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# Number Systems

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## Common Number Systems

Used by humans? Yes No No No Used in computers? No Yes No No

## System Decimal Binary Octal Hexadecimal

Base Symbols 10 2 8 16 0, 1, 9 0, 1 0, 1, 7 0, 1, 9, A, B, F

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Quantities/Counting (1 of 3)
Decimal
0 1

Binary
0 1

Octal
0 1

## Decimal Binary Octal

0 1

2
3 4 5

10
11 100 101

2
3 4 5

2
3 4 5

1000

10

9
10 11 12

1001
1010 1011 1100

11
12 13 14

9
A B C

6
7

110
111

6
7

6
7

13
14 15

1101
1110 1111

15
16 17

D
Ep. 33 F

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Quantities/Counting (2 of 3)
Decimal Binary

HexaOctal decimal 10
11 12 13 14

8
9 10 11 12

1000
1001 1010 1011 1100

8
9 A B C

13
14 15

1101
1110 1111

15
16 17

D
E F

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Quantities/Counting (3 of 3)
Decimal Binary

HexaOctal decimal 20
21 22 23 24

16
17 18 19 20

10000
10001 10010 10011 10100

10
11 12 13 14

21
22 23

10101
10110 10111

25
26 27

15
16 17

Etc.

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## Conversion Among Bases

The possibilities:
Decimal Octal

Binary

pp. 40-46

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Quick Example

Base

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Decimal

Octal

Binary

Next slide

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## Weight 12510 => 5 x 100 = 2 x 101 = 20 1 x 102 = 100 125 5

Base

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Binary to Decimal
Decimal Octal

Binary

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Binary to Decimal
Technique
Multiply each bit by 2n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results

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Example
Bit 0 1010112 => 1 x x x x x x 21 22 23 24 25 20 = 1 = 2 = 0 = 8 = 0 = 32 4310

1 0 1 0 1

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Octal to Decimal
Decimal Octal

Binary

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Octal to Decimal
Technique
Multiply each bit by 8n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results

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Example

## 7248 => 4 x 80 = 2 x 81 = 16 7 x 82 = 448 46810

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Decimal Octal

Binary

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Technique
Multiply each bit by 16n, where n is the weight of the bit The weight is the position of the bit, starting from 0 on the right Add the results

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Example

ABC16 =>

## C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810

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Decimal to Binary
Decimal Octal

Binary

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Decimal to Binary
Technique
Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.

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Example
12510 = ?2

2 125 2 62 2 31 2 15 7 2 3 2 1 2 0

1 0 1 1 1 1 1

12510 = 11111012

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Octal to Binary
Decimal Octal

Binary

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Octal to Binary
Technique
Convert each octal digit to a 3-bit equivalent binary representation

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Example
7058 = ?2

## 111 000 101

7058 = 1110001012

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Decimal Octal

Binary

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Technique
Convert each hexadecimal digit to a 4-bit equivalent binary representation

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Example
10AF16 = ?2

## 0001 0000 1010 1111

10AF16 = 00010000101011112

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Decimal to Octal
Decimal Octal

Binary

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Decimal to Octal
Technique
Divide by 8 Keep track of the remainder

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Example
123410 = ?8 8 8 8 8 1234 154 19 2 0

2 2 3 2

123410 = 23228

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Decimal Octal

Binary

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Technique
Divide by 16 Keep track of the remainder

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Example
123410 = ?16 16 16 16 1234 77 4 0

2 13 = D 4

123410 = 4D216

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Binary to Octal
Decimal Octal

Binary

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Binary to Octal
Technique
Group bits in threes, starting on right Convert to octal digits

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Example
10110101112 = ?8

## 1 011 010 111

10110101112 = 13278

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Decimal Octal

Binary

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Technique
Group bits in fours, starting on right Convert to hexadecimal digits

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Example
10101110112 = ?16

10 1011 1011

10101110112 = 2BB16

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Decimal Octal

Binary

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Technique
Use binary as an intermediary

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Example
10768 = ?16 1 0 7 6

001

000

111

110

10768 = 23E16

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Decimal Octal

Binary

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Technique
Use binary as an intermediary

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Example
1F0C16 = ?8 1 F 0 C

0001 1

1111 7 4

0000 1

1100 4

1F0C16 = 174148

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## Exercise Convert ...

Decimal 33 1110101 703 1AF
Dont use a calculator!

Binary

Octal

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Exercise Convert

## Hexadecimal 21 75 1C3 1AF

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Common Powers (1 of 2)
Base 10
Power
10-12 10-9

Symbol p n

## Value .000000000001 .000000001 .000001

10-6
10-3 103 106 109 1012

milli
kilo mega giga tera

m
k M G T

.001
1000 1000000 1000000000 1000000000000

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Common Powers (2 of 2)
Base 2
Power
210 220 230

Symbol k M G

## Value 1024 1048576 1073741824

What is the value of k, M, and G? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
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Example
In the lab 1. Double click on My Computer 2. Right click on C: 3. Click on Properties

/ 230 =

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## Exercise Free Space

Determine the free space on all drives on a machine in the lab
Free space

Drive
A: C: D: E: etc.

Bytes

GB

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ab ac = ab+c

## 26 210 = 216 = 65,536

or
26 210 = 64 210 = 64k MELJUN P. CORTES

Two 1-bit values
A 0 0 1 1 B 0 1 0 1 A+B 0 1 1 10

two
pp. 36-38

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Two n-bit values
Add individual bits Propagate carries E.g., 10101 + 11001 101110
1 1

21 + 25 46

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Multiplication (1 of 3)
Decimal (just for fun)
35 x 105 175 000 35 3675
pp. 39

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Multiplication (2 of 3)
Binary, two 1-bit values
A 0 0 1 1 B 0 1 0 1 AB 0 0 0 1

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Multiplication (3 of 3)
Binary, two n-bit values
As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010 MELJUN P. CORTES

Fractions
Decimal to decimal (just for fun)

## 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14

pp. 46-50

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Fractions
Binary to decimal
10.1011 => 1 0 1 0 1 x x x x x 1 x 2-3 = 2-2 = 2-1 = 20 = 21 = 2-4 = 0.0625 0.125 0.0 0.5 0.0 2.0 2.6875

pp. 46-50

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Fractions
Decimal to binary
3.14579
.14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc.

11.001001...

p. 50

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## Exercise Convert ...

Decimal 29.8 101.1101 Binary Octal

3.07
C.82
Dont use a calculator!

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Exercise Convert

## Decimal 29.8 5.8125

Binary 101.1101

Octal 5.64

1D.CC 5.D

11101.110011 35.63

3.109375
12.5078125

11.000111
1100.10000010

3.07
14.404

3.1C
C.82

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Thank you

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