Lesson Thirteen: In Which We Learn Our Place

Up to this point, everything we have talked about has been true of mathematics in general. If you construct numbers in the same way as we have, all things that we have found to be true will still be true. However, for a moment I must take a quick side trip into an idiosyncrasy of our particular number system.

The important fact in this discussion is that we have ten symbols that we use to represent our numbers. They are 0,1,2,3,4,5,6,7,8, and 9. That means that we can count from zero to nine just fine and dandy, but if we want to represent the number ten we must innovate. What we did was invent the ten's place.

The Place System:

The numbers 0-9 represent single things, so 4 apples means that you have four individual apples. What happens when you put another number to the left of the single things is that it represents sets of ten. So if you have a dozen apples, you have 12 apples, which means one set of ten apples, represented by the 1 in the "ten's place," and two individual apples, represented by the 2 in the "one's place." If you happen to be 25 years old, it means you have been alive for two sets of ten years and 5 individual years.

Of course, if you have nine sets of ten and nine individual things, 99, and then you get one more, you have ten sets of ten. Since we do not have a symbol for ten of something, we need to expand our number to the left and create the "hundred's place." Thus 100 represents one set of a hundred, no sets of ten, and no sets of individuals. When we need to represent ten sets of hundreds, we add a new place for thousands, and so on and so forth, allowing us to represent any integer, no matter how large, using our ten symbols.

Base:

Because we have ten symbols, our number system is called base 10. All this means is that we write all our numbers using ten basic symbols. If you are familiar with how computers work, you may have heard of base 2. As you might imagine, base 2 means that every number is represented with two symbols. Zero is still 0 and one is still 1, but if you want the next number, you have run out of symbols already, so you need a new place. Thus the sequence 10 in base 2 represents the number 2, because it is one set of two and no individuals.

Historically there have been variations on what number is the base. Nowadays base 10 is used for mathematics, and most societal calculations, while base 2 and base 16 are used by some computer scientist. How would you express the number nine in base 6?

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.