Last week, I took a bit of a pedagogical break, and Fundamental Understanding of Mathematics XI was about doing a bit of math, ie: problem solving. A few weeks before that, in Fundamental Understanding of Mathematics VIII, I wrote about the additive identity, zero, and how special a number that was. It turns out there's another way of looking at zero, which, if we have counting numbers, gives us negative numbers. Rather than taking some number, adding zero, and getting the same number, instead we will take some number, add a second number and get zero.
I think this is the first time we look at the "existence" of some particular number. It's like an if/then statement:
If a is some number, then there is some other number, call it b, such that a + b = 0.
Now, b is some particular number, not just any number. It is the opposite of a. By definition, opposite numbers add up to zero. By saying b is opposite of a we simply mean a + b = 0.
One way to think about these opposite numbers is using a number line. Let's make a some counting number, say, 3. 3 is represented on a number line by a line segment, three units long, starting at zero and extending to the right. If we are going to add some number to 3 and wind up back at zero, it should be obvious that the new number must be the same length, but it must point, not to the right, but to the left. It points in the opposite direction.
We call that number: negative three.
This property, that there is, or exists, the number negative three to be the opposite of three, is called the Additive Inverse property.
This property comes in very handy when solving equations or working with adding or subtracting positive and negative numbers.
One thing we know about equations is that the stuff on the left side of the equal sign is the same value as the stuff on the right side of the equal sign. A couple of weeks ago we saw that we could add zero to (or subtract zero from) a number and not change its value. At the time, this didn't seem too exciting, but the Additive Inverse Property lets us play around with the zero that we are adding to one side or the other.
Zero might be (+3)+(-3). Or (+172)+(-172). Or (+a)+(-a). All of those additions of opposite numbers add up to zero. There's even a special name for them: Zero Pairs.
Suppose you have an equation:
x + 17 = y
and you want to find x. You can plop a (+17)+(-17) Zero Pair onto the right side of the equation, and be confident that you haven't changed the value, since all you've done is added zero to that side.
x + 17 = y + (+17)+(-17)
But now you have a +17 on both sides of the equal sign, and you can remove them both without affecting the balance, both sides will still have the same value (although it will be a different value than the one you started with.)
x = y +(-17)
Most people who are good at algebra will look at this and say, well, you're just subtracting the same thing from both sides, and, this is true. But if you've ever wondered why that rule worked, the additive inverse and additive identity properties are the reasons behind it. You're just adding zero, but you can make that zero look almost any way you want it to look. Cool.
Have fun in the comments.