suppose that an officer in the U.S. Air Force who is stationed at Vandenberg Air Force base must drive to visit three other California Air Force bases before returning to Vandenberg. The officer needs to visit each base once. The vertices in the graph in [link] represent the four U.S. Air Force bases, Vandenberg, Edwards, Los Angeles, and Beale. The edges are labeled to with the driving distance between each pair of cities.

Hamilton Paths

Hamilton Paths

Just as circuits that visit each vertex in a graph exactly once are called Hamilton cycles (or Hamilton circuits), paths that visit each vertex on a graph exactly once are called Hamilton paths. As we explore Hamilton paths, you might find it helpful to refresh your memory about the relationships between walks, trails, and paths by looking at [link]. We know that paths are walks that don’t repeat any vertices or edges. So, a Hamilton path visits every vertex without repeating any vertices or edges. [link] shows a path from vertex *A* to vertex *E* and a Hamilton path from vertex *A* to vertex *E*.

Existence of a Hamilton Path

It turns out that there is no Hamilton path between vertices *A* and *E* in Graph *G* in [link]. To understand why, let’s imagine there is a red apple tree on one side of a bridge and a green apple tree on the other side of the bridge. Now suppose someone asked you to pick up all the fallen apples under each tree without crossing the bridge more than once, and making sure that the first apple you pick up and the last apple you pick up are both red. You would say, that is impossible! To have the first and last apple be red would either require leaving the green apples on the ground or crossing the bridge twice.

Hamilton Path or Euler Trail?

We learned in Euler Trails that an Euler trail visits each edge exactly once, whereas a Hamilton path visits each vertex exactly once. Let’s practice distinguishing between the two.

Traveling Salesperson Problem

Brute Force and Greedy Algorithms

An algorithm is a sequence of steps that can be used to solve a particular problem. We have solved many problems in this chapter, and the procedures that we used were different types of algorithms. In this section, we will use two common types of algorithms, a brute force algorithm and a greedy algorithm. A brute force algorithm begins by listing every possible solution and applying each one until the best solution is found. A greedy algorithm approaches a problem in stages, making the apparent best choice at each stage, then linking the choices together into an overall solution which may or may not be the best solution.

The Traveling Salesperson Problem

Now let’s focus our attention on the graph theory application known as the traveling salesperson problem (TSP) in which we must find the shortest route to visit a number of locations and return to the starting point.

Finding Weights of All Hamilton Cycles in Complete Graphs

Notice that we listed all of the Hamilton cycles and found their weights when we solved the TSP about the officer from Vandenberg. This is a skill you will need to practice. To make sure you don't miss any, you can calculate the number of possible Hamilton cycles in a complete graph. It is also helpful to know that half of the directed cycles in a complete graph are the same cycle in reverse direction, so, you only have to calculate half the number of possible weights, and the rest are duplicates.

The Brute Force Method

The method we have been using to find a Hamilton cycle of least weight in a complete graph is a brute force algorithm, so it is called the brute force method. The steps in the brute force method are:

The Nearest Neighbor Method

When the brute force method is impractical for solving a traveling salesperson problem, an alternative is a greedy algorithm known as the nearest neighbor method, which always visit the closest or least costly place first. This method finds a Hamilton cycle of relatively low weight in a complete graph in which, at each phase, the next vertex is chosen by comparing the edges between the current vertex and the remaining vertices to find the lowest weight. Since the nearest neighbor method is a greedy algorithm, it usually doesn’t give the best solution, but it usually gives a solution that is "good enough." Most importantly, the number of steps will be the number of vertices. That’s right! A problem with 10 vertices requires 10 steps, not 362,880. Let’s look at an example to see how it works.

Trees

What Is A Tree?

Whether we are talking about a family tree or a tree in a forest, none of the branches ever loops back around and rejoins the trunk. This means that a tree has no cyclic subgraphs, or is acyclic. A tree also has only one component. So, a tree is a connected acyclic graph. Here are some graphs that have the same characteristic. Each of the graphs in [link] is a tree.

Types of Trees

Mathematicians have had a lot of fun naming graphs that are trees or that contain trees. For example, the graph in [link] is not a tree, but it contains two components, one containing vertices *a* through *d*, and the other containing vertices *e* through *g*, each of which would be a tree on its own. This type of structure is called a forest. There are also interesting names for trees with certain characteristics.

Characteristics of Trees

As we study trees, it is helpful to be familiar with some of their characteristics. For example, if you add an edge to a tree graph between any two existing vertices, you will create a cycle, and the resulting graph is no longer a tree. Some examples are shown in [link]. Adding edge *bj* to Graph *T* creates cycle (*b*, *c*, *i*, *j*). Adding edge *rt* to Graph *P* creates cycle (*r*, *s*, *t*). Adding edge *tv* to Graph *S* creates cycle (*t*, *u*, *v*).

Spanning Trees

Suppose that you planned to set up your own computer network with four devices. One option is to use a “mesh topology” like the one in [link], in which each device is connected directly to every other device in the network.

