# Binary Number System

The word binary comes from ‘Bi’ means 2. Thus, a binary number system consists of 2 numbers, 0 and 1. It starts with 0 and ends in 1 and, therefore, has a base 2. The base-2 system in the positional notation is represented as (11101)2.

It is widely used in making the latest computers and computer-based devices in their electronic circuits using logic gates. Each digit in the machine is referred to as a bit.

The numbers 0 to 20 in decimal are written in binary as follows:

## Binary to Decimal Conversion

The binary number is converted to the decimal number by expressing each digit as the product of each number (1 or 2) to the power of 2 based on its place value.

If a binary number has n digits an-1….a3a2a1a0, its corresponding decimal number is obtained as:

(a0×20) + (a1×21) + (a2×22) +….

Let us convert the binary number 11101 to its corresponding decimal number.

The binary number 11101 is expressed as:

(11101)2 = (1 × 24) + (1 × 23) + (1 × 22) + (0 × 21) + (1 × 20)

= (29)10

## Decimal to Binary Conversion

A decimal number is converted to its corresponding binary number by dividing the number by 2 until we get 1 as the quotient. The quotients are written from down to up.

Let us convert the decimal number 20 to its corresponding binary number.

Dividing the number by 2 in each step, we get:

Thus, the decimal number (20)10 is expressed as (10100)2

## Operations on Binary Numbers

Like decimal numbers, binary numbers are used to perform mathematical operations: addition, subtraction, multiplication, and division.

Binary numbers are added digit by digit to obtain the result of addition.

We will use the following 4 rules for addition:

• 0 + 0 = 0
• 1 + 0 = 1
• 0 + 1 = 1
• 1 + 1 = 0

### Binary Subtraction

Like addition, binary subtraction is done digit by digit to obtain the result.

• 0 – 0 = 0
• 1 – 0 = 1
• 0 – 1 = 0
• 1 – 1 = 0

### Binary Multiplication

The rules for multiplying 2 binary numbers are given below:

• 0 × 0 = 0
• 0 × 1 = 0
• 1 × 0 = 0
• 1 ×1 = 1

### Binary Division

The rules for dividing 2 binary numbers are given below:

• 0 ÷ 1 = 0
• 1 ÷ 1 = 1
• 1 ÷ 0 = Meaningless
• 0 ÷ 0 = Meaningless

## Complement of a Binary Number

• 1’s complement is obtained by inverting the digits of the binary number. For example, the complement of (110)2 is (001)2.
• 2’s complement is given by inverting the digits of the number and adding 1 to the least significant bit. For example, 2’s complement of (111)2 is (001)2.

## Solved Examples

Convert the binary number (111001)2 to its decimal number.

Solution:

The given binary number is (111001)2
To find the corresponding decimal number:
= (1 × 20) + (1 × 21) + (1 × 22) + (0 × 23) + (0 × 24) + (1 × 25)
= 1 + 2 + 4 + 8 + 16 + 32
= 57
Thus, the corresponding decimal number is (57)10.

Convert the decimal number (134)10 to its binary number.

Solution:

The given decimal number is (134)10
To find the corresponding binary number:
134 ÷ 2 = 81, R = 0
81 ÷ 2 = 40, R = 1
40 ÷ 2 = 20, R = 0
20 ÷ 2 = 10, R = 0
10 ÷ 2 = 5, R = 1
5 ÷ 2 = 2, R = 1
2 ÷ 2 = 1, R = 0
1 ÷ 2 = 0, R = 1
Thus, the corresponding binary number is (10110010)2