How to Subtract r-base Numbers Using Complements

 

In the decimal numbering system, the radix complement is called the ten's complement and the diminished radix complement the nines' complement. In binary, the radix complement is called the two's complement and the diminished radix complement the ones' complement.

Subtraction Using Complements can be done by two ways:

Subtraction Using r's Complement:

The subtraction of two positive numbers (M-N) both of base r can be done as follows;

1.  Add the minuend M to the r's complement of subtracted N.
2.  Inspect the result obtained in step 1 for end carry.

• If end carry occurs, discard it.

• If end carry doesn't occur, take r's complement if the number obtained in step 1 and place a negative sign in front.

For decimal numbers:

Example: Subtract (72532-3250) using 10's complement.

M= 72532

N= 03250

10's complement of N= 105-3250 = 96750

Now, adding 10's complement of N to M,

72532+96750=169282

Here, end carry occurs as 1. So, 10's complement= 69282

For binary numbers:

Example: Subtract (1000100-1010100) using 2's complement.

M= 1000100

N= 11010100

2's complement of N=10110

Now, adding 2's complement of N to M

10110+1000100= 110000

Here, end carry occurs as 1, so 2's complement = -10000

 

Subtraction Using r-1's Complement:

 

The subtraction of two positive numbers (M-N) both of base r-1 can be done as follows;

1.  Add the minuend M to the r-1's complement of subtracted N.
2.  Inspect the result obtained in step 1 for end carry.

• If end carry occurs, add 1 to the least significant bit, i.e. end-around carry.

• If end carry doesn't occur, take r-1's complement of the number obtained in step 1 and place a negative sign in front.

For decimal number:

Example: Subtract (453.35-321.17) using 9's complement.

9's complement of 321.17= 678.82

Now, Adding 9's complement of M to N,

453.35+678.82= 1132.17

Here, end carry occur, so 9's complement = 132.18

For binary numbers:

Example Subtract (1000100-1010100) using 1's complement.

M= 1000100

N= 11010100

1's complement of N=101011

Now, adding 1's complement of N to M

101011+1000100= 1101111

Here, end carry doesn't occur as 1, so 1's complement of 1101111= -1000 which is the required answer.

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