Binary Adders

Within these castle walls be forged Mavens of Computer Science ...

— Merlin, The Coder

Half Adder

Prerequisites

• W1011 Number Systems
• W1012 Alternative Base Addition
• W1013 Boolean Algebra
• W1014 Logic Gates
• W1015 Bitwise Operations
• W1016 Logic Composition

Introduction

One of the most fundamental operations performed by computers, aside from the logical operations that we've already discussed, is the arithmetic operation of addition.

Half-Adder

Let's consider what's required to add two, single-bit binary integers. We'll need one bit to represent the sum of the integers, and another to handle the carry. Representing this in the form of a truth table yields:

Inputs		Outputs
$a$	$b$	$C$	$S$
0	0	0	0
0	1	0	1
1	0	0	1
1	1	1	0

This is formally termed a half-adder, a logic circuit capable of adding two bits.

Observe

Observe, Ponder, and Journal: Section 1

1. What truth table do you recognize that produces the output of the Carry column?
2. What truth table do you recognize that produces the output of the Sum column?

Full-Adder

In order to add two single-bit binary integers PLUS a carry, we need an adder capable of adding three single-bit binary numbers. Again, we'll need one bit to represent the sum of the integers, and another to handle the carry. Representing this in the form of a truth table yields:

Inputs			Outputs
$a$	$b$	$c$	$C$	$S$
0	0	0	0	0
0	0	1	0	1
0	1	0	0	1
0	1	1	1	0
1	0	0	0	1
1	0	1	1	0
1	1	0	1	0
1	1	1	1	1

This is formally termed a full-adder, a logic circuit capable of adding three bits.