Digital Circuits and Systems Number Systems Shankar Balachandran* - - PowerPoint PPT Presentation

Shankar Balachandran* Associate Professor, CSE Department Indian Institute of Technology Madras

*Currently a Visiting Professor at IIT Bombay

Digital Circuits and Systems

Spring 2015 Week 2 Module 12

Number Systems

 Digits = powers of 10

… 100, 10, 1,

1/10, 1/100, 1/1000 …

… 102, 101, 100, 10-1, 10-2, 10-3 … Ex: (36.25)10 = 3*10 + 6*1 + 2*1/10 + 5* 1/100

 Bits = powers of 2

… 8, 4, 2, 1,

1/2, 1/4, 1/8 …

… 23, 22, 21, 20, 2-1, 2-2, 2-3 … Ex: (100100.01)2 = 1*32 + 1*4 + 1*1/4

Digits vs. bits

Number Systems 2

Binary to decimal

 add powers that have a 1

(101001)2 = 32 + 8 + 1 (0.1001)2 = 1/2 + 1/16 (10110.1011)2 = ? 16 + 4 + 2 + 1/2 + 1/8 + 1/16

= 22.6875

Number Systems 3

Decimal to binary

 Left of decimal point

Repeatedly divide integer part by 2 until you

get 0

Read remainders bottom to up

22 = (?)2 22 11 R 0 5 R 1 2 R 1 1 R 0 R 1 (10110)2

Number Systems 4

Decimal to binary

 Right of decimal point

Repeatedly multiply fractional part by 2 until

you get 1

Read integer portion top to bottom

0.8125 = (?)2 0.8125 1.6250 1.25 0.5 1.0 (0.1101)2

Number Systems 5

Decimal to binary

 What if there are both left and right of the decimal

point?

 Do them separately and combine

22.8125 = (?)2 22 11 R 0 5 R 1 2 R 1 1 R 0 R 1 0.8125 1.6250 1.25 0.5 1.0 (10110.1101)2

up down

Number Systems 6

Exercises



3.25 = (?)2

 0.0101 = (?)10  0.3 = (?)2  114.25 = (?)2  1010001.01101 = (?)10  0101.0101 = (?)10  1010.0101 = (?)10

Number Systems 7

Hexadecimal base

 Hex digits = powers of 16

… 256, 16, 1,

1/16, 1/256 …

… 162, 161, 160, 16-1, 16-2 …

use digits 0-9, A-F

 A=10, B=11, C=12, D=13, E=14, F=15

 Often preceded by 0x  Also with a base explicitly written (24.4)16

Ex: (24.4)16 = 2*16 + 4*1 + 4*1/16

Number Systems 8

Hexadecimal base

 Hex (hexadecimal)

 Hex digit is a group of 4 bits  Memorize this table!!

dec. hex binary 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

Number Systems 9

 Hex (hexadecimal)

Group from decimal point outward Pad with zeros to get groups of 4

(1101101001010.101001)2 (0001 1011 0100 1010 . 1010 0100)2 1 B 4 A . A 4 (1101101001010.101001)2 = (1B4A.A4)16

Binary to Hex

Number Systems 10

 Octal digits = powers of 8

… 64, 8, 1,

1/8, 1/64 …

… 82, 81, 80, 8-1, 8-2 …

use digits 0-7 sometimes preceded by 0 Also with a base explicitly written(24.4)8

Ex: (44.2)8 = 4*8 + 4*1 + 2*1/8

Octal base

Number Systems 11

 Octal

 Octal digits are groups of 3 bits  Pad with zeros

dec.

• ctal binary

000 1 1 001 2 2 010 3 3 011 4 4 100 5 5 101 6 6 110 7 7 111

Octal base

Number Systems 12

 Octal

Group from decimal point outward Pad with zeros to get groups of 3

(1101101001010.101001)2 (001 101 101 001 010 . 101 001)2 1 5 5 1 2 . 5 1 (1101101001010.101001)2 = (15512.51) 8

Binary to Octal

Number Systems 13

 What about other conversions such as:

 Octal  Hex  Decimal  Hex  …

 Use other conversions you already know

 Octal  Binary  Hex  Decimal  Binary  Hex

Other conversions

Number Systems 14

End of Week 2: Module 12

Thank You

Number Systems 15