The full adder

The missing input

The half adder from the last lesson adds two bits and can send a carry out to the left. What it cannot do is take a carry in from the right. Think of adding 19 and 13 on paper: the ones column makes 12, so you write 2 and carry 1, and the tens column then has three things to add, 1 and 1 and the carried 1. Every column except the rightmost works like that. A circuit that can sit in any column needs three inputs.

Three inputs make eight combinations, so the table has eight rows. Adding three bits gives a total of 0, 1, 2 or 3, and the two outputs are just that total written in binary: 0 is 00, 1 is 01, 2 is 10 and 3 is 11. The left digit is the carry out and the right digit is the sum. Click through the rows and watch the total.

Two rules that describe the table

Look at the sum column. It is 1 whenever an odd number of the inputs are 1: one input, or all three. That is what you get by XORing all three inputs together, a ⊻ b ⊻ c, because XOR flips the answer every time it meets another 1. Now the carry out column. It is 1 whenever the total is 2 or more, in other words whenever at least two of the three inputs are 1. That rule has a name, the majority: the output agrees with whatever most of the inputs are doing. As gates it is "a and b, or the carry in and exactly one of a and b", which the reference page writes as (a ∧ b) ∨ (c ∧ (a ⊻ b)).

Building it from two half adders

You do not have to design the full adder from scratch. The neat way is to use the chip you already made. The first half adder adds a and b, giving a first sum and a first carry. The second half adder adds that first sum to the carry in, giving the final sum and a second carry. The final carry out is the first carry OR the second carry. That is the whole circuit: two half adders and one OR gate.

Count the gates and it comes to two XORs, two ANDs and one OR, five in all. Whether you draw it as five gates or as two half adder chips and an OR, it is the same truth table, and the truth table is the thing that matters.

Build it

In the simulator, place two of your half adder chips and an OR gate. Wire toggles a and b into the first chip. Wire its sum output and a third toggle, the carry in, into the second chip. The second chip's sum is the full adder's sum. Wire both chips' carry outputs into the OR, and the OR's output is the carry out. Check all eight rows against the table above, then package the whole thing as a chip called "full adder". In the next lesson you will place four of them in a row.

What to remember

• A full adder adds three bits: a, b and the carry in from the column to its right.
• The sum is a XOR b XOR carry in: 1 when an odd number of inputs are 1.
• The carry out is 1 when at least two of the three inputs are 1, the majority rule.
• Two half adders and an OR gate make a full adder; the two carries are never both 1.