+ CS 325: CS Hardware and SoftwareOrganization and Architecture

Integers and Arithmetic

+Outline

Number Representation Decimal Binary Hexadecimal

Decimal vs. Hexadecimal vs. Binary

Number Conversions Dec Bin, Dec Hex Bin Dec, Bin Hex Hex Dec, Hex Bin

+Decimal Numbers: Base 10

Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Example: 4923 =

(4x103) + (9x102) + (2x101) + (3x100)

+Number Base:

Number with base x x digits:Base 10 (Decimal): 0, 1, 2, 3, 4 ,5 ,6 ,7 , 8,

9Base 2 (Binary): 0, 1

Number representation:d31d30d29….d2d1d0 is a 32 digit number

4326210 is a 5 digit base 10 (Dec) number

101011010112 is a 11 digit base 2 (Bin) number

+Binary Numbers: Base 2

Digits: 0, 1

Example:101011=

(1x25) + (0x24) + (1x23) + (0x22) + (1x21) + (1x20)

= 4310

What about a base that converts to binary easily?

32 16 8 4 2 1

25 24 23 22 21 20

1 0 1 0 1 1

+Hexadecimal Numbers: Base 16

Digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Decimal digits + A – F

Example:12E = (1x162) + (2x161) + (Ex160) = 30210

A B C D E F

10 11 12 13 14 15

+Decimal vs. Hexadecimal vs. BinaryDEC HEX BIN

0 0 0000

1 1 0001

2 2 0010

3 3 0011

4 4 0100

5 5 0101

6 6 0110

7 7 0111

8 8 1000

9 9 1001

10 A 1010

11 B 1011

12 C 1100

13 D 1101

14 E 1110

15 F 1111

DEC HEX BIN

16 10 0001 0000

17 11 0001 0001

18 12 0001 0010

19 13 0001 0011

20 14 0001 0100

21 15 0001 0101

22 16 0001 0110

23 17 0001 0111

24 18 0001 1000

25 19 0001 1001

26 1A 0001 1010

27 1B 0001 1011

28 1C 0001 1100

29 1D 0001 1101

30 1E 0001 1110

31 1F 0001 1111

+Number Conversion: Dec Bin

Converting from base 10 to base 2: Continue dividing decimal number by 2 and keep the remainder

Example: 3510

1000112

35/2 17 1 LSB

17/2 8 1

8/2 4 0

4/2 2 0

2/2 1 0

1/2 0 1 MSB

+Number Conversion: Dec Bin

Example: Convert 42310 to Bin

1101001112

+Number Conversion: Dec Hex

Converting from base 10 to base 16:

Example: 3510

2316

35/16 2 3 LSB

2/16 0 2 MSB

+Number Conversion: Dec Hex

Example: Convert 21010 to Hex

D216

+Number Conversion: Bin Dec

Converting from base 2 to base 10:

Example: 110102

(1x24) + (1x23) + (0x22) + (1x21) + (0x20)

16 + 8 + 0 + 2 + 0 = 2610 16 8 4 2 1

1 1 0 1 0

MSB LSB

+Number Conversion: Bin Dec

Example: Convert 101011102 to Dec

17410

+Number Conversion: Bin Hex

Converting from base 2 to base 16:

Example: 110101102 1 Hex digit represents 16 Decimal values 4 Binary digits represent 16 Decimal values 1 Hex digit replaces 4 Binary digits

D616

1 1 0 1 0 1 1 0

13 D 6

+Number Conversion: Bin Hex

Example: Convert 110011112 to Hex

CF16

+Number Conversion: Hex Dec

Converting from base 16 to base 10:

Example: 8E316

(8x162) + (Ex161) + (3x160)

2048 + 224 + 3 = 227510

+Number Conversion: Hex Dec

Example: Convert 63F16 to Dec

159910

+Number Conversion: Hex Bin

Converting from base 16 to base 2:

Example: 9A2E16

10011010001011102

9 1001 MSB

A 1010

2 0010

E 1110 LSB

+Number Conversion: Hex Bin

Example: Convert 26FA16 to Bin

100110111110102

+What to do with representations of numbers?

add, subtract, multiply, divide, compare

Example: 8 + 6 = 14

1 0 0 0

+0 1 1 0

1 1 1 0

Simple enough to add in binary that we can build circuits to do it.

+Which base do we use?

Decimal: Great for human, especially when doing arithmetic

Hex: Easier for humans to read than long strings of binary numbers. Easy to convert to binary, each hex decimal = 4 binary bits.

Binary: used by all computers. Bin represents an abstraction…but and abstraction of what?

+The Transistor

A controlled switch.Collector – positive leadEmitter – negative leadBase – control lead

A binary “1” represents an active transistor.

+The Transistor

+Limits of Computer Numbers

Bits can represent anythingCharacters

‘a’, ‘F’7 bit ASCII, 8 bit Extended ASCII

Logical Values0 False, 1 True

Colors?Locations/addresses? Commands?But N bits only 2N things