Tags: #◇Computer Science #◇CSARCH1 #◇Computer Architecture #◇Data Representation #◇Complements

---

Complements

◇Overview

This module discusses how to represent different types of complements for any number system and how to perform subtraction using addition and complements. Complements are used in digital computers for logic manipulation and to simplify subtraction operations.

◇Types of Complements

For each base-$r$ system, there are two types of complements:

1. Diminished Radix Complement: $(r-1)$'s complement.
   • Binary ($r=2$): 1's complement.
   • Decimal ($r=10$): 9's complement.
2. Radix Complement: $r$'s complement.
   • Binary ($r=2$): 2's complement.
   • Decimal ($r=10$): 10's complement.

◇Important Formulas

Diminished Radix Complement ($(r-1)$'s complement)

Given a number $N$ in base $r$ having $n$ digits:
$(r^n - 1) - N$

• Example: 9's complement of $1230_{10}$ ($r=10, n=4$):
  • $(10^4 - 1) - 1230 = 9999 - 1230 = 8769_{10}$
• Example: 1's complement of $0110_2$ ($r=2, n=4$):
  • $(2^4 - 1) - 0110 = 1111 - 0110 = 1001_2$

Radix Complement ($r$'s complement)

Given a number $N$ in base $r$ having $n$ digits:
$r^n - N \quad \text{for } N \neq 0, \text{ and } 0 \text{ for } N = 0$

• Practically, this is equivalent to: $(r-1)$'s complement + 1.
• Example: 10's complement of $1230_{10}$:
  • 9's complement is 8769, plus 1 = $8770_{10}$.
• Example: 2's complement of $0110_2$:
  • 1's complement is 1001, plus 1 = $1010_2$.

◇Subtraction with Complements

Subtraction of two $n$-digit unsigned numbers ($M - N$) in base $r$:

1. Add the minuend $M$ to the $r$'s complement of the subtrahend $N$.
2. If $M \geq N$: The sum will produce an end carry. Discard the carry. The remaining result is $M - N$.
3. If $M < N$: The sum produces no end carry. The result is the $r$'s complement of $(M - N)$. To view the answer in a familiar format, take the $r$'s complement of the sum and place a negative sign in front (though negative numbers technically don't exist for strictly unsigned formats).

Worked Examples

Example 1: Using 10's complement, perform $72532 - 3250$

• Pad $N$ to match the number of digits in $M$: $N = 03250$.
• $M = 72532$
• 10's complement of $N = 96750$
• $M + N' = 72532 + 96750 = 169282$
• Discard end carry -> $69282$

Example 2: Using 10's complement, perform $3250 - 72532$

• $M = 03250$
• 10's complement of $N = 27468$
• $M + N' = 03250 + 27468 = 30718$
• No carry -> Take 10's complement of $30718$ and add negative sign: $-69282$

Example 3: Using 2's complement, perform $1010100_2 - 1000011_2$

• $M = 1010100$
• 2's complement of $N = 0111101$
• $M + N' = 1010100 + 0111101 = 10010001$
• Discard end carry -> $0010001_2$

◇References & Navigation

• Previous Topic: ◇Binary Codes for Decimal Digits
• Next Topic: ◇Number System Base Conversion