A binary number system is a method of representing the number with the base 2, it uses the digits 1 and 0. As it uses only two digits 0 and 1 and has a base of 2, which is called binary.

All digital devices use a binary number system in their electronic circuit. The input 0 indicates off state and the input 1 indicates the state. Because of these implementations binary number systems are most widely used in modern computer technology.

Each digit is referred to as a bit.

There are no 2, 3, 4, 5, 6, 7, 8, 9 in the binary number system.

Example of binary number system is (101)2 = 1 x 22 + 0 x 21+ 1 x 20

= 4 + 0 + 1

=5

Binary Arithmetic Operations

As we perform arithmetic operations on the decimal number system, similarly we can perform all arithmetic operations on binary numbers.Operations such as

• Subtraction
• Multiplication and
• Division

Let us study what is binary addition in detail

Binary addition is the addition of binary numbers. Binary addition is as similar in decimal number system the difference is only of the base. The decimal number system has base 10 and uses digits from 1 to 9 while the binary number system has the base 2 and uses only digits 0 and 1. Binary numbers consist of only two digits 0 and 1.

Most of the digital devices understand only binary number systems, so before adding two binary number systems we should understand the state of the digits. Each bit represents a voltage signal. The bit 0 indicates ‘off’ state while the bit 1 indicates the ‘ on’ state.

Addition of two binary numbers as easy as the decimal number system. Just we have to take note of some rules while adding two binary numbers. There are  four rules associated with binary addition. The binary addition rules are as follows.

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 =10 ( carry 1 to the next significant bit)

As binary numbers include only two digits i.e. 0 and 1, these four rules are all the possible conditions for the addition of binary numbers.

Here is the stepwise procedure of how to add two binary numbers.

Steps to Add Two Binary Numbers

While  two numbers of decimal number systems what we do is if the addition of two numbers is greater than 1 we carry the digit to the next column and then add.

Let us consider an example add 63  to 37

1

6 3

+3 7

——–

1 0 0

Here what we do is we add first ones column here the addition is 10 which is the two-digit number and greater than 9 as 9 is the largest number in decimal number system, we carry 1 to next column tens column, here too the result is 10, as there is no more column to carry we can write the number as it is.

A similar procedure is undertaken to add two binary numbers. Let us add 111 and 110

Step 1: First let us add the first column.

As per the binary rules for the addition we have 1 + 1 = 10. Now write 0 in the same column and carry 1 to the next column that is the tens column.

1

1 0 1

+ 1 1 1

————

0

Step 2:Now add digits in tens place we have  0 + 1 = 1 and add the carry 1. I.e 1 +1 =     10. We get 10. Again write 0 in the tens column and carry one to the next column that is hundreds columns.

1 1

1 0 1

+ 1 1 1

———–

0  0

Step 3: Now add digits in hundreds place i.e 1 + 1 = 10 and add the carry 1 that will be 10 + 1 = 11. So write 1 in hundreds column and take carry to the thousands place.

1 1 1

1 0 1

+ 1 1 1

———–

1  0  0

Step 4: Write the last digit 1 as it is in thousands columns.

1 1 1

1 0 1

+ 1 1 1

———–

1  1  0  0

So the result of the addition of 101 and 111 is 1100

We can cross-check these with the decimal number system.

• 1 0 1 in decimal number system is 5
• 1 1 1 in decimal number system is 7
• Adding 5 + 7 we get 12
• And 1 1 0 0 in the decimal number system is 12.

Let us practice some more binary addition examples.

Example 1:  Add 10102 and 11112

Solution: 1 1

1 0 1 0

+ 1 1 1 1

—————————

1 1 0  0  1

Solution

1       1 1

1 0 0 1 1

+   1 1 0 0 0 1

——————————

1 0 0 0 1 0 0

Quiz Time

1. How to Add Two Binary Numbers

Answer: Here is the step wise procedure to add two binary numbers.

Step 1: First let us add the first column.

As per the binary rules for the addition we have 1 + 1 = 10. Now write 0 in the same column and carry 1 to the next column that is the tens column.

1

1 0 1

+   1 1 1

————

0

Step 2:Now add digits in tens place we have  0 + 1 = 1 and add the carry 1. I.e 1 +1 = 10. We get 10. Again write 0 in the tens column and carry one to the next column that is hundreds columns.

1 1

1 0 1

+ 1 1 1

———–

0  0

Step 3: Now add digits in hundreds place i.e 1 + 1 = 10 and add the carry 1 that will be 10 + 1 = 11. So write 1 in hundreds column and take carry to the thousands place.

1 1 1

1 0 1

+    1 1 1

———–

1  0  0

Step 4: Write the last digit 1 as it is in thousands columns.

1 1 1

1 0 1

+     1 1 1

———–

1  1  0  0

So the result of the addition of 101 and 111 is 1100

We can cross-check these with the decimal number system.

1. 1 0 1 in decimal number system is 5

2. 1 1 1 in decimal number system is 7

3. Adding 5 + 7 we get 12

4. And 1 1 0 0 in the decimal number system is 12.

2. How to Convert Binary Numbers to Decimal Number Systems?

Answer: Binary numbers are the number with digits 0 and 1 and base 2 while the decimal numbers are the numbers with digits 0 to 9 and base 10.

The formula for converting binary number to decimal number is

For binary number with n digits:

dn-1 … d3 d2 d1 d0

The decimal number can be obtained by the sum of the product of binary digits     ( dn) and their power of 2 (2n):

decimal number = d0×20 + d1×21 + d2×22 + d3 x 23+ ……

For Example

Find the decimal value of 1110012:

Binary number: 1 1 1 0 0 1

And their power of 2: 25 24 23 22 21 20

1110012  =  1 x 25 + 1 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20 = 5710

3. How to Convert Decimal Numbers to Binary Numbers?

Steps to convert decimal number to binary number

Divide the given number by 2.

Take the quotient for the next iteration.

And the remainder for the binary digit.

Divide the obtained quotient again by 2

Repeat the steps until we get a quotient equal to 0.

For Example

Convert 1310 to binary:

Solution: divide 13 by 2

13/2 = 6 and remainder 1

6/2 = 3 and remainder is 0

3/2 = 1 and remainder is 1

1/2 = 0 and remainder is 1

So we collect the reminders in the order we get 10112

1310 = 10112