# Number System

**Computers** understand **machine language.** Every letter, symbol etc. that we write in the instructions given to computer, it gets converted into machine language. This machine language comprises of numbers. In order to understand the language used by computers and other digital system it is crucial to have a better understanding of number system

Types of Number System

There are various types of the number system in mathematics. The four most common number system types are:

- Decimal number system (Base- 10)
- Binary number system (Base- 2)
- Octal number system (Base-8)
- Hexadecimal number system (Base- 16)

Decimal Number System (Base 10)

In this number system, the digits 0 to 9 represents numbers. As it uses 10 digits to represent a number, it is also called the base 10 number system. Each digit has a value based on its position called place value. The value of the position increases by 10 times as we move from right to left in the number.

For example, the value of 786 is

= 7 x 10^{2} + 8 x 10^{1} + 6 x 10^{0}

= 700 + 80 + 6

Binary Number System (Base 2)

A computer can understand only the “on” and “off” state of a switch. These two states are represented by 1 and 0. The combination of 1 and 0 form binary numbers. These numbers represent various data. As two digits are used to represent numbers, it is called a binary or base 2 number system.

The binary number system uses positional notation. But in this case, each digit is multiplied by the appropriate power of two based on its position.

For example, (101101)_{2} in decimal is

= 1 x 2^{5} + 0 x 2^{4} + 1 x 2^{3} + 1 x 2^{2} + 0 x 2^{1} + 1 x 2^{0}

= 1 x 32 + 0 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 1 x 1

= 32 + 8 + 4 + 1

= (45)_{10}

Octal Number System (Base 8)

This system uses digits 0 to 7 (i.e. 8 digits) to represent a number and the numbers are as a base of 8.

For example, (24)_{8} in decimal is

= 2×8^{1}+4×8^{0}

= (20)_{10}

Hexadecimal Number System (Base 16)

In this system, 16 digits used to represent a given number. Thus it is also known as the base 16 number system. Each digit position represents a power of 16. As the base is greater than 10, the number system is supplemented by letters. Following are the hexadecimal symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

To take A, B, C, D, E, and F as part of the number system is conventional and has no logical or deductive reason.

### Number System Chart

Name | Base | Symbols | Example |

Decimal | 10 | 0,1,2,3,4,5,6,7,8,9 | (2795)_{10} |

Binary | 2 | 0,1 | 111000010 |

Octal | 8 | 0,1,2,3,4,5,6,7 | (1576)_{8} |

Hexadecimal | 16 | 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F | 3DB |