# Number Systems in Digital Electronics

Many number systems are in use in digital technology that represents the digits in various forms.
The different number systems in digital electronics are:

1. Decimal number system
2. Binary number system
3. Octal number system and

DECIMAL NUMBER SYSTEM:

The decimal system uses ten digits 0 to 9. The base or radix of a number system is defined as the number of digits it uses to represent numbers. Since the decimal system uses 10 digits (0 to 9) its base or radix is 10. The weight of each digit of a decimal number depends on its relative position within the number.

 103 102 101 100 . 10-1 10-2 10-3
1000             100            10                 1                  .                     0.1             0.01              0.001

Example:

3472.65 = 3 x 103 + 4 x 102 + 7 x 101 + 2 x 100 + 6 x 10-1 + 5 x 10-2
= 3 x 1000 + 4 x 100 + 7 x 10 + 2 x 1 + 6 x 0.1 + 5 x 0.01

Symbols used: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.
Base (or radix, or weight): 10
Place value: ….. 103, 102, 101, 100 . 10-1, 10-2, 10-3 …..

BINARY NUMBER SYSTEM:

Electronic circuits can be designed easily with one of the two stable states, either ON (High) or OFF (LOW). ON is represented by a high voltage and OFF by a low voltage. These two states can be represented by the digits 1 (for HIGH) and 0 (for LOW). Thus we have a number system with only 2 digits 0 and 1 and is called the binary number system. These binary digits are called ‘bits’. The electronic elements of a computer system can easily understand the two states 0 and 1. Hence digital computers make use of binary number system in representing information. This is also a positional number system.

 23 22 21 20 . 2-1 2-2 2-3
8                  4                  2                  1                      .                0.5                0.25               0.125

Example:
1011.01 = 1 x 23 + 0 x 22 + 1 x 21 + 1 x 20 + 0 x 2-1 + 1 x 2-2 = 1 x 8 + 0 x 4 + 1 x 2 + 1 x 1 + 0 x 0.5 + 1 x 0.25 = 11.25 (in decimal)

Symbols used: 0 and 1
Base (or radix, or weight): 2
Place value: ….. 23, 22, 21, 20 . 2-1, 2-2, 2-3 …..

OCTAL NUMBER SYSTEM:

The base of octal number system is 8. It uses 8 digits - 0, 1, 2, 3, 4, 5, 6 and 7. It is also a positional number system. The places to the left of the octal points are positive powers of 8 and places to the right of the octal point are negative powers of 8.

 83 82 81 80 . 8-1 8-2 8-3
512             64                   8                 1                       .                   0.125       0.015625   0.001953125

Example:

72.2 = 7 x 81 + 2 x 80 + 2 x 8-1
= 7 x 8 + 2 x 1 + 2 x 0.125
= 58.25 (in decimal)

Symbols used: 0, 1, 2, 3, 4, 5, 6 and 7
Base (or radix, or weight): 8
Place value: ….. 83, 82, 81, 80 . 8-1, 8-2, 8-3 …..

This has a base of 16. It uses 16 digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, E and F. Each digit to the left of the hex point has successive negative powers of 16 and to the right of the hex point has successive negative powers of 16. To convert the given hex number to decimal we multiply each digit by its weight and take the sum.

 163 162 161 160 . 16-1 16-2 16-3
4096             256                16               1                       .            0.0625       0.00390625   0.000244140625

Example:

3C.A = 3 x 161 + 12 x 160 + 10 x 16-1
= 3 x 16 + 12 x 1 + 10 x 0.0625
= 60.625 (in decimal)

Symbols used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F
Base (or radix, or weight): 16
Place value: ….. 163, 162, 161, 160 . 16-1, 16-2, 16-3 …..