Lesson Two: In Which We Add

So far, whenever we have changed whole numbers we have done so by the smallest amount possible, incrementing them 1. However, it is not entirely without precedent that two largish groups of whole numbers should collide and need to be combined. When this happens it would be convenient if we didn't have to do so each 1 at a time. This is the problem which we create addition to solve.

Addition and the Next Number Generator:

Given what we already know, the easiest to define addition is to use the Next Number Generator. Any number plus one is the next number, so 1+1 is 2 and 2+1 is 3. Notice that since 1+1 and 2 are the same thing and 2+1 and 3 are the same thing, 3 must also be 1+1+1. Using the Next Number Generator in this way, any whole number can be considered as an addition of units. Thinking about whole numbers this way means that we already know how to add them together.

Addition:

Consider 3+4, while you may know that it is 7, we do not yet know exactly what 3+4 means. However, we do know that 3 means 1+1+1, and that 4 means 1+1+1+1, so 3+4 must also be 1+1+1+1+1+1+1. Aha, that we know the meaning of! Start with the unit, then find the next number six times, so 3+4 must be 7. Hmm, but if you look at 4+3, converting it to 1's you also get 1+1+1+1+1+1+1 so it looks like 4+3 is the same as 3+4. In fact, adding together any two numbers will be the same, no matter which order you add them in, because of this we say that addition is commutative.