1's & 2's Complement Calculator

Binary Complements

Binary complements are mathematical operations used to represent negative numbers in binary form. They are fundamental to computer arithmetic and digital system design.

1's Complement

Definition

The 1's complement of a binary number is obtained by inverting all bits - changing every 0 to 1 and every 1 to 0.

Calculation Method

1. Take the original binary number
2. Invert each bit (0 → 1, 1 → 0)
3. The result is the 1's complement

Example: 1's Complement of 1010

• Original: 1010
• Invert: 0101
• Result: 0101 (1's complement)

2's Complement

Definition

The 2's complement is obtained by adding 1 to the 1's complement. It's the standard method for representing signed integers in computers.

Calculation Method

1. Calculate the 1's complement
2. Add 1 to the result
3. The sum is the 2's complement

Example: 2's Complement of 1010

• Original: 1010
• 1's complement: 0101
• Add 1: +0001
• Result: 0110 (2's complement)

Signed Number Representation

Sign-Magnitude

Uses the MSB as sign bit (0=positive, 1=negative). Simple but has two zeros.

1's Complement Representation

Negative numbers are 1's complement of positive. Still has two zeros problem.

2's Complement Representation

Most common method. Negative numbers are 2's complement of positive. Single zero.

Range of Numbers

For n-bit 2's Complement

• Range: -2^(n-1) to +2^(n-1) - 1
• 4-bit: -8 to +7
• 8-bit: -128 to +127
• 16-bit: -32,768 to +32,767

Advantages of 2's Complement

Arithmetic Simplicity

• Addition and subtraction use same circuitry
• No special handling for sign bit
• Overflow detection is straightforward

Unique Zero

• Only one representation for zero
• Eliminates ambiguity in comparisons
• Simplifies hardware design

Applications

Computer Arithmetic

• Integer representation in CPUs
• Signed arithmetic operations
• Memory addressing
• Assembly language programming

Digital Signal Processing

• Audio and video processing
• Filter implementations
• Mathematical computations
• Control systems

Examples & Practice

Example 1: 8-bit Representation of -5

1. +5 in binary: 00000101
2. 1's complement: 11111010
3. Add 1: 11111011
4. Result: -5 = 11111011

Example 2: Convert 11110000 to Decimal

1. MSB = 1, so it's negative
2. 1's complement: 00001111
3. Add 1: 00010000
4. Decimal: 16, so original = -16

Common Pitfalls

Overflow

Adding two positive numbers that result in a negative, or two negative numbers that result in a positive indicates overflow.

Sign Extension

When extending bit width, copy the sign bit to maintain the same value (e.g., 4-bit 1111 becomes 8-bit 11111111).

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