Transcript Introduction

Number Systems
ELEC 311
Digital Logic and Circuits
Dr. Ron Hayne
Images Courtesy of Cengage Learning
Number Systems
 Decimal (Base 10)
953.7810 = 9x102 + 5x101 + 3x100 + 7x10-1 + 8x10-2
 Binary (Base 2)
1011.112 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2
= 8 + 0 + 2 + 1 + 1/2 + 1/4
= 11.7510
311_01
2
Number Systems
Decimal
Binary
Hex
0
0000
0
1
0001
1
2
0010
2
3
0011
3
4
0100
4
5
0101
5
6
0110
6
7
0111
7
8
1000
8
9
1001
9
10
1010
A
11
1011
B
12
1100
C
13
1101
D
14
1110
E
15
1111
311_01
F
3
Negative Numbers
311_01
4
2’s Complement
 MSB serves as sign bit (fixed-width)


0 => positive
1 => negative
 Negation Operation


Complement all bits
Add 1
 Alternate Negation Operation

Starting from right to left


Copy up to and including the first 1
Complement the rest
311_01
5
Overflow
 Operation produces a result that exceeds the
number system
 Example (4-bits)

Range -8 to +7
 Detection Rule

Overflow occurs if the addends’ signs are the
same, but the sum’s sign is different from the
addends’
311_01
6
Binary Codes
Decimal
Digit
0
8-4-2-1
6-3-1-1
Code
Code
(BCD)
0000
0000
Excess-3 2-out-of-5
Code
Code
Gray
Code
0011
00011
0000
1
0001
0001
0100
00101
0001
2
0010
0011
0101
00110
0011
3
0011
0100
0110
01001
0010
4
0100
0101
0111
01010
0110
5
0101
0111
1000
01100
1110
6
0110
1000
1001
10001
1010
7
0111
1001
1010
10010
1011
8
1000
1011
1011
10100
1001
9
1001
1100
1100
11000
1000
311_01
7