Table of contents

Introduction

Complements

1’s complement

2’s complement

FAQs

Key Takeaways

Last Updated: Mar 27, 2024

Complements

Author Malay Gain

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Introduction

Complements are used in digital computers for simplifying the manipulation of logical operations. Subtraction operation in binary can be simplified by using complements as signed binary numbers can be expressed using the complement.

Here, we will describe basic concepts of complements and its types.

Complements

There are two types of complements for each r-based number system. These are the radix complement and diminished radix complement.

Let’s discuss each type of complement with examples.

Diminished Radix Complement or (r-1)’s complement

This complement is defined as if there is a number N in base r having n digits, the

Diminished Radix Complement or (r-1)’s complement of N is (r^n - 1) - N .

Example

So for a 6-digit decimal number( base 10) N, Diminished Radix Complement or (r-1)=9’s complement is (10^6 - 1) - N = 999999 - N

For, a 6-digit binary number N, (2-1)= 1’s complement is (2^6 - 1) - N = 111111 - N

So, 1’s complement of 101100 is (111111 - 101100) =010011

Also Read - Shift Registers in Digital Electronics

Characteristic

While computing the Diminished Radix Complement, subtraction from (r^n - 1) will never require a borrow.

This complement can be computed digit by digit.

Radix Complement or r’s complement

Radix complement of an n-digit number N in base r is defined as (r^n - N) for N ≠ 0 and as 0 for N = 0.

Example

So for a 6-digit decimal number( base 10) N, Radix Complement or r =10’s complement is (10^6 - N ).

For a 6-digit binary number N, 2’s complement is (2^6 - N)

So, 2’s complement of 1101100 is 0010100.

Characteristic

r’s complement can be obtained by adding 1 to the (r-1)’s complement as

(r^n - N) =[ (r^n - 1) - N ] +1

1’s complement

For binary numbers, 1’s complement can be expressed as Diminished Radix Complement as the base is 2.

For binary numbers we can easily compute its 1’s complement without following the steps of Diminished Radix Complement.

Step1: replace all 0’s by 1’s

Step 2: replace all 1’s by 0’s

For example, 1’s complement of (1101100)2 is (0010011)2 .

2’s complement

For binary numbers, 2’s complement can be expressed as Radix Complement for binary numbers as the base is 2.

For binary numbers, we can easily compute its 2’s complement without following the steps of Radix Complement.

Step1: Compute 1’s complement the binary number.

Step2: add 1 to the 1’s complement.

For example, 2’s complement of (1101100)2 is (0010011)2 + 1 = (0010100)2 .

Must read decimal to binary c++

FAQs

Q1. What is 9’s complement?

The 9’s complement of a decimal digit is the number that must be added to it to generate 9 i.e., the complement of 6 is 3 while the complement of 4 is 5 etc.

Q2. State shortcut method of computing 9’s complement for an n-digit number.

A quick technique to find the 9’s complement of a larger (n-digit) number is to replace each digit in the number by its 9’s complement.

Key Takeaways

This article covered different types of compliments describing the computation process of each type.

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