A binary adder is a digital circuit that performs the addition of two binary numbers. In binary representation, numbers are expressed using only two symbols: 0 and 1. Binary addition follows similar rules to decimal addition, with a few differences due to the base-2 system. There are different types of binary adders, but we'll focus on a simple yet commonly used one called a "half adder" and then expand it to a "full adder."
Half Adder:
A half adder takes two binary inputs, A and B, and produces two outputs: the sum (S) and the carry (C). The carry output indicates whether there is a carry-over to the next higher bit position.
Truth table of a half adder:
A B Sum (S) Carry (C)
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
As you can see, the sum (S) is the XOR (exclusive OR) of A and B, and the carry (C) is the AND of A and B.
Full Adder:
A full adder is an extension of the half adder, but it also considers an additional input, the carry-in (Cin), from the previous lower bit position. It generates two outputs: the sum (S) and the carry-out (Cout) to the next higher bit position.
Truth table of a full adder:
A B Cin Sum (S) Carry-out (Cout)
0 0 0 0 0
0 0 1 1 0
0 1 0 1 0
0 1 1 0 1
1 0 0 1 0
1 0 1 0 1
1 1 0 0 1
1 1 1 1 1
The sum (S) output is now determined by the XOR of A, B, and Cin, while the carry-out (Cout) is generated by any two of the three inputs (A, B, Cin) using the majority function (sum of at least two 1's).
Cascading Full Adders:
To add multi-bit binary numbers, you can cascade multiple full adders together. Each full adder takes care of adding individual bits and propagates any carry to the next higher bit position.
In this way, binary adders can perform arithmetic operations on binary numbers, enabling digital computers to execute calculations and perform complex tasks using binary arithmetic and logic.