Complements in Binary Arithmetic

Binary arithmetic is an essential aspect of computer science and digital electronics. Complements play a crucial role in simplifying operations like subtraction. This page explains the concepts of 1's and 2's complements and their application in binary addition and subtraction.

1's Complement

The 1's complement of a binary number is obtained by flipping all the bits in the number. This means converting all 1s to 0s and all 0s to 1s.

Example:

For the binary number 1010:

1. Original number: 1010

2. 1's complement: 0101

Characteristics:

1. Used primarily in signed binary numbers to represent negative values.

2. For n-bit binary numbers, the 1's complement ranges from -2^{n-1} + 1 to 2^{n-1} -1.

2's Complement

The 2's complement of a binary number is obtained by adding 1 to its 1's complement. This method is widely used in computers to perform subtraction and represent negative numbers.

Steps:

1. Find the 1's complement of the number.

2. Add 1 to the result.

Example:

For the binary number 1010:

1. 1's complement: 0101

2. Add 1: 0101 + 1 = 0110

3. 2's complement: 0110

Characteristics:

1. Eliminates the ambiguity of two representations for zero.

2. For n-bit binary numbers, the range is from -2^{n-1} to 2^{n-1} - 1.

Binary Addition Using Complements

Example:

Add 5 (0101) and -3 (2's complement of 3):

1. Convert 3 to binary: 0011

2. Find the 2's complement of 3:

1. 1's complement: 1100

2. Add 1: 1100 + 1 = 1101

3. Add 5 and -3:

1. 0101 + 1101 = 10010

4. Discard the carry (if working with 4 bits):

1. Result: 0010 (which is 2 in decimal).

Binary Subtraction Using Complements

Example:

Subtract 5 (0101) from 9 (1001) using 2's complement:

1. Convert 5 to binary: 0101

2. Find the 2's complement of 5:

1. 1's complement: 1010

2. Add 1: 1010 + 1 = 1011

3. Add 9 and -5:

1. 1001 + 1011 = 10100

4. Discard the carry (if working with 4 bits):

1. Result: 0100 (which is 4 in decimal).

Advantages of Using Complements

1. Simplifies subtraction by converting it into addition.

2. Reduces hardware complexity in digital circuits.

3. Efficiently handles negative numbers in binary systems.

Summary Table:

Operation	Binary Number	1's Complement	2's Complement
Positive number 5	0101	1010	1011
Negative number   -5	-	1010	1011

By mastering 1's and 2's complements, you can perform binary arithmetic with ease, an essential skill in computer science and digital logic design.