Constructing a Spanning Tree Using Paths

Suppose that you wanted to find a spanning tree within a graph. One approach is to find paths within the graph. You can start at any vertex, go any direction, and create a path through the graph stopping only when you can’t continue without backtracking as shown in [link].

Revealing Spanning Trees

Another approach to finding a spanning tree in a connected graph involves removing unwanted edges to reveal a spanning tree. Consider Graph *D* in [link].

Kruskal’s Algorithm

In many applications of spanning trees, the graphs are weighted and we want to find the spanning tree of least possible weight. For example, the graph might represent a computer network, and the weights might represent the cost involved in connecting two devices. So, finding a spanning tree with the lowest possible total weight, or minimum spanning tree, means saving money! The method that we will use to find a minimum spanning tree of a weighted graph is called Kruskal’s algorithm. The steps for Kruskal’s algorithm are:

Math and...

Introduction

Math and Art

Golden Ratio

The golden ratio, also known as the golden proportion, is a ratio aspect that can be found in beauty from nature to human anatomy as well as in golden rectangles that are commonly found in building structures. The golden ratio is expressed in nature from plants to creatures such as the starfish, honeybees, seashells, and more. It is commonly noted by the Greek letter *ϕ* (pronounced “fee”). $\varphi =\frac{1+\sqrt{5}}{2}$, which has a decimal value approximately equal to 1.618.

Fibonacci Sequence and Application to Nature

The Fibonacci sequence can be found occurring naturally in a wide array of elements in our environment from the number of petals on a rose flower to the spirals on a pine cone to the spines on a head of lettuce and more. The Fibonacci sequence can be found in artistic renderings of nature to develop aesthetically pleasing and realistic artistic creations such as in sculptures, paintings, landscape, building design, and more. It is the sequence of numbers beginning with 1, 1, and each subsequent term is the sum of the previous two terms in the sequence (1, 1, 2, 3, 5, 8, 13, …).

Golden Ratio and the Fibonacci Sequence Relationship

Mathematicians for years have explored patterns and applications to the world around us and continue to do so today. One such pattern can be found in ratios of two adjacent terms of the Fibonacci sequence.

Golden Rectangles

Turning our attention to man-made elements, the golden ratio can be found in architecture and artwork dating back to the ancient pyramids in Egypt ([link]) to modern-day buildings such as the UN headquarters. The ancient Greeks used golden rectangles—any rectangles where the ratio of the length to the width is the golden ratio—to create aesthetically pleasing as well as solid structures, with examples of the golden rectangle often being used multiple times in the same building such as the Parthenon, which is shown in [link]. Golden rectangles can be found in twentieth-century buildings as well, such as the Washington Monument.

Math and the Environment

Making a Positive Impact on Climate Change—Water Usage

Our use of water is one element that impacts climate change. Having access to clean, potable water is critical for not only our health but also for the health of our ecosystem. About 1 out of 10 people on our planet do not have easy access to clean water to drink. As each of us conserves water, we prolong the life span of fresh water from our lakes and rivers and also reduce the impact on sewer systems and drainage in our communities. Additionally, as we conserve water, we also conserve electricity that is used to bring water to and in our homes. So, what can we do to help conserve water?

History of Solar Energy

In the mid-1800s, Willoughby Smith discovered photoconductive responsiveness in selenium. Shortly thereafter, William Grylls Adams and Richard Evans Day discovery that selenium can produce electricity if exposed to the sun was a major breakthrough. Less than 10 years later, Charles Fritts invented the first solar cells using selenium. Jumping a mere 100 years later, Bell Labs in the United States produced the first practical photovoltaic cells in the mid-1950s and developed versions used to power satellites in the same decade.

Compute Power Needs for Common Home Devices

A kilowatt (kW) is 1,000 watts (W). A kilowatt-hour (kWh) is a measurement of energy use, which is the amount of energy used by a 1,000-watt device to run for an hour. Using the definition of a kilowatt-hour, to calculate how long it would take to consume 1 kWh of power, we divide 1,000 by the watts use of a device.

Solar Advantages

There are multiple advantages that solar power can offer us today including reducing greenhouse gas and CO2 emissions, powering vehicles, reducing water pollution, reducing strain on limited supply of other power options such as fossil fuels. We will look further at reducing greenhouse gas and CO2 emissions.

Math and Medicine

Concentrations and Dosages of Drugs

Consider any drug and the recommended dosage varies based on several factors such as age, weight, and degree of illness of a person. Hospitals and medical dispensaries do not stock every possible needed concentration of medicines. Drugs that are delivered in liquid form for intravenous (IV) methods in particular can be easily adjusted to meet the needs of a patient. Whether administering anesthesia prior to an operation or administering a vaccine, calculation of the concentration of a drug is needed to ensure the desired amount of medicine is delivered.

