Last week, in Number Sense 013, we took a look at a basic property of addition: commutativity. The ol' switcheroo. A + B is the same as B + A. This week, we will explore a particular number on the number line: zero.
Zero occupies a unique place on the number line, and has some properties no other number has because of that position.
When we built the number line, zero was the first number we put on it. Later, when we defined number, we said that the numbers' value was the distance of that number from zero on the number line. So, how far away is zero from itself?
We can also show the value of a number by a bar where the length of the bar is the distance on the number line:
As the numbers get smaller, the bars get shorter. Can we model zero this way? With a very, very short bar?
After all, that's how we show zero on a number line...
Unfortunately, that won't work. If we look at that very very short bar under high magnification, we would see something like this:
No matter how short we make that bar, when we look at it closely, we see it has a non-zero length.
If we want to show number values as bars, it really would look something like this:
No length at all. Not even a very short one.
But if it blows up, doesn't that mean it's not a number? If you recall our definition of number, we said a number was simply a point on a number line. Zero is definitely a point on a number line, so it's still a number. It has the no length property because of our definition of a numbers' value: the distance from zero.
Since it has no length, no value, we can do an interesting thing with it.
7 + 0 = 7
5 + 0 = 5
Unknown number + 0 = same number (still unknown) (aka x + 0 = x )
Adding zero to a number does not change that number's distance from zero, does not change it's value, leaves you with the same number after you finish the addition. (This, by the way, is why I don't like to call unknown numbers “variables.” Variable means the value changes. When solving problems involving an unknown number, the value of that unknown number does not change.)
There is, of course, a name for this zero behavior. It's called the additive identity. Zero is sometimes called the identity element for addition. It all means the same thing: adding zero gives you back the number you started with. The same identical number.
Have fun in the comments.