digital circuits and digital systems mooc · 2017-08-04 · hexadecimal base hex (hexadecimal) hex...
TRANSCRIPT
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
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 05 R 12 R 11 R 00 R 1
0.8125
1.62501.250.51.0
(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
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 1111Number 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. octal binary
0 0 000
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