# Number Systems in Digital Electronics

A system that is used for representing numbers is called the number system. In digital electronics, the numbers are used to represent the information. Hence, it is important to learn and understand different types of number systems so we can easily represent and interpret the information in the form of numbers.

There are several types of number systems and the basis of this classification is the base or radix of the number system. The base or radix of the number system is the total number of symbols used to denoted the numbers in the number system.

## Types of Number Systems

Depending on the base or radix, number systems can be classified into the following four major types −

• Decimal Number System
• Binary Number System
• Octal Number System

## Decimal Number System

The system of numbers which has base or radix 10, i.e. uses total 10 symbols to represent numbers of the system is called decimal number system. The symbols used in the decimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; where each of these symbols assigned a specific value.

The decimal number system is a position value system, which means the value of the digit depends on the position in the number. To understand the concept of position value system, consider the following example.

Let a decimal number 1234 which has total four digits, this number can also be written as follows −

$$\mathrm{(1 \times 10^{3})+(2 \times 10^{2})+(3 \times 10^{1})+(4 \times 10^{0})}$$

$$\mathrm{(1 \times 1000)+(2 \times 1000)+(3 \times 1000)+(4 \times 1)}$$

$$1000 + 200 + 30 + 4 = 1234$$

Hence, from this example, we can see that the value of different digits of the number depends on their respective position in the number.

## Binary Number System

A number system with base or radix 2 is called binary number system. The binary number system uses only 2 symbols (0 and 1) to represent binary numbers. All modern digital devices like computers, combinational circuits, sequential circuits, etc. use the binary number system to operate.

We can convert a binary number into its equivalent decimal number as follows −

Let a binary number 1101 and we have to convert it into an equivalent decimal number, then −

$$\mathrm{(1 \times 2^{3})+(1 \times 2^{2})+(0 \times 2^{1})+(1 \times 2^{0})}$$

$$8 + 4 + 0 + 1 = 13$$

$$\mathrm{∴(1101)_{2}=(13)_{10}}$$

## Octal Number System

A number system which has base 8 is called an octal number system. Therefore, the octal number system uses 8 symbols, (0, 1, 2, 3, 4, 5, 6, 7) to represent the number.

An octal number can be converted into an equivalent decimal number as follows −

Let an octal number 124 and we need to find its equivalent in decimal, then

$$\mathrm{(1 \times 8^{2})+(2 \times 8^{1})+(4 \times 8^{0})}$$

$$64 + 16 + 4 = 84$$

$$\mathrm{∴(124)_{8}=(84)_{10}}$$

The number system with base or radix 16 is called as hexadecimal number system. Thus, the hexadecimal number system uses 16 symbol to represent numbers. These symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Where, A = 10; B = 11; C = 12; D = 13; E = 14; F = 15.

The hexadecimal number system is extensively used in microprocessors and microcontrollers. The most significant advantage of hexadecimal number system over binary number system is that the hexadecimal numbers are much shorter in size than the binary numbers, which makes these hexadecimal numbers more readable.

Though, just like binary and octal numbers, we may convert the hexadecimal numbers into decimal numbers. To understand the conversion process, consider the following example.

Let 1AF is a hexadecimal number, we can convert it into the equivalent decimal number as follows −

$$\mathrm{(1 \times 16^{2})+(A \times 16^{1})+(F \times 16^{0})}$$

$$\mathrm{(1 \times 16^{2})+(10 \times 16^{1})+(15 \times 16^{0})}$$

$$256 + 160 + 15 = 431$$

$$\mathrm{∴(1AF)_{16}=(431)_{10}}$$

## Conversion Table

The following table shows the decimal numbers from 0 to 15 and their equivalent binary, octal and hexadecimal numbers −

Decimal Number Binary Number Octal Number Hexadecimal Number
0 0000 0 0
1 0001 1 1
2 0010 2 2
3 0011 3 3
4 0100 4 4
5 0101 5 5
6 0110 6 6
7 0111 7 7
8 1000 10 8
9 1001 11 9
10 1010 12 A
11 1011 13 B
12 1100 14 C
13 1101 15 D
14 1110 16 E
15 1111 17 F

## Conclusion

Thus, in this article we discussed about number system, and different types of number systems and their applications. All digital systems require to implement only two states, i.e., low (off) and high (on) that can be easily implemented using the binary number system. Hence, in digital electronics, the binary number system is most widely used because it uses the minimum number of digits. However, the hexadecimal number system is also used in some specialized digital devices such microprocessors and microcontrollers.