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Just to review where we've gotten so far:
We have even numbers E and F, and since we know even numbers are 2 times some number, we let 2 times M equal E, and 2 times N equal F. (Equal here means that 2 times M is just another way of writing the number E; they may look different, but they are different ways of representing the same number.) So we can rewrite the sum E plus F.
We got this far, and I suggested that this is a chance to apply what we know about the Distributive Property.
Here's what I had in mind. The Distributive Property gives us permission to add first, then multiply the sum, or multiply first and then add the products. Whichever way we do it, whichever way helps us more with our particular problem, the answer stays the same.
So, we have:
and the Distributive Property gives us permission to say that:
Then we can string together all these things that are equal to each other, and leave out some of the stuff in the middle, and end up with:
Now if M and N are just some whole numbers, then the sum, M plus N, is just some other whole number. (The mathematical word for this idea is closure, and we say that the whole numbers are closed under addition).
This means that E + F equals 2 times some whole number. In other words, the sum of E plus F is an even number. That's what we wanted to prove, and we just did it!
Now along the way, we needed to decide what makes even numbers even (that an even number is equal to 2 times something), and we needed the Distributive Property, and we needed the closure of the whole numbers.
Those are the stones that we can build our arch with. Those are the stepping stones we line up to get across the stream. We just made the journey from point A (guessing that something might be true) to B (PROVING that something IS true). This is the essential and fundamental work of the mathematician, and we just did it. Now you give yourself a big pat on the back, and I'll stand here and cheer.
--- You mean that's it? That's a proof? That's not what the proofs look like in my math book.
Well, I did say that this was the scratch paper, remember? We could go back and make it a little neater.
But, I'll let you in on another mathematician's secret. First, you have to promise not to tell anybody, okay? A lot of proofs in math books really stink! Oh, most of them are technically correct, but they are written like Mozart's music, sparkling little gems with not a wasted note. Well, you know what, notes are cheap. Don't write your proofs in shorthand. Your job is to connect all those stepping stones clearly, so that someone else coming along behind you can follow. If you have a reason for your next step, then say so. This will make it easier for those who follow. It will also mean that you really understand your proof, and it will make it easier to spot any gaps you've left, that you need to fill in.
So, don't be afraid of writing too much. Don't write in shorthand. The proofs in a lot of math books are written by professors to other professors, to impress each other with how much they can leave out and still be technically correct. Those kinds of proofs never helped me learn mathematics.
Take a few moments to look at what we have just done. We have just proved that any even number plus any even number equals an even number. Now, this may not sound like much, but it's a big deal. In some advanced proofs in number theory, one of the first things we do is separate the odds and the evens. Just dividing the problem in half makes each piece that much easier to work with.
So, once we have proved this, it's proved forever. We put it in our pocket until we need it in some other problem, and then we can just plunk it down where we need a stepping stone. Andrew Wiles' proof of Fermat's Last Theorem was far longer, and far more complicated, but it started at point A, and Dr. Wiles lined up all the stepping stones he needed to get to point B. Many of these intermediate steps were already proved by other mathematicians, and Dr. Wiles had to prove some for himself as he went along. The process was bigger and more complicated, but it was the same process that we just went through.
--- So, what's next, Spud?
Practice. Practice. Practice. Writing proofs is a lot like playing the piano, or ballroom dancing. Once you learn the basic steps, or notes, you practice new ways of combining them.
So, go back to those examples we started with. What happens when we add an odd number to an odd number? Take a guess at what happens, and then see if you can prove it. If you send me your proof, I'll publish it here, and send you a gold star.
Meanwhile, there are a few other dance steps I want to show you. Click on the link below when you're ready to continue.
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