Last week, in Fundamental Understanding of Mathematics XV we took a look at solving an equation by keeping it balanced while we manipulated it using some properties developed in earlier Fundamental Understanding of Mathematics diaries.
"There is no try," says Yoda, "there is only do, or not do."
Updating Yoda to the 21st century, we shall say that there is do, and undo.
This week we are going to venture away from addition and multiplication by considering inverses.
Undoing something you've done on a computer takes you back to the way things were before you did whatever it was you want to undo. In mathematics, undo is called inverse, and it also gets you back to where you were before. Unlike dealing with computers, though, using an inverse operation does not imply you did something wrong that you need to fix. In fact, it isn't even necessary to do the thing in the first place. You can use an inverse operation all by itself.
Let's get specific. Subtraction is the inverse of addition. If you add three pebbles to the pot, taking three pebbles out of the pot will get you back to where you were before you added the three pebbles. Subtraction undoes the addition.
Let's try that the other way around. Take three pebbles out of the pot (we assume the pot starts, in both examples, with some number of pebbles in it.) We can then undo that subtraction by adding three pebbles to the pot. Addition is the inverse of subtraction.
Addition and subtraction are inverses of each other. We may like to think that addition comes first, and subtraction is its inverse, but if we start with subtraction, then addition is its inverse. That's why it makes sense that we can use either one as a "stand alone" operation, without have to do the other one first.
It's a chicken and egg problem -- which one came first? Mathematicians say: neither came first. One is simply the inverse of the other, so use whichever makes sense for the problem at hand, without worrying about whether you're going to have an omlette or chicken soup.
One definition of subtraction holds that
a - b = a + (-b)
Subtraction is simply the addition of the subtracted number's opposite. Subtracting b is the same as adding negative b.
We know that a zero pair is any number added to it's opposite, and the sum is zero
b + (-b) = 0
We also know from the identity property for addition that we can add zero (or a zero pair) to anything without changing it.
a - b + b + (-b) = a + (-b)
I've taken the definition of subtraction, and added the zero pair for b to the left side of the equation. I get to do this without doing anything to the right side of the equation because I am actually adding zero, which does not change the balance of the equation.
Now begin on the left side of the equation: we have a pebbles in the pot. Then we remove b pebbles. Then we put b pebbles back in the pot. It should be clear that a - b + b is just a by itself. So we can make a substitution in the equation.
a + (-b) = a + (-b)
Now, we already know that something is equal to itself, which is exactly what we have here. So this last equation is true. Since we got to the last equation by making valid changes to the original equation (the definition of subtraction) we can see that the definition is true as well.
I'll let the usual suspects argue whether this proof is informal, semi-formal or formal in the comments. Have fun! :)