Digital Logic & Design
Adil Waheed
Lecture 03
Range of Binary Numbers
• Processors can handle 64-bit unsigned binary
values.
• Maximum unsigned decimal number is 18.446 x
1018
• How to represent larger numbers?
• How to represent very small numbers?
• How to represent numbers with integer part and
fraction part?
Hexadecimal Number System
• Base 16
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Representing Binary in compact form
• 11011000001102 = 1B06 H
Counting in Hexadecimal
Decimal Binary
Hexadecimal
Decimal Binary
Hexadecimal
0
0000
0
8
1000
8
1
0001
1
9
1001
9
2
0010
2
10
1010
A
3
0011
3
11
1011
B
4
0100
4
12
1100
C
5
0101
5
13
1101
D
6
0110
6
14
1110
E
7
0111
7
15
1111
F
Binary-Hexadecimal Conversion
• Binary to Hexadecimal Conversion
• 11010110101110010110
• 1101 0110 1011 1001 0110
•D
6
B
9
6
• Hexadecimal to Binary Conversion
• FD13
• 1111 1101 0001 0011
Decimal-Hexadecimal Conversion
• Decimal to Hexadecimal Conversion
• Indirect Method
• Decimal →Binary → Hexadecimal
• Repeated Division by 16
Decimal-Hexadecimal Conversion
• Hexadecimal to Decimal Conversion
• Indirect Method
• Hexadecimal →Binary → Decimal
• Sum-of-Weights
Hexadecimal Addition & Subtraction
• Hexadecimal Addition
• Carry generated
• Hexadecimal Subtraction
• Borrow weight 16
Repeated Division by 16
Number
Quotient
Remainder
2096
131
0
131
8
3
8
0
8
Sum-of-Weights
CA02
(C x 163) + (A x 162) + (0 x 161) + (2 x 160)
(12 x 163) + (10 x 162) + (0 x 161) + (2 x 160)
(12 x 4096) + (10 x 256) + (0 x 16) + (2 x 1)
49152 + 2560 + 0 + 2
51714
Hexadecimal Addition
Carry
1
+
2AC6
92B5
BD7B
6+5=11d Bh
C+B=23d 17h
A+2+1=13d Dh
2+9=11d Bh
Hexadecimal Subtraction
Borrow
-
111
92B5
2AC6
67EF
21-6=15d Fh
26-C=14d Eh
17-A=7d 7h
8-2=6d 6h
Octal Number System
• Base 8
• 0, 1, 2, 3, 4, 5, 6, 7
• Representing Binary in compact form
• 11011000001102 = 154068
Counting in Octal
Decimal
0
1
2
3
4
5
6
7
Binary
000
001
010
011
100
101
110
111
Octal
0
1
2
3
4
5
6
7
Counting in Octal
Decimal Octal Decimal Octal
8
10
16
20
9
11
17
21
Decimal Octal
24
30
25
31
10
11
12
13
14
15
26
27
28
29
30
31
12
13
14
15
16
17
18
19
20
21
22
23
22
23
24
25
26
27
32
33
34
35
36
37
Binary-Octal Conversion
• Binary to Octal Conversion
• 11010110101110010110
• 011 010 110 101 110 010 110
•3 2
6
5 6 2
6
• Octal to Binary Conversion
• 1726
• 001 111 010 110
Decimal-Octal Conversion
• Decimal to Octal Conversion
• Indirect Method
• Decimal →Binary → Octal
• Repeated Division by 8
Decimal-Octal Conversion
• Octal to Decimal Conversion
• Indirect Method
• Octal →Binary → Decimal
• Sum-of-Weights
Octal Addition & Subtraction
• Octal Addition
• Carry generated
• Octal Subtraction
• Borrow weight 8
Repeated Division by 8
Number
Quotient
Remainder
2075
259
3
(O0)
259
32
3
(O1)
8
4
0
(O2)
4
0
4
(O3)
Sum-of-Weights
4033
(4 x 83) + (0 x 82) + (3 x 81) + (3 x 80)
(4 x 512) + (0 x 64) + (3 x 8) + (3 x 1)
2048 + 0 + 24 + 3
2075
Octal Addition
Carry 1
7602
+ 5771
15573
2+1=3d 3O
0+7=7d 7O
6+7=13d 15O
1+7+5=13d 15O
Octal Subtraction
Borrow
-
11
7602
5771
1611
2-1=1d 1O
8-7=1d 1O
13-7=6d 6O
6-5=1d 1O
Alternate Representations
• BCD Code
• BCD Addition
• Gray Code
Alternate Representations
• BCD (Binary Coded Decimal) Code
Decimal
0
1
2
3
4
BCD
0000
0001
0010
0011
0100
Decimal
5
6
7
8
9
BCD
0101
0110
0111
1000
1001
BCD Addition
• Multi-digit BCD numbers can be added together
23
0010 0011
45
0100 0101
68
0110 1000
23
0010 0011
48
0100 1000
71
0110 1011
• 1011 is illegal BCD number
BCD Addition
• Add a 0110 (6) to an invalid BCD number
• Carry added to the most significant BCD digit
23
48
71
0010 0011
0100 1000
0110 1011
0110
0111 0001
Alphanumeric Code
• Numbers, Characters, Symbols
• ASCII 7-bit Code
• American Standard Code for Information
Interchange
• 10 Numbers (0-9)
• 26 Lower Case Characters (a-z)
• 26 Upper Case Characters (A-Z)
• Punctuation and Symbols
ASCII Code
•
•
•
•
•
•
•
•
Numbers 0 to 9
ASCII 0110000 (30h) to 0111001 (39h)
Alphabets a to z
ASCII 1100001 (61h) to 1111010 (7Ah)
Alphabets A to Z
ASCII 1000001 (41h) to 1011010 (5Ah)
Control Characters
ASCII 0000000 (0h) to 0011111 (1Fh)
Error Detection
• Digital Systems are very Reliable
• Errors during storage or transmission
• Parity Bit
• Even Parity
• Odd Parity
parity
• Odd parity:
• The number of 1-bit must add up to an odd number
• Even parity:
• The number of 1-bit must add up to an even number
Summary
• Hexadecimal Number System
• Binary-Hexadecimal Conversion
• Decimal-Hexadecimal Conversion
• Octal Number System
• Binary-Octal Conversion
• Decimal-Octal Conversion
Summary
• Alternate Representations
• BCD Code
• Gray Code
• Alphanumeric Codes
• ASCII
• Error Detection
• Parity Bit