Lesson Four: In Which Less Than Nothing Happens

Continuing our example from Lesson Three, suppose the school day ends and students begin leaving. We are now faced with the challenge of making the total number of students at the school smaller. Suppose just one student leaves, when she/he arrived we added 1 to the total number of students at the school. Now we need a number that we can add to undo adding 1 earlier. To address this problem, we introduce the negative numbers.

Negative Numbers:

Negative one, which we write -1, is the number that we add to something when we want to undo adding 1 to it. This means that 1+(-1)=0 since the -1 cancels out the 1 so nothing should happen, and the way we write nothing numerically is 0. Since 2=1+1, if we want to undo 2 we need to use (-1)+(-1) which we call -2, or negative two. Just like we could use 1 repeatedly added to itself to form the whole numbers, we can build the negative whole numbers by repeatedly adding -1 to itself. Should we encounter 3+(-2) we might pause momentarily, for there is no 2 for the -2 to undo, but if we write everything in terms of 1 and -1 things are resolves. We obtain 1+1+1+(-1)+(-1) which allows two of the 1/-1 pairs to be cancelled, leaving only 1. Since adding by 0 changes nothing, -0, which undoes addition by 0, also changes nothing. Thus -0=0, since 0 is our symbol for nothing changing.

"Subtraction":

Assuming this is not your first time learning math, you probably have heard of something called subtraction. Subtraction is a way to make numbers smaller without needing to introduce the negative numbers. Basically you have two choices on how to make things smaller, introduce negative numbers and leave everything as addition, or leave everything as whole numbers and introduce subtraction. Subtraction has two very big problems. The first is that the nice properties of associativity and commutativity that addition has are not true for subtraction. That is, 3-(2-1) is not (3-2)-1, and 3-2 is not 2-3. The second problem is that it is perfectly natural to wonder, once the notion of subtraction is on the table, what IS 2-3? Then, of course, we must introduce the negative numbers after all, making all the inconvenience we went through to keep them out of things wasted effort. This is our first real lesson from thinking about math systematically and logically, subtraction is an inconvenient lie!

Integers:

Just like the numbers you get by adding together 1's has a special name, the whole numbers, the numbers you get by adding combinations of 1's and -1's has its own name, the integers. The integers are made up of the whole numbers, zero, and the negative whole numbers. Please feel free to explore for yourself that you can make all of these by adding 1's and -1's. Slightly more tricky is that these are the only things you can make by adding 1's and -1's, but it is true, and I invite you to ponder upon it.