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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 endaround 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 and1
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:
 Two’s Complement(2’s):



 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)
 For example:

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