Complements

complements are one of the most important aspects of arithmetic operations in digital systems. As we are pretty much familiar with the fact that SUBTRACTION is nothing but a complementED ADDITION, we can carry out a subtraction operation by using the following steps:

• Compute the complement of the SUBTRAHEND.
• Add it with the MENUEND to get the difference.

Complement of any data is needed primarily to perform SUBTRACTION operations. The following example better clarifies the concept:

Suppose we have to calculate: a = (10 - 6)

We can do it via any of the two ways stated below:

In Digital Architecture Complements are of following two types:

1. (r-1)'s complement
2. r's complement

Since complements are of two types, they are significantly different for different number systems. Consider the following table:

Basically in a number system, r represents RADIX or simply the BASE. Thus r's complement means to the complement w.r.t. BASE, similarly (r-1)'s complement means to the complement w.r.t. (BASE-1).

legends	(r-1)'s	r's
Formula
Examples
Decimal Numbers
Octal Numbers
Hexadecimal Numbers
Decimal Numbers

Shortcuts for (r-1)'s complement:

Subtract every digit in the number from (r-1) or (base-1). In contrast:

1. Subtract every digit in the number from 9 to calculate 9's compliment
2. Subtract every digit in the number from 1 to calculate 1's compliment
3. Subtract every digit in the number from 7 to calculate 7's compliment
4. Subtract every digit in the number from 15 to calculate 15's compliment

Shortcuts for r's complement:

Retain every LSB zero, then subtract the first non-zero digit from r or Base and finally subtract every digit from (r-1) or (Base-1). In Contrast:

1. Subtract first non-zero LSB digit from 10 and others from 9 to calculate 10's compliment
2. Subtract first non-zero LSB digit from 2 and others from 1 to calculate 2's compliment
3. Subtract first non-zero LSB digit from 8 and others from 7 to calculate 8's compliment
4. Subtract first non-zero LSB digit from 16 and others from 15 to calculate 16's compliment