Gates and the Adder

What this is

A truth table lists the output for each combination of 0 and 1 inputs. A gate implements a rule such as AND, OR or XOR. Join those rules and we can describe something that adds binary numbers.

By the end, you can trace a one-bit adder and follow its carry through three columns. The interactive drawings use an ideal Boolean model; understanding that model is a useful step towards understanding physical hardware.

Who it’s for

Anyone who has ever wondered what a processor is.

Not what it does, which you can read anywhere. What it is made of, at the bottom, where the explanations usually stop and say “and then it adds the numbers”.

What you need first

You will use 0/1 values, small sums and binary place values: 1, 2, 4, 8. We recall the rules as they appear. Truth Tables and Counting in Twos offer more practice; prior attendance is not required.

This gates nothing in turn. Nothing later in the curriculum requires it. It is here because it is one of the most satisfying things a beginner can learn, and because after it the word “hardware” stops sounding like a different subject.

The shape

• The same table, drawn — a gate’s Boolean rule, drawn with wires. Set the inputs and watch the output follow.
• The half adder — add two bits and you need two answers, a sum and a carry. One is XOR and the other is AND. That is the entire circuit.
• The full adder and the ripple — accept a carry as well as produce one, then chain them together and add whole numbers.

The model we use

We treat inputs and outputs as settled 0 or 1 values. These drawings do not measure voltages, power or the time an output takes to settle. A physical part needs electrical and timing specifications as well as a Boolean table.

BASIC and assembly remain independent entry points. Use this module when you want to look inside an arithmetic operation; completing it is not a condition for starting a system track.