Addition, from scratch

So we need to produce two answers: the carry and the sum. Let's find a gate for each.

The carry first. It should be 1 only when both inputs are 1 (that's the 1+1 case). We have already met a gate that does exactly that: AND.

Now the sum. Look at when it should be 1: for 0+1 and 1+0, but not for 1+1. It's almost OR, except OR also fires on 1+1, and we don't want that.

So we use one new gate, XOR. It means "one or the other, but not both." 0 and 1 gives 1. 1 and 1 gives 0. That is the sum column, exactly.

Wire AND and XOR to the same two inputs, and you have a circuit that adds two bits.

Two gates. That's addition, built from nothing but logic.

Addition, from scratch

So we need to produce two answers: the carry and the sum. Let's find a gate for each.

The carry first. It should be 1 only when both inputs are 1 (that's the 1+1 case). We have already met a gate that does exactly that: AND.

Now the sum. Look at when it should be 1: for 0+1 and 1+0, but not for 1+1. It's almost OR, except OR also fires on 1+1, and we don't want that.

So we use one new gate, XOR. It means "one or the other, but not both." 0 and 1 gives 1. 1 and 1 gives 0. That is the sum column, exactly.

Wire AND and XOR to the same two inputs, and you have a circuit that adds two bits.

Two gates. That's addition, built from nothing but logic.