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Just to review where we've gotten so far:
We started by looking at sums of even and odd numbers in different combinations. I suggested that it seems that an odd number plus an odd number equals an even number, an odd number plus an even number equals an odd number, and an even number plus an even number equals an even number.
Since I can only prove one thing at a time, I started off by claiming that the sum of any even number added to any even number is an even number.
To make a mathematical statement that covers every even number I checked the definition, and then, after stating the claim in plain language, I introduced some symbols.
I let E and F be even numbers, and M and N be whole numbers, so that E equals 2 times M and F equals 2 times N. Then I rewrote the statement this way:
Let's stop here for a minute, because this simple statement is really quite incredible. This is an example of the power of mathematical abstraction. I don't know what number E is. It could be any even number. That is the only restriction we're putting on E, that it's even. But, once we say that E is even, we know that E is 2 times something. This is how I can make a statement about all the even numbers without having to write them all out. I use a representative. This idea is incredibly powerful, and is the foundation of the algebra we study in school.
To finish laying out the symbols, I need a couple more. Let G be another whole number. Then I can say:
which says in symbols that the sum of one even number plus another even number equals a whole number.
The keystone here is that I'm claiming that G must be even. That is, G equals 2 times some whole number.
Now it's time to apply what you already know. Have you learned about the distributive property yet? If so, you can skip this part. Otherwise, we need a little more background information.
Don't let the fancy name scare you. All we're talking about here is a simple fact about how addition and multiplication fit together when we're doing common, everyday arithmetic.
Suppose I want to add 3 plus 4, and multiply that sum by 5. Well, I can add first, and then multiply. That is, the sum of 3 plus 4 is 7, and then 7 times 5 is 35. We can dress that up like this:
What the distributive property says is that instead of adding first, and then multiplying, we can multiply first, and then add. I can say 3 times 5 is 15, and 4 times 5 is 20, and then 15 plus 20 is 35. We get the same answer!
The Distributive Property says that this is always true when we do our usual addition and multiplication with our usual numbers. (When you study abstract algebra, you will find that the distributive property is true with lots of unusual addition and multiplication with unusual numbers.)
Now when you see this in your algebra book, it will probably look something like:
If that looks like just a bunch of squiggles, go back and work out an example, like we just did with (3 plus 4) times 5. If you get familiar with the Distributive Property, and recognize when you can (and cannot) use it, you can make many problems easier to figure out.
So, back to our odds and evens. We have even numbers E and F, and since we know even numbers are 2 times some number, so we let 2 times M equal E, and 2 times N equal F. So we can rewrite the sum E plus F.
And this is a chance to apply what we know about the Distributive Property! You give it some thought now. Think about what we're trying to prove, that E plus F equals an even number, and how the Distributive Property can help us here. When you're ready, click on the link below to continue on the the exciting conclusion of our first proof.
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