Binary Number System

Binary Number Definition:

According to digital electronics and mathematics, a binary number is defined as a number that is expressed in the binary system or base 2 numeral system that describes numeric values by two separate symbols; typically 1 (one) and 0 (zero). The base-2 system is the positional notation with 2 as a radix. The binary system is applied internally by almost all latest computers and computer-based devices because of its direct implementation in electronic circuits using logic gates. Every digit is referred to as a bit.

What is Bit in Binary Number?

A single binary digit is called a “Bit”. The number below has 6 bits.

110110

Binary numbers chart with decimal values is organized in the below column. To understand binary math, you should know how the number system works. Let’s start with the decimal system since it is easier to deal with.

For example, the number to be operated is 1235.  

Thousands Hundreds Tens Ones
1 2 3 5

This indicates,

1235 = 1 × 1000 + 2 × 100 + 3 × 10 + 5 × 1

Given,

1000 = 103 = 10 × 10 × 10
100 = 102 = 10 × 10
10 = 101 = 10
1 = 100 (any value to the exponent zero is one)

The above table can be described as,

Thousands Hundreds Tens Ones
103 102 101 100
1 2 3 5

Hence,

1235 = 1 × 1000 + 2 × 100 + 3 × 10 + 5 × 1

= 1 × 103 + 2 × 102 + 3 × 101 + 5 × 100

The decimal number system operates in base 10 wherein the digits 0-9 represent numbers. In binary system operates in base 2 and the digits 0-1 represent numbers and the base is known as radix. Put differently, the above table can also be shown in the following manner.

Thousands Hundreds Tens Ones
Decimal 103 102 101 100
Binary 23 22 21 20

We place the digits in columns 100, 101 and so on in base 10. When there is a need to put a value higher than 9 in the form of 10(n+1) for instance, to add 10 to column 100, you need to add 1 to the column 101.

We place the digits in columns 20, 21 and so on in base 2. To place a value that is higher than 1 in 2n, you need to add 2(n+1). For instance, to add 3 to column 20, you need to add 1 to column 21.

Position in Binary Number System

In the Binary system, we have ones, twos, fours etc…

For example 1011.110

It is shown like this:

1 × 8 + 0 × 4 + 1 × 2 + 1 + 1 × ½ + 1 × ¼ + 0 × 1⁄8

= 11.75 in

Decimal

To show the values greater than or less than one, the numbers can be placed to the left or right of the point.

10.1

10 is a whole number on the left side of the decimal and as we move more left, the number place gets bigger (Twice).

The first digit on the right is always Halves ½ and as we move more right, the number gets smaller (half as big).

In the example given above:

  • “10” shows ‘2’ in decimal.
  • “.1” shows ‘half’.
  • So, “10.1” in binary is 2.5 in decimal.

Binary Number System Table

Some of the binary notations of decimal numbers are mentioned in the below list.

Decimal number Binary value
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000
9 1001
10 1010

Example Question

Let us practice some of the problems for better understanding:

Question 1: What is binary number 1.1 in decimal?

Solution:

Step 1: 1 on the left-hand side is on the one’s position, so it’s 1.

Step 2: The one on the right-hand side is in halves, so it’s

× ½

Step 3: so, 1.1 = 1.5 in decimal.

Question 2:  Write 10.112 in Decimal?

Solution: 

10.11 =  1 x (2)1 + 0 (2)0 + 1 (½)1  + 1(½)2  

= 2 + 0 + ½ + ½

= 2.75

So, 10.11 is 2.75  in Decimal.

Keep visiting BYJU’S to explore and learn more such maths related topics in a fun and engaging way.

Practise This Question

The value of 1000 is 1000.

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