Complements: 1’s complement, 2’s complement

  

Complements

·       Methods used to simplify arithmetic operations (especially subtraction) and represent negative numbers in binary systems.

·       Helpful when working with signed binary numbers.

 

Importance of Complements

1. Representing Negative Numbers: Negative numbers are represented in binary systems without separate "negative markers."
2. Simplifying Arithmetic: Operations like subtraction can be performed as addition by adding complements.
3. Efficient Hardware Design: use for simpler and faster arithmetic circuits.

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Types of Complements

1. Radix Complement (2's Complement for binary numbers)
2. Diminished Radix Complement (1's Complement for binary numbers)

 

Note:- These complements are named based on the base (radix) of the number system. For binary numbers, the base is 2.

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1.    1's Complement (Diminished Radix Complement)

Obtained by inverting all the bits (changing 1 to 0 and 0 to 1) of binary number.

• Representation:
• Positive numbers remain unchanged.
• Negative numbers are represented as the 1's complement of their absolute value.

Steps to Find 1's Complement:

1. Write the binary representation of the number.
2. Invert all bits (flip 1 to 0 and 0 to 1).

Example:

For a 4-bit representation:

• +5 = 0101
• -5 = 1's complement of 0101 = 1010

Advantages:

• Simple to compute.

Disadvantages:

• There are two representations for 0:
• Positive 0: 0000
• Negative 0: 1111

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2's Complement (Radix Complement)

Obtained by inverting all the bits (1's complement) of a binary number and then adding 1 to the least significant bit (LSB).

• Representation:
• Positive numbers remain unchanged.
• Negative numbers are represented as the 2's complement of their absolute value.

Steps to Find 2's Complement:

1. Write the binary representation of the number.
2. Invert all bits (1's complement).
3. Add 1 to the result.

Example:

For a 4-bit representation:

• +5 = 0101
• -5:
1. Take 1's complement of 0101: 1010
2. Add 1: 1010 + 1 = 1011

 

Use 2's Complement

1. Single Representation for Zero: Only one representation (0000).
2. Simplified Arithmetic: Subtraction is performed by adding the 2's complement of a number.
3. Sign Bit: The most significant bit (MSB) serves as the sign bit (0 for positive, 1 for negative).

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Comparison Table:- Key Differences Between 1's Complement and 2's Complement

Feature	1's Complement	2's Complement
Definition	Base - 1 Complement	Base Complement
Representation	Invert all bits	Invert all bits, then add 1
Zero Representation	Two (positive and negative)	Single representation
Use in Modern Systems	Rarely used	Widely used

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Example of Signed Numbers (4-bit)

Decimal	Binary (1's Complement)	Binary (2's Complement)
+5	0101	0101
-5	1010	1011

In modern computing systems, 2's complement is the preferred representation because it simplifies hardware design and arithmetic operations.

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