Binary Number System

Binary Number System: 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. It describes numeric values by two separate symbols; basically 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

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

What is Bit in Binary Number?

A single binary digit is called a “Bit”. The binary 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.

Decimal to Binary Number Conversion

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.

Binary Arithmetic Operations

Like we perform the arithmetic operations in numerals, in the same way, we can perform addition, subtraction, multiplication and division operations on Binary numbers. Let us learn them one by one.

Binary Addition

Adding two binary numbers will give us a binary number itself. This is the simplest method. Addition of two single-digit binary number is given in the table below.

Binary Numbers Addition
0 0 0
0 1 1
1 0 1
1 1 0; Carry →1

Let us take an example of two binary numbers and add them.

For example: Add 11012 and 10012.
Solution: Binary Addition

Binary Subtraction

Subtracting two binary numbers will give us a binary number itself. This is also an easy method. Subtraction of two single-digit binary number is given in the table below.

Binary Numbers Subtraction
0 0 0
0 1 1; Borrow 1
1 0 1
1 1 0

Let us take an example of two binary numbers and subtract them.

Example: Subtract 11012 and 10102.

Solution: Binary Subtraction

Binary Multiplication

The multiplication process is the same for the binary numbers as it is for numerals. Let us understand it with example.

Example: Multiply 11012 and 10102.

Solution: Binary Multiplication

Binary Division

Binary division is similar to the decimal number division method. We will learn with an example here.

Example: Divide 10102 by 102

Solution: Binary Division

Learn more about binary division here.

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.

For 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.

Uses of Binary Number System

Binary numbers are commonly used in computer applications. All the coding and languages in computers such as C, C++, Java. etc. uses binary digits 0 and 1 to write a program or encode any digital data. Basically, the computer understands only coded language. Therefore these 2-digit number system is used to represent a set of data or information in discrete bits of information.

Binary Number System Examples

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 math-related topics in a fun and engaging way.

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