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# Section 1.1: Integer Operations and the Division Algorithm - PowerPoint PPT Presentation

Section 1.1: Integer Operations and the Division Algorithm. MAT 320 Spring 2008 Dr. Hamblin. Addition. “You have 4 marbles and then you get 7 more. How many marbles do you have now?”. 4. 11. 7. Subtraction. “If you have 9 toys and you give 4 of them away, how many do you have left?”. 5.

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### Section 1.1: Integer Operations and the Division Algorithm

MAT 320 Spring 2008

Dr. Hamblin

• “You have 4 marbles and then you get 7 more. How many marbles do you have now?”

4

11

7

• “If you have 9 toys and you give 4 of them away, how many do you have left?”

5

4

9

• “You have 4 packages of muffins, and each package has 3 muffins. How many total muffins do you have?”

4

12

3

• “You have 12 cookies, and you want to distribute them equally to your 4 friends. How many cookies does each friend get?”

3

12

• As you can see, division is the most complex of the four operations

• Just as multiplication is repeated addition, division can be thought of as repeated subtraction

• 28 – 4 = 24

• 24 – 4 = 20

• 20 – 4 = 16

• 16 – 4 = 12

• 12 – 4 = 8

• 8 – 4 = 4

• 4 – 4 = 0

• Once we reach 0, we stop. We subtracted seven 4’s, so 28 divided by 4 is 7.

• 92 – 12 = 80

• 80 – 12 = 68

• 68 – 12 = 56

• 56 – 12 = 44

• 44 – 12 = 32

• 32 – 12 = 20

• 20 – 12 = 8

• We don’t have enough to subtract another 12, so we stop and say that 92 divided by 12 is 7, remainder 8.

• Since 28 divided by 4 “comes out evenly,” we say that 28 is divisible by 4, and we write 28 = 4 · 7.

• However, 92 divided by 12 did not “come out evenly,” since 92  12 · 7. In fact, 12 · 7 is exactly 8 less than 92, so we can say that 92 = 12 · 7 + 8.

remainder

dividend

quotient

divisor

• Subtracting 13 one at a time would take a while

• 3409 – 100 · 13 = 2109

• 2109 – 100 · 13 = 809

• 809 – 50 · 13 = 159

• 159 – 10 · 13 = 29

• 29 – 13 = 19

• 19 – 13 = 3

• So 3409 divided by 13 is 262 remainder 3.

• All in all, we subtracted 262 13’s, so we could write 3409 – 262 · 13 = 3, or 3409 = 13 · 262 + 3.

• Start with dividend a and divisor b (“a divided by b”)

• Repeatedly subtract b from a until the result is less than a (but not less than 0)

• The number of times you need to subtract b is called the quotient q, and the remaining number is called the remainder r

• Once this is done, a = bq + r will be true

• Let a and b be integers with b > 0. Then there exist unique integers q and r, with 0  r < b and a = bq + r.

• This just says what we’ve already talked about, in formal language

• Method 2: Guess and CheckFill in whatever number you want for q, and solve for r. If r is between 0 and b, you’re done. If r is too big, increase q. If r is negative, decrease q.

• Method 3: CalculatorType in a/b on your calculator. The number before the decimal point is q. Solve for r in the equation a = bq + r

Negative Numbers Theorem)

• Notice that in the Division Algorithm, b must be positive, but a can be negative

• How do we handle that?

-30 divided by 8 Theorem)

• “You owe me 30 dollars. How many 8 dollar payments do you need to make to pay off this debt?”

• Instead of subtracting 8 from -30 (which would just increase our debt), we add 8 repeatedly

-30 divided by 8, continued Theorem)

• -30 + 8 = -22

• -22 + 8 = -14

• -14 + 8 = -6 (debt not paid off yet!)

• -6 + 8 = 2

• So we made 4 payments and had 2 dollars left over

• -30 divided by 8 is -4, remainder 2

• Check: -30 = 8 · (-4) + 2

Caution! Theorem)

• Negative numbers are tricky, be sure to always check your answer

• Be careful when using the calculator method

• Example: -41 divided by 7The calculator gives -5.857…, but if we plug in q = -5, we get r = -6, which is not a valid remainder

• The correct answer is q = -6, r = 1
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