Binary Number System

A binary number system is type of number system that is used to represent a number by using two symbols only. The word bi is used because the total number of symbols used to represent any number in counting is only two. The symbols used are “0” and “1” only. Hence, the base-2 numeral system is used to represent binary numbers. For example, (1101)2 is a binary number where 2 represents base number system used for representing above mentioned number.

This number system is widely used in computers. All inputs given to a computer are decoded by it into a series of 0’s or 1’s before being processed further since a computer can only understand binary information, which is represented by the numbers 0 or 1. It is simple to convert a decimal number into a binary number and vice-versa. The notations for decimal numbers and binary numbers are different. For example, a decimal is represented as (15)10 where 10 is the base of the decimal number, and the corresponding binary number is represented as (1111)2 where 2 is the base of a binary number.

Binary Number Table

Decimal Number

Binary Number

1

001

2

010

3

011

4

100

5

101

6

110

7

111

8

1000

9

1001

10

1010

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.

Binary arithmetic is essential part of all the digital computers and many other digital system.

Binary Addition

There are four steps in binary addition, they are written below

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 0 (carry 1 to the next significant bit)
• An example will help us to understand the addition process. Let us take two binary numbers 10001001 and 10010101

The above example of binary arithmetic clearly explains the binary addition operation, the carried 1 is shown on the upper side of the operands.

Binary Subtraction

Here are too four simple steps to keep in memory

• 0–0 = 0
• 0–1 = 1, borrow 1 from the next more significant bit
• 1–0 = 1
• 1–1 = 0

A binary arithmetic example is given to understand the operation more clearly

The operation shows the binary subtraction clearly.

Binary Multiplication

Binary multiplication may sound like it would be more difficult than binary addition or subtraction — but is actually a simple process. Here are the four steps to be followed, using the same binary numbers 10001001 and 10010101:

• 0×0=0
• 1×0=0
• 0×1=0
• 1×1=1 (there is no carry or borrow for this)

The arithmetic of multiplying binary numbers is shown below:

Binary Division

Binary division is comprised of other two binary arithmetic operations, multiplication and subtraction; an example will explain the operation more easily.

Here 101 is the quotient and 1 is the remainder.