THE HALF ADDER
by Isai Damier

Introduction

The Half-Adder is the basic building block of all arithmetic circuits. Every microchip or machine that can perform addition, subtraction, multiplication, or division has Half-Adder blocks inside. Because the Half-Adder is so material to our ability to do math on computers, this article will show you how to design the Half-Adder from scratch.

DESIGN

There are three steps to designing a digital circuit:

Step 1.   Convert the problem description into a truth table.
Step 2.   Find a switching function defined by the truth table.
Step 3.   Use the logic of the function to build the circuit.

Truth Table

If you think back a little, you will remember that back in grade school you did a lot of homework converting English statements into algebraic expressions. As an instance, for Angie is eight years older than Dina, you had to write
A= 8+D. Converting a problem description into a truth table is a lot easier than that.

For our purpose, we need to design a circuit that is capable of adding two single-bit binary numbers. Hence, our circuit should be able to perform the following four possible calculations:

0		0		1		1
+ 0		+1		+ 0		+ 1
0		1		1		10

And this line of four operations is essentially your truth table! All that's left is to make it look like a table, with labels. So first you rotate the line of operations from the current horicontal position to a vertical position, and then you label each column. In addition, since one of the result is two bit, we turn all the answers into two-bit format —like below:

A	B		RESULT
0	0		00
0	1		01
1	0		01
1	1		10

Now for the finishing touch. Every bit in a truth table must have its own label. So you must split the result column in two and label the first bit C and the second bit S. (Of course you can label them whatever you want!)

A	B		C	S
0	0		0	0
0	1		0	1
1	0		0	1
1	1		1	0

Voila! This is the Half-Adders's truth table!

Switching Function

We label each bit of the truth table to make it easy to use the labels to create a switching function. Instead of trying to solve for two or more bits at once, we can solve one bit at a time. For example, notice that the relationship between inputs A and B and output C describes an AND gate. This quick observation is easy because we label the leading bit C. Notice further that the relationship between the inputs A and B and output S describes the XOR gate. Hence, the compound function that defines the Half-Adder is

C = A • B
S = A B.

Of course, extracting a switching function from a truth table is not always so easy. So we will illustrate a more systematic approach. We start by pretending that you don't know what the XOR truth table looks like (it's on the XOR gate page). Consequently we will form the S bit switching function as follows: for every row where S = 1, we will write the relationship between the inputs A, B and the S bit—like below.

A	B		S	S
0	0		0
0	1		1	A • B
1	0		1	A •B
1	1		0

Then we add the terms to get S=A • B + A • B. If you checkout the XOR gate page, you will learn that
A B= A • B + A • B.

Circuit

Based on the compound function C = A • B and S = A B, we have two input signals (A, B), two output signals (C, S) and we need two gates: an AND gate and an XOR gate. Go back to the top of this page to see the Half-Adder.