Complement Number

Introduction

• A complement number is a mathematical concept used in various number systems, including binary, octal, decimal, and hexadecimal to represent negative numbers and perform arithmetic operations.

Definition

• In digital systems, the complement of a number refers to the binary representation of the number’s inverse with respect to a base.
• The complement numbers refer to the inverse representation of a binary number.

Characteristics

• Complement numbers are essential for computer arithmetic, especially in representing and manipulating negative integers.
• They allow efficient handling of arithmetic operations and minimize the complexity of hardware implementations for addition and subtraction.

Types

• There are two commonly used complements: the one’s complement and the two’s complement.

  • One’s Complement(1’s): 

• • • The one’s complement of a binary number is obtained by flipping/reversing all the bits (changing 0s to 1s and 1s to 0s).
    • One’s complement is often used for error detection and correction in digital systems.
    • It is the bitwise negation of the original number.
    • For example:

• • • • Original number: 101101
      • One’s complement(1’s) : 010010
    • The ones’ complement has a drawback called the “ones’ complement arithmetic” problem, where addition and subtraction of numbers require extra steps to handle end-around carry.

• • Two’s Complement(2’s):
    • The two’s complement of a binary number is used in computing for signed number representation.
    • It is obtained by adding 1 to the one’s complement of the number.
    • The two’s complement representation allows both positive and negative numbers to be represented.
    • In the two’s complement representation, arithmetic operations like addition and subtraction can be performed with regular binary arithmetic. The leftmost bit indicates the sign of the number, where 0 represents a positive number and 1 represents a negative number.
    • In the two’s complement representation, the leftmost bit is the sign bit (0 for positive, 1 for negative), and the remaining bits represent the magnitude of the number.
    • Two’s complement is widely used in computer arithmetic because it simplifies addition and subtraction operations and allows negative numbers to be represented without a separate sign bit.
    • The two’s complement representation is widely used because it simplifies arithmetic operations.
    • To calculate the two’s complement:

1. 1. 1. • Find the one’s complement of the given number.
         • Add 1 to the least significant bit (LSB) of the one’s complement result and finally, we got the output of 2’s.

• • • For example:
      • Original number: 101101
      • One’s complement: 010010
      • Two’s complement: 010011 (one’s complement + 1)

Use

• Complement numbers are used in digital computers in order to simplify the subtraction operation and for logical manipulations.