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 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.
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
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
