Why are complements (such as 1's and 2's complement) used in digital number…

Complements simplify arithmetic operations in digital systems by enabling subtraction through addition. They also allow for a unique representation of zero and simplify hardware design in computers.

Note for more understanding

In digital electronics, manufacturing space on a silicon chip is precious. Building a circuit that can perform addition requires logic gates configured as Full Adders. If a computer had to perform subtraction the way humans do (borrowing from the next column), engineers would have to design and build an entirely separate, complex circuit called a Full Subtractor.

Complement arithmetic elegantly solves this hardware bloat through a mathematical trick:

1. Transforming Subtraction into Addition: Mathematically, subtracting a binary number B from A is identical to adding a negative B:

   A - B = A + (-B)

3. The Role of 2's Complement: In a 2's complement system, a negative number is represented by inverting all its bits (which is the 1's complement) and adding 1.

4. Hardware Efficiency: Because of this representation, when the processor needs to execute a subtraction instruction, it can simply route the second operand ($B$) through an inverter (NOT gates) to flip the bits, set the carry-in bit of the initial adder to 1 (completing the 2's complement transformation), and run it through the exact same Arithmetic Logic Unit (ALU) addition hardware.

This single unified design reduces the number of transistors required on a chip, lowers power consumption, and speeds up computation times.