Validating Effectiveness of a New Vaccine

The process to develop a new vaccine and be able to offer it to the public typically takes 10 to 15 years. In the United States, the system typically involves both public and private participation in a process. During the 1900s, several vaccines were successfully developed, including the following: polio vaccine in the 1950s and chickenpox vaccine in the 1990s. Both of these vaccines took years to be developed, tested, and available to the public. Knowing the typical timeline for a vaccine to move from development to administration, it is not surprising that some people wondered how a vaccine for Covid-19 was released in less than a year’s time.

Mathematical Modeling to Track the Spread of a Vaccine

With a large number of people receiving a Covid-19 vaccine, the concern at this time is how to create an affordable vaccine to reach people all over the world. If a world solution is not found, those without access to a vaccine will serve as incubators to variants that might be resistant to the existing vaccines.

Math and Music

Basics of Frequency as It Relates to Sound

Every sound is created by an object vibrating and these vibrations travel in waves that are captured by our ears. Some vibrations we may be able to see, such as a plucked guitar string moving, whereas other vibrations we may not be able to see, such as the sound created when we hold our breath when accidentally dropping our cell phone on a hard floor. We don’t see the vibrations of our cell phone hitting the floor; however, any audible sound created in the fall is the result of vibrations in the form of sound waves, which can be pictured similarly to waves moving through the ocean.

Basics of Pitch

When considering the various sound levels the human ear can hear, the ear perceives sound both from the frequency level and the pitch of a sound. The quality of the sound is referred to as pitch, the tonal quality of a sound and how high or low the tone. Sounds with a high frequency have a high pitch, such as 900 Hz, and sounds with a low pitch have a low frequency, such as 50 Hz.

Note Values, Half-Steps, Whole Steps, and Octaves

*“There are not more than five musical notes, yet the combinations of these five give rise to more melodies than can ever be heard.”*

Frequencies of Octaves

Notes that are one octave apart have the same name and are related in frequency values. Given the frequency of any note, the frequency of same note one octave higher is doubled and this pattern continues as you move up and down the notes on a keyboard or any other musical instrument. Song writers and singers use this knowledge to change the pitch of a note up or down to align with a person’s vocal range. Regardless of which C is played or sung, the pitch is the same and the frequency is related by a power or two.

Math and Sports

Data Analytics (Statistics) Is Crucial to Advance a Team’s Success

Analyzing the vast data that today’s world has amassed to find patterns and to make predictions for future results has created a degree field for data analytics at many colleges, which is in high demand in places that might surprise you. One such place is in sports, where being able to analyze the available data on your team’s players, potential recruits, opposing team strategies, and opposing players can be paramount to your team’s success.

Single Round-Robin Tournaments

A common tournament style is single round-robin tournaments ([link]), where each team or opponent plays every other team or opponent, and the champion is determined by the team that wins the most games. Ties are possible and are resolved based on league rules.

Single-Elimination Tournaments

When desiring a more efficient tournament style to determine a champion, one option is single-elimination tournaments ([link]), where teams are paired up and the winner advances to the next round of play. The losing team is defeated from tournament play and does not advance in the tournament, although some leagues offer consolation matches.

Co-Req Appendix: Integer Powers of 10

Nonnegative Integer Powers of 10

The phrase **nonnegative integers** refers to the set containing 0, 1, 2, 3, … and so on. In the expression ${10}^{5}$, 10 is called the **base**, and 5 is called the **exponent**, or **power**. The exponent 5 is telling us to multiply the base 10 by itself 5 times. So, ${10}^{5}=10×10×10×10×10=100,000$. By definition, any number raised to the 0 power is 1. So, ${10}^{0}=1$.

Negative Integer Powers of 10

The **reciprocal** of a number is 1 divided by that number. For example, the reciprocal of 10 is $\frac{1}{10}$. We use negative exponents to indicate a reciprocal. For example, ${10}^{-1}=\frac{1}{{10}^{1}}=\frac{1}{10}$. Similarly, any expression with a negative exponent can be written with a positive exponent by taking the reciprocal. Several negative powers of 10 have been simplified in the table that follows. What do you notice about the number of zeros in the denominator (bottom) of each fraction?

Multiplying Integers by Positive Powers of 10

Did you know that the distance from the sun to Earth is over 90 million miles? This value can be represented as 90,000,000, or we can write it as a product: $9×10,000,000=9×{10}^{7}$, which is actually a more compact way of writing 90 million. Notice that the power of 7 reflects the number of zeros in 90 million. Several products of positive integers and powers of 10 are given in the table that follows. Notice that the number of zeros is the same as the exponent except in one case.

Multiplying by Negative Powers of 10

As we have seen, negative powers of 10 are decimals. Several products of positive integers and powers of 10 are given in the table below. Notice that multiplying an integer by 10 raised to a negative integer power results in a smaller number than you started with. Also, the number of decimal places to the right of the decimal point is the same as the exponent except in one case.

Moving the Decimal Place

A helpful shortcut when multiplying a number by a power of 10 is to “move the decimal point.” The following table shows several powers of 10, both positive and negative. Compare the location of the decimal point in the original number to the location of the decimal point in the product. How has it changed?