Saturday, June 9, 2012

Simple Math Facts Finally Proven

Before I begin, remember that I posted about the mathematical game of KenKen a few weeks ago. If you didn't see it, go to I gave a sample puzzle, which I will give the answer to next week. Please make sure to try it, so you can see if you are correct. You can also go to for more puzzles.

There are some things in math that are so basic that we commonly just take it for granted. For example, that an odd number + an odd number = an even number.

1 + 1 = 2
3 + 5 = 8
31 + 87 = 118

Or that an even x an even = an even.

2 x 2 = 4
8 x 6 = 48
14 x 18 = 252 (to do this in your head, check out the very first blog post)

But why are these facts true? Well, let's try to prove them with some handy-dandy algebra,

First off, odd + odd = even. An odd number can be written algebraically as:

2n + 1 (assuming that n has no decimal, or in other words, n is an integer)

This is because 2n is even (an integer times two is always even), and adding one to an even number always gives you an odd number.

So, we have:

(2n + 1) + (2m + 1) =

Pretend m is another integer. Since it is all addition, we can ignore the parentheses (this is known as the associative law) and add the two ones together. This gives us:

2n + 2m + 2 =

We can factor out a 2 to get:

2(n + m + 1) =

Since n, m, and 1 are all integers, they must be an integer when added together. Therefore, we can say that:

2(integer) =

As we said before, 2 times an integer is an even number, so we have just seen that odd + odd = even.

What about the second fact, that even x even = even. Let's use the same exact logic.

(2n) x (2m) =
4mn =
2(2mn) =

