Lecture Slides on Computer System Design unit 1

= .1001101

Convert Decimal 106.6015625 to Binary
106.6015625 = 1101010.1001101

2|985 . 2|492 - 1 2|246 - 0 2|123 - 0 2| 61 - 1 2| 30 - 1 2| 15 - 0 2| 7 - 1 2| 3 - 1 2| 1 - 1 2| 0 - 1   = 1111011001

0.78357 x 2 1.56714 x 2 - 1 1.13428 x 2 - 1 0.26856 x 2 - 0 0.53712 x 2 - 0 1.07424 x 2 - 1 0.14848 x 2 - 0 0.29696 x 2 - 0 0.59392 x 2 - 0 1.18784 x 2 - 1 0.37568 x 2 - 0 0.75136 x 2 - 0 1.50272 x 2 - 1 1.00544 x 2 - 1 0.01088 x 2 - 0 0.02176 x 2 - 0 0.04352 x 2 - 0 = .1100100010011000

Convert Decimal 985.78357 to Binary
985.78357= 1111011001.1100100010011000

Binary Representation

• Base 2 Number Representation
• Represent 1521310 as 111011011011012
• Represent 1.2010 as 1.0011001100110011[0011]…2
• Represent 1.5213 X 104 as 1.11011011011012 X 213

Determine x in 104 = 2x
104 = 2x
log2104 = log2 2x
4 log210 = x log2 2 where log2 2 = 1
4 * 3.3219 = x where log210 = 3.32192809489
x = 13
Or 104 = 213
Determine x in 10x = 213
10x = 213
log1010x = log10 213
x log1010 = 13 log10 2 where log1010 = 1
x = 13*0.301 where log10 2 = 0.301029995664
x = 4
Or 104 = 213

Hexadecimal Number System

• Hexadecimal Numbers :

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

• Each hexadecimal digit corresponds to four binary digits

Hex Number Binary coded hex

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

Example : Binary ( 10110101)2 is Hex (B5)16

Hexadecimal Number System

• Convert decimal 215 to Hex
• (215)10 = (D7)16
• Convert Hex D7 to Binary
• (D7)16 = (11010111)2
• Convert Hex D7 to Decimal
• (D7)16 = D x 161 + 7 x 160

= 13 x 16 + 7 x 1 = (215)10

16|215 .
16| 13 - 7
16| 0 - D

Encoding Byte Values

• Byte = 8 bits
• Binary 000000002 to 111111112
• Decimal: 010 to 25510
• Hexadecimal 0016 to FF16
• Base 16 number representation
• Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’
• Write FA1D37B16 in C as
• 0xFA1D37B
• 0xfa1d37b

a = 0xfa1d37b; b=100; decimal

b=0x100; hexadecimal = 256 decimal

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
A
10
1010
B
11
1011
C
12
1100
D
13
1101
E
14
1110
F
15
1111
Hex
Decimal
Binary

Octal Number System

• Octal Numbers : 0,1, 2, 3, 4, 5, 6, 7
• Each octal digit corresponds to three binary digits

Octal Number Binary coded Octal

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

Example : Binary (100111101)2 is octal (475)8

Octal Number System

• Convert decimal 215 to Octal
• (215)10 = (327)8
• Convert octal 327 to Binary
• (327)8 = (011010111)2
• Convert octal 327 to Decimal
• (327)8 = 3 x 82 + 2 x 81 + 7 x 80

= 192 + 16 + 7 = (215)10

8|215 .
8| 26 - 7
8| 3 - 2
8| 0 - 3
Convert 0x879CEF to (i) Binary (ii) Decimal & (iii) Octal
(i) HEX2BIN(0x879CEF) = (1000011110011100 11101111)2
 

• HEX2DEC(0x879CEF)

= 8 x 165 + 7 x 164 + 9 x 163 + C x 162 + E x 161 + F x 160
= 8 x 1048576 + 7 x 65536+ 9 x 4096 + 12 x 256 + 14 x 16 + 15 x 1
= 8388608 + 458752 + 36864 + 3072 + 224 + 15
= 8887535
(iii) HEX2OCT(0x879CEF) = 100001111001110011101111 = (41716357)8

Convert 0x4BAD6CAB to (i) Binary (ii) Decimal & (iii) Octal
(i) HEX2BIN(0x4BAD6CAB) = 01001011101011010110110010101011
 

• HEX2DEC(0x4BAD6CAB)

= 4 x 167 + B x 166 + A x 165 + D x 164 + 6 x 163 + C x 162 + A x 161 + B x 160
= 4 x 268435456 + 11 x 16777216 + 10 x 1048576 + 13 x 65536 + 6 x 4096 + 12 x 256 +
10 x 16 + 11 x 1
= [phone] + 184549376 + 10485760 + 851968 + 24576 +3072 + 160 +11
= 1269656747
 
(iii) HEX2OCT(0x4BAD6CAB)
= 01001011101011010110110010101011= (11353266253)8

Convert Octal 36745412 to (i) Binary (ii) Decimal & (iii) Hexadecimal
(i)OCT2BIN(36745412) = 011110111100101100001010
(ii)OCT2DEC(36745412)
= 3 x 87 + 6 x 86 + 7 x 85 + 4 x 84 + 5 x 83 + 4 x 82 + 1 x 81 + 2 x 80
= 3 x 2097152+6 x 262144+ 7 x 32768+4 x 4096+5 x 512 + 4 x 64 + 1x8 + 2
= 6291456 + 1572864 + 229376 + 16384 + 2560 + 256 + 10
= 8112906
(iii) OCT2HEX(36745412) = 0111 1011 1100 1011 0000 1010= 0x7BCB0A
Convert Octal 753642532452 to (i) Binary (ii) Decimal & (iii) Hexadecimal

• OCT2BIN(753642532452) = 111101011110100010101011010100101010



• OCT2DEC(753642532452)

= 7 x 811 + 5 x 810 + 3 x 89 + 6 x 88 + 4 x 87 + 2 x 86 + 5 x 85 + 3 x 84 + 2 x 83 + 4 x 82 +
5 x 81 + 2 x 80
= 7*8589934592 + + 5*1073741824 + 3* 134217728 + 6 x 16777216 + 4 x 2097152 +
2 x 262144 + 5 x 32768 + 3 x 4096 + 2 x 512 + 4 x 64 + 5 x 8+ 2
= 60129542144 + 5368709120 + 402653184 + 100663296 + 8388608 + 524288+ 163840
+ 12288 + 1024 + 256 + 40 +2
= 66010658090


• OCT2HEX(753642532452)

= 111101011110100010101011010100101010 = 0xF5E8AB52A

Data Representations

C Data Type	Typical 32-bit	Typical 64-bit	x86-64