Binary Arithmetic Operations – Subtraction

This operation is similar to the basic arithmetic subtraction performed on decimal numbers in Maths. Hence, when we subtract 1 from 0, we need to borrow 1 from the next higher-order digit, to reduce the digit by 1 and the remainder left here is also 1.

When you borrow a number during subtraction, it becomes the number of the base you are subtracting from.

For example, in Base 2 if you borrow a number the number you borrow becomes 2.

Worked Example 2.2.1:

Subtract the following:

i. 1001_two from 11001_two
ii. 11011_two  from  111010_two
iii. 1101_two – 111_two
iv. 10001_two – 1110_two
v. 1111_two – 101_two

Solution

i. 1001_two from 11001_two

Explanation

1st Column (from right to left): 1 – 1 = 0. Write 0.

2nd Column: 0 – 0 = 0. Write 0.

3rd Column: 0 – 0 = 0. Write 0.

4th Column: 1 – 1 = 0. Write 0.

5th Column: 1 – 0 = 1. Write 1.

ii. 11011_two  from  111010_two

1st Column (from right to left): 0 is smaller than 1 so we will have to borrow from the 2nd column. Since it is in base 2 any number borrowed is 2. So borrowing two from the second column will give (2 + 0) – 1 = 1

2nd Column: This reduces 1 to 0 in the second column. 0 is smaller than 1 so we have to borrow again from the third column. We now have 2 – 1 = 1

3rd Column: This reduces 0 to -1 in the third column. This too is smaller than 1 so have to borrow from the 4th column. = 2 – 1 – 0 = 1

4th Column: This reduces 1 to 0 in the fourth column. We have to borrow from the 5th column. Remember that since we are subtracting in base 2 the number we borrow is 2. so we have 2 + 0 – 1 = 1

5th Column: This reduces 1 to 0 in the fifth column. We have to borrow from the 5th column.
So we have 2 + 0 – 1 = 1

6th Column: 1 is reduced to 0. The number in the 6th column is 0 so we can ignore it.

Answer = 11111₂

iii. 1101 – 111

1st Column (from right to left) = 1 – 1 = 0. Write 0.

2nd Column: 0 is smaller than 1 so we have to borrow from the third column. We are subtracting in base 2 so the number we borrow is 2. We therefore have 2 + 0 – 1 = 0

3rd Column: This reduces 1 to 0 in the third column. This too is smaller than 1 so have to borrow from the 4th column. We now have 2 + 0 – 1 = 1

4th Column: This reduces 1 to 0 in the fourth column. We don’t have to write anything down since what we have is 0.

Answer = 110₂

iv. 10001_two – 1110_two

v. 1111_two – 101_two