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.

Lesson One: In Which We Create Some Numbers

Before we do some mathematics, most of you would probably be more comfortable if we had some numbers with which to do it. All the numbers that we will need for now can be created using two simple concepts, a unit, and a next number generator.

The Unit:

If you are forced to do math by some circumstance of life, odds are that the numbers you are using represent some real world objects, be they cars, meters, pizza boxes, or people. The unit, which we denote by '1', represents a single object. In some sense, it is the smallest amount of a thing that you can have. If you have a parking lot full of cars, you can use your bulldozer to push some off a cliff and still have cars left. But, if you have 1 car and you use your plasma cutter to get rid of some of it, you no longer have any cars left, although some of it may remain, if it is less than 1 car it is no longer a car, in Oregon we call such things redneck flowerpots.

When 1 isn't out representing some real world object, it will perform a similar duty for our numbers. That makes 1 the smallest whole number that you can have, cut it up for scrap and you no longer have a whole number, you have something else.

The Next Number Generator:

Since 1 is the least amount of a thing that you can have, the next smallest amount of a thing that you can have is two 1's, which we denote as 2. Consider, if you have a room with 1 person in it and you want to have more people in the room, you must have at least 2 people in the room. Anything less and you don't have more people in the room, you have 1 person and an organ donation, which is something entirely different than a person. So, since 1 is the smallest amount of whole number we can have, the next smallest is 2, and we say that 2 is the next whole number after 1.

In fact, if we have any whole number, we can arrive at the next whole number by increasing our amount by 1. This is all that the Next Number Generator does, it takes a whole number and makes the next one by increasing it by 1, the smallest amount that it can be increased.

The Whole Numbers:

I while back I casually italicized the term, "whole number," assuming that you had a notion what those might be. However, here is a nice quick way of defining them in terms of things we already understand. Starting with our unit, 1, and using the Next Number Generator to make other numbers, we can make all the numbers we shall need for now. Any number that can be made in this way shall be called a whole number.