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:

Decimal	Binary
0	0
1	1
2	10
3	11
4	100
5	101
6	110
7	111
8	1000
9	1001
10	1010
11	1011
12	1100
13	1101
14	1110
15	1111
16	10000
17	10001
18	10010
19	10011
20	10100

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 a_n-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:

Dividend	Quotient	Remainder
20 ÷ 2	10	0
10 ÷ 2	5	0
5 ÷ 2	2	1
2 ÷ 2	1	0
1 ÷ 2	0	1

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 Addition

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

Click here to learn more about addition, subtraction, multiplication, and division in the binary number system.

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