Arithmetic Logic with Adders and Subtractors

You will construct half and full adders to perform binary arithmetic and cascade them to create multi-bit ripple carry adders.

In this lesson, you will learn how to harness basic logic gates to perform binary addition, the fundamental operation behind all computer calculations.

The Half Adder: Adding Two Bits

When we add two single binary digits (bits), we encounter two possible outcomes: the sum and a carry-out. For example, 1 + 1 equals 0 with a carry of 1.

A Half Adder takes two inputs, A and B, and produces two outputs: Sum (S) and Carry (C). Using the logic gates we’ve mastered:

Sum is an XOR operation: S = A ⊕ B
Carry is an AND operation: C = A · B

This works for the first column of an addition, but what about columns further to the left that need to include a carry from a previous step?

The Full Adder: Incorporating the Carry

To handle three inputs (bit A, bit B, and an incoming carry Cin), we use a Full Adder. This is essentially two Half Adders joined together. It calculates the sum of three bits, resulting in:

Sum = A ⊕ B ⊕ Cin
CarryOut = (A · B) + (Cin · (A ⊕ B))

The Ripple Carry Adder

To add multi-bit numbers—like two 4-bit integers—we "cascade" these adders. We connect the CarryOut of one Full Adder to the CarryIn of the next.

       C3      C2      C1      C0 (Cin)
       |       |       |       |
[FA3]--[FA2]--[FA1]--[FA0]

This is called a Ripple Carry Adder. It is intuitive but has a physical limitation: the carry signal must "ripple" through every stage before the final sum is accurate. If you want to subtract, we use a trick: by inverting the bits of the second number (using NOT gates) and setting the initial CarryIn to 1, we can turn our adder into a subtractor using two's complement arithmetic!

Key Takeaway: By cascading Full Adders, we create a Ripple Carry Adder that can process multi-bit numbers, forming the arithmetic backbone of your computer's processor.