Last week, in Number Sense 014, we saw that zero had an interesting property, we can add it to anything and get the same number. Zero is the identity element for addition. At first glance, it seems obvious, and not too useful. Let's see what we can do with such a property.
Here is our number line, with zero on it.
That's one way of looking at it. Here is another:
I've taken a number, in this case 4, and added it to it's opposite number, -4.
As with our tightrope walking goat, we end up where we started: at zero. But hold on! How do we know our goat started at zero? What would happen if the goat started somewhere else?
Let's suppose he started walking at some not zero spot on the number line. Call the spot a. He walks in one direction for a certain distance, turns and walks back the same distance.
Our goat winds up back at spot a! So, what just happened? We started at a, which, since it is on the number line, is some number. The goat's journey represents two opposite numbers, added together. The result of this adding the goat's journey is that we get back to a, the same number.
So, a number + the goat's journey = the same number. The goat's journey is the identity element for addition, in other words, zero.
The goat's journey, a number plus it's opposite, is zero.
This then, is another way to think of zero, as a sum of opposite numbers. Zero, then, instead of being nothing, can be the sum of a whole lot of something.
Let's make some practical use of this idea. First, we will use a standard size arrow. 1 is a convenient length when dealing with integers, so our arrows will all be length 1. Here are a number of them.
Second, we will use color to tell them apart, rather than use the arrowhead. Green will mean +1 and red will mean -1. We no longer need to draw tedious arrows, we will replace them with simple circles, red and green.
Finally, since we've decided to make each circle exactly one unit (either positive or negative) we really don't need the number line any more.
Here, then, is a collection of +1s and -1s that add up to zero. It's a bit easier to see if we rearrange them slightly:
This is a picture of zero, as a whole lot of something. The simplest picture we could draw would be one of each:
And we call this a “zero pair.” A +1 and a -1, adding up to zero.
We can use these to do some arithmetic. Suppose we want to add 3 + 4
Put three greens on the table, representing +3.
Put four more greens on the table. Then count: 3+4=7
Let's do a subtraction: 3 – 4. We begin as before, putting three greens on the table.
But now, we have a puzzle. We want to subtract four greens, that is to say, remove four greens from the table, but we only have three on the table. Zero pairs to the rescue. Besides the three greens on the table, there is nothing. Nothing is zero. Zero can also be a zero pair. Our table could look like this:
And now it is easy to remove four greens from the table.
And we are left with a red, or -1, on the table. So 3 – 4 = -1. We can subtract large numbers from smaller numbers!
Have fun in the comments.