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Binary Arithmetic

- CPSC 101 Chp 2
- John Lamertina

Early Need to Count

One-to-One Sets

27 pebbles 27 sheep

Counting in 10s

- Up to TEN
- More than TEN?

Countable Sets of 10

10

10

7

Why Decimal Numbers?

- 10 fingers
- 10 digits (0,1,2,3,,9)
- Countable Sets of 10
- Ones
- Tens
- Hundreds
- Thousands
- etc

Simple Decimal Number

- 27
- 20 7
- 2 tens 7 ones
- (210) (71)
- (2101) (7100)

Anything to the zero power is 1

Powers of 10

Another Decimal Number

9327 9000 300 20 7 9 thousands 3

hundreds 2 tens 7 ones (91000) (3100)

(210) (71) (9103) (3102) (2101)

(7100)

Powers of 10

A Decimal Number with Zeros

7009 7000 0 0 9 7 thousands 0

hundreds 0 tens 9 ones (71000) (0100)

(010) (91) (7103) (0102) (0101)

(9100)

Powers of 10

Decimal vs Binary

- Decimal numbers are powers of 10
- Deci 10
- Binary numbers are powers of 2
- Bi 2

Why Binary Numbers?

- Electronic circuits have two possible states or

values Off and On - Off and On Zero and One (0, 1)
- Two digits (0, 1) binary number system
- Thus computers operate on the binary number system

Simple Binary Number

1012 1002 02 12 (122) (021) (120)

Read 101, base 2 not 101 squared.

Powers of 2

Simple Binary Number

1012 1002 02 12 (122) (021)

(120) 410 010 110 510

1012 510

Simple Decimal Number vs Simple Binary Number

15 10 5 1 ten 5 ones (110) (51)

(1101) (5100)

11112 10002 1002 102 12 (123) (122)

(121) (120) 810 410 210 110 1510

Powers of 10

Powers of 2

Convert Binary to Decimal

101101012 (127) (026) (125) (124)

(023) (122) (021) (120) 128 0 32

16 0 4 0 1 18110

Digit 1 0 1 1 0 1 0 1

Exponent 7 6 5 4 3 2 1 0

Power of 2 27 26 25 24 23 22 21 20

Decimal 128 64 32 16 8 4 2 1

Convert Decimal to BinarySuccessive Division

by Two

Example Decimal Number 29

Divide by 2 Remainder Explanation

29 29 starting decimal number

14 29/2 14, remainder of 1

7 14/2 7, remainder of 0

3 7/2 3, remainder of 1

3/2 1, remainder of 1

1

0

1

1

1

Binary Result

1

1

1

0

1

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