Last week, in Fundamental Understanding of Mathematics VII we began an exploration of the number line, and determined that it was useful for both counting and measuring. Counting and measuring both depend on two special spots on the number line, zero and one, because these two locations define the entire rest of the line.
Zero is special. It's the only number on the line that doesn't have an opposite anywhere on the line. It's right in the middle. Take some other number, a non-zero number, call it n. There's another number the same distance from zero, except in the opposite direction, call it - n. n and - n are in different places on the number line, they are different numbers.
That's not the case with zero.
Zero is zero distance from zero.
Let's suppose you had a positive zero, so you move to the right along the line, starting at zero, and go until you've gone zero distance. Where are you? At zero. Do the same thing in the opposite direction, and where do you get to? Same place. Zero. Not even infinitesimally close to zero but just barely on the left side of zero. Sorry. You haven't moved a bit, despite your initial intention to move to the right for a positive move or to the left for a negative one.
Not moving is not moving. You're at zero. There's only one spot on the number line, no opposite number. This fact gives zero a very special property. I can add zero to any number, and not change that number! This is called the additive identity, since it deals with addition. In symbols:
a + 0 = a
It really doesn't look like much. Adding nothing to a number doesn't change it. Big deal. So what. I'm bored already. Can we move on to something interesting?
Ooo! Ooo! I know! Subtracting nothing from a number doesn't change it either!
It's like a magician pulling an invisible rabbit out of his hat. You can't see the rabbit, so the trick really doesn't seem very impressive. Just wait! When we start solving equations, that ability to pull invisible rabbits out of a hat will prove powerful enough to solve half our algebra problems. Damn, that's some rabbit! I'd call him Harvey, but he already has a name, and his name is Zero, the additive identity.
So, Zero defines the middle of our number line. Since a line, geometrically speaking, goes forever in either direction, any spot we choose is automatically in the center of the line, having exactly an infinite length on either side. When we draw the line on paper, though, we don't have enough room to draw for an infinite distance on either side of zero, so, we put zero where ever it's convenient. Sometimes off to the left side of the paper, sometimes near the center, sometimes off to the right. But where ever we mark it, once we label the mark "zero," we know that spot is the middle of the number line.
But a line with a zero mark in the middle isn't very useful, neither for counting nor for measuring. In order to make the line useful, we need to decide on the length of one interval.
Like the location of zero, the actual length of this interval is up to us, and, also like zero, once we've decided, we can't change our mind.
We use this distance, called the "unit" distance, to lay out the regular tic marks seen on most number lines. But, initially, we show the distance we've chosen by marking the number 1 on the number line.
The space between zero and one is the unit distance for our number line. And that's all we need to completely define our number line. All the decision making is done, the rest is mechanical -- we could build a machine to do it. In yardstick factories, that's exactly what we've done.
One also has some special properties that make it a very powerful number for solving equations. But we'll save that for later. This diary is getting long enough.
Have fun in the comments.