Binary Number System uses two digits, ‘0′ and ‘1’, and is the foundation for all modern computing. The word binary is derived from the word “bi,” which means two. But what makes it so essential, and how does it work? This article will dive deep into binary numbers, binary decimal number conversion and vice versa, 1’s and 2’s complements, and how they are used in computer systems.
There are generally various types of number systems, and among them, the four major ones are,
- Binary Number System (Number system with Base 2)
- Octal Number System (Number system with Base 8)
- Decimal Number System (Number system with Base 10)
- Hexadecimal Number System (Number system with Base 16)
In the Binary Number System, we have a base of 2. The base of the Binary Number System is also called the radix of the number system.
In a binary number system, we represent the number as,
In the above example, a binary number is given in which the base is 2. In a binary number system, each digit is called the “bit”. In the above example, there are 5 digits.
Binary Number Table
|
Decimal Number
|
Binary Number
|
Decimal Number
|
Binary Number
|
|
1
|
001
|
11
|
1011
|
|
2
|
010
|
12
|
1100
|
|
3
|
011
|
13
|
1101
|
|
4
|
100
|
14
|
1110
|
|
5
|
101
|
15
|
1111
|
|
6
|
110
|
16
|
10000
|
|
7
|
111
|
17
|
10001
|
|
8
|
1000
|
18
|
10010
|
|
9
|
1001
|
19
|
10011
|
|
10
|
1010
|
20
|
10100
|
Binary to Decimal Conversion
A binary number is converted into a decimal number by multiplying each digit of the binary number by the power of either 1 or 0 to the corresponding power of 2. Let us consider that a binary number has n digits, B = an-1…a3a2a1a0. Now, the corresponding decimal number is given as
D = (an-1 × 2n-1) +…+(a3 × 23) + (a2 × 22) + (a1 × 21) + (a0 × 20)
Let us go through an example to understand the concept better.
Example: Convert (10011)2 to a decimal number.
Solution:
The given binary number is (10011)2.
(10011)2 = (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (1 × 20)
= 16 + 0 + 0 + 2 + 1 = (19)10
Hence, the binary number (10011)2 is expressed as (19)10.
Decimal to Binary Conversion
A decimal number is converted into a binary number by dividing the given decimal number by 2 continuously until we get the quotient as 1, and we write the numbers from downwards to upwards.
Let us go through an example to understand the concept better.
Example: Convert (28)10 into a binary number.
Solution:
Hence, (28)10 is expressed as (11100)2.
Arithmetic Operations on Binary Numbers
We can easily perform various operations on Binary Numbers. Various arithmetic operations on the Binary number include,
Now, let’s learn about the same in detail.
Binary Addition
The result of the addition of two binary numbers is also a binary number. To obtain the result of the addition of two binary numbers, we have to add the digits of the binary numbers by digit. The table below shows the rule of binary addition.
|
Binary Number (1)
|
Binary Number (2)
|
Addition
|
Carry
|
|
0
|
0
|
0
|
0
|
|
0
|
1
|
1
|
0
|
|
1
|
0
|
1
|
0
|
|
1
|
1
|
0
|
1
|
Binary Subtraction
The result of the subtraction of two binary numbers is also a binary number. To obtain the result of the subtraction of two binary numbers, we have to subtract the digit of the binary numbers by digit. The table below shows the rule of binary subtraction.
|
Binary Number (1)
|
Binary Number (2)
|
Subtraction
|
Borrow
|
|
0
|
0
|
0
|
0
|
|
0
|
1
|
1
|
1
|
|
1
|
0
|
1
|
0
|
|
1
|
1
|
0
|
0
|
Binary Multiplication
The multiplication process of binary numbers is similar to the multiplication of decimal numbers. The rules for multiplying any two binary numbers are given in the table,
|
Binary Number (1)
|
Binary Number (2)
|
Multiplication
|
|
0
|
0
|
0
|
|
0
|
1
|
0
|
|
1
|
0
|
0
|
|
1
|
1
|
1
|
Binary Division
The division method for binary numbers is similar to that of the decimal number division method. Let us go through an example to understand the concept better.
Example: Divide (11011)2 by (11)2.
Solution:
1’s and 2’s Complement of a Binary Number
- 1’s Complement of a Binary Number is obtained by inverting the digits of the binary number.
Example: Find the 1’s complement of (10011)2.
Solution:
Given Binary Number is (10011)2
Now, to find its 1’s complement, we have to invert the digits of the given number.
To find the 1’s complement of a binary number, you simply flip all the bits:
Thus, 1’s complement of (10011)2 is (01100)2
- 2’s Complement of a Binary Number is obtained by inverting the digits of the binary number and then by adding 1 to the least significant bit.
Example: Find the 2’s complement of (1011)2.
Solution:
Given Binary Number is (1011)2
To find the 2’s complement, first find its 1’s complement, i.e., (0100)2
Now, by adding 1 to the least significant bit, we get (0101)2
Hence, the 2’s complement of (1011)2 is (0101)2
Uses of Binary Number System
Binary Number Systems are used for various purposes, and the most important use of the binary number system is,
- Binary Number System is used in all Digital Electronics for performing various operations.
- Programming Languages use the Binary Number System for encoding and decoding data.
- Binary Number System is used in Data Sciences for various purposes, etc.
Read More,
Solved Example of Binary Number System
Example 1: Convert the the Decimal Number (98)10 into Binary.
Solution:
Thus, Binary Number for (98)10 is equal to (1100010)2
Example 2: Convert the Binary Number (1010101)2 to decimal Number.
Solution:
Given Binary Number, (1010101)2
= (1 × 20) + (0 × 21) + (1 × 22) + (0 × 23) + (1 × 24) + (0 × 25) + (1 ×26)
= 1 + 0 + 4 + 0 + 16 + 0 + 64
= (85)10
Thus, Binary Number (1010101)2 is equal to (85)10 in decimal system.
Example 3: Divide (11110)2 by (101)2.
Solution:
Example 4: Add (11011)2 and (10100)2.
Solution:
Hence, (11011)2 + (10100)2 = (101111)2
Example 5: Subtract (11010)2 and (10110)2.
Solution:
Hence, (11010)2 – (10110)2 = (00100)2
Example 6: Multiply (1110)2 and (1001)2.
Solution:
Thus, (1110)2 × (1001)2 = (1111110)2
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