 Binary addition is one of the binary operations. To recall, the term “Binary Operation” represents the basic operations of mathematics that are performed on two operands. Basic arithmetic operations like addition, subtraction, multiplication, and division, play an important role in mathematics. In this lesson, all the concepts about binary addition are explained, which includes:

The binary addition operation works similarly to the base 10 decimal system, except that it is a base 2 system. The binary system consists of only two digits, 1 and 0. Most of the functionalities of the computer system use the binary number system. The binary code uses the digits 1’s and 0’s to make certain processes turn off or on. The process of the addition operation is very familiar to the decimal system by adjusting to the base 2.

Before attempting the binary addition process, we should have complete knowledge of how the place works in the binary number system. Because most of the modern digital computers and electronic circuits perform the binary operation by representing each bit as a voltage signal. The bit 0 represents the “OFF” state, and the bit 1 represents the “ON” state.

Binary addition is much easier than the decimal addition when you remember the following tricks or rules. Using these rules, any binary number can be easily added. The four rules of binary addition are:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 =10

### How To Do Binary Addition?

Now, look at the example of the binary addition:101 + 101

Procedure for Binary Addition of Numbers:

101

(+) 101

• Step 1: First consider the 1’s column, and add the one’s column,( 1+1 ) and it gives the result 10 as per the condition of binary addition.
• Step 2: Now, leave the 0 in the one’s column and carry the value 1 to the 10’s column.

1

101

(+) 101

————–

0

• Step 3: Now add 10’s place, 1+( 0 + 0 ) = 1. So, nothing carries to the 100’s place and leave the value 1 in the 10’s place

1

101

(+) 101

————-

10

• Step 4: Now add the 100’s place ( 1 + 1 ) = 10. Leave the value 0 in the 100’s place and carries 1 to the 1000’s place.

1

101

(+) 101

————-

1010

So, the resultant of the addition operation is 1010.

When you cross-check the binary value with the decimal value, the resultant value should be the same.

The binary value 101 is equal to the decimal value 5

So, 5 + 5 = 10

The decimal number 10 is equal to the binary number 1010.

The table of adding two binary numbers 0 and 1 is given below:

 x y x+y 0 0 0 0 1 1 1 0 1 1 1 0      (where 1 is carried over)

You can see from the above table, x and y are the two binary numbers. So when we give the input for x = 0 and y = 0, then the output is equal to 0. When x = 0 or 1 and y = 1 or 0, then x+y = 1. But when both x and y are equal to 1, then their addition equals to 0, but the carryover number will equal to 1, which means basically  1 + 1 = 10 in binary addition, where 1 is carry forwarded to the next digit.

A few examples of binary additions are as follows:

Example 1: 10001 + 11101

Solution:

1

1 0 0 0 1

(+) 1 1 1 0 1

———————–

1 0 1 1 1 0

Example 2: 10111 + 110001

Solution:

1 1 1

1 0 1 1 1

(+) 1 1 0 0 0 1

———————–

1 0 0 1 0 0 0

### Binary Addition Using 1’s Complement

• The number 0 represents the positive sign
• The number 1 represents the negative sign

Addition of Positive and Negative Number

Case 1: When a positive number has a greater magnitude

Take the 1’s complement of the negative number, and the carry is added to the resultant sum at the 1’s place. When you add the carry with the resultant, you will get the sum value.

Example:

+ 1111 and -1101

+ 1 1 1 1 = 0 1 1 1 1

– 1 1 0 1 = 1 0 0 1 0 (taking 1’s complement)

——————-

0 0 0 0 1

1

———————

0 0 0 1 0

Therefore, the solution is + 0010.

• Case 2: When a negative number has a greater magnitude

Take the 1’s complement of the negative number, and there will be no end-around carrying in this case. Finally, the sum is obtained by taking the 1’s complement of the resultant.

Example:

+ 1111 and -1101

– 1 1 1 1 = 1 0 0 0 0 (taking 1’s complement)

+1 1 0 1 = 0 1 1 0 1

—————-

1 1 1 0

——————

1 0 0 1 0 (taking 1’s complement)

Take the 1’s complement of both the negative numbers and then add. The end around carrying will appear, and it will generate a number 1 in the sign bit. The sum value can be obtained by taking the 1’s complement of the resultant.

Example:

• -1010 and – 0011
• 1 0 1 0 = 1 0 1 0 1 (taking 1’s complement)
• 0 0 1 1 = 1 1 1 0 0 (taking 1’s complement)

————————–

1 0 0 0 1

1

—————————–

1 0 0 1 0

—————————-

1 1 1 0 1 (taking 1’s complement)

Therefore the solution is – 1101

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