Early computers used sign-and-magnitude to represent signed numbers:

• Leftmost bit = sign (0 = +, 1 = –)
• Remaining bits = magnitude

+5 = 0000 0101     -5 = 1000 0101

But this system had major problems:

1. Two zeros → 0000 0000 (+0) and 1000 0000 (–0)
2. Addition doesn’t work naturally – you need special rules depending on the signs
3. Comparison is messy – negative zero creates confusion

To fix this, computers use two’s complement, which:

• Has only one representation of zero
• Makes addition and subtraction work with ordinary binary circuits
• Represents negative numbers by inverting the bits and adding 1

To calculate the negative value of a number:

1. invert (flip) all the bits
2. and add 1 to the resulting number.

This process converts the positive number into its negative counterpart.

Notes

• when you add one to the number you perform a carry just like in normal binary addition.
• Ignore any binary overflow from beyond the digits you use
• A positive number needs to begin with a zero before conversion, so if it doesn't start with zero you will need to add a zero to the left.

1. Simplicity

Two's complement is a simple method of representing both positive and negative numbers using binary digits. It allows for easy arithmetic operations and logical manipulations.

2. Range of Representation

With two's complement, a wider range of values can be represented using a fixed number of bits compared to other number systems. This allows for efficient storage and manipulation of data.

3. Efficiency in Circuit Design

Two's complement is widely used in digital circuit design for arithmetic operations in processors, microcontrollers, and other electronic devices. Its efficient representation of negative numbers simplifies circuit design and reduces the complexity of hardware implementation.

4. Compatibility

Two's complement is compatible with various arithmetic algorithms and logic circuits, making it suitable for use in both hardware and software implementations. It is widely supported in computers, calculators, and digital systems.