# Binary Number System

Binary Number System is a number system that is used to represent various numbers using only two symbols “0” and “1”. The word binary is derived from the word “bi” which means two. Hence, this number system is called Binary Number System. Thus, the binary number system is a system that has only two symbols.

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)

Here, we are only going to learn about Binary Number System. This number system is very useful for explaining tasks to the computer. In the Binary Number System, we have two states “0” and “1” and these two states are represented by two states of a transistor. If the current passes through the transistor then the computer reads “1” and if the current is absent from the transistor then it read “0”. Thus, alternating the current the computer reads the binary number system. Each digit in the binary number system is called a “bit”.

In this article, we will learn about the Binary Number System, the Conversion of the Binary Number System, the Binary Table, the Operation of Binary Numbers, Examples, and others in detail.

## Binary Number System

Binary Number System is the number system in which we use two digits “0” and “1” to perform all the necessary operations. 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,

• (11001)2

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.

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 Ã— 2n-1) +…+(a3 Ã— 23) + (a2 Ã— 22) + (a1 Ã— 21) + (a0 Ã— 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 Ã— 24) + (0 Ã— 23) + (0 Ã— 22) + (1 Ã— 21) + (1 Ã— 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,

• Binary Subtraction
• Binary Multiplication
• Binary Division

Now let’s learn about the same in detail.

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 digit of the binary numbers by digit. The table added below shows the rule of binary addition.

Binary Number (1)

Binary Number (2)

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 added 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 (101101)2 by (110)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.

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 uses  Binary Number System for encoding and decoding data.
• Binary Number System is used in Data Sciences for various purposes, etc.

## Binary Number System Example

Example 1: Convert Decimal Number (98)10 into Binary.

Solution:

Thus, Binary Number for (98)10 is equal to (1100010)2

Example 2: Convert Binary Number (1010101)2 to Decimal Number.

Solution:

Given Binary Number, (1011101)2

= (1 Ã— 20) + (0 Ã— 21) + (1 Ã— 22) + (0 Ã— 23) + (1 Ã— 24) + (0 Ã— 25) + (1 Ã—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 &#x2013 (10110)2 = (00100)2

Example 6: Multiply (1110)2 and (1001)2.

Solution:

Thus, (1110)2 Ã— (1001)2 = (1111110)2

## FAQs on Binary Number System

### 1. What is a Binary Number System?

Binary Number System is one of the four number system that is used to represent the numbers using only two digits, “0” and “1”. In binary number system the digits are called ‘bits’. Binary Number System is used by computers to perform various calculations.

### 2. What is a Bit?

A bit in Binary Number System is defined as a individual digits that holds the value ‘0’ or ‘1’.

### 3. What is a Nibble?

A group of four digits is called the Niblle.

### 4. What is the Binary Value of 10?

The binary value of 10 is (1010)2

### 5. What are Types of Number Systems?

There are various types of number systems and some of them are,

• Binary Number System
• Octal Number System
• Decimal Number System

### 6. How to Calculate Binary Numbers?

Binary numbers are calculated from dicmal numbers by dividing the decimal number with 2 and writing the remainder. Then we arrange all the remainders from newest to oldest to get the binary number.

### 7. How to Add Binary Numbers?

Binary numbers are added by using the formulas written below,

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (carry 1)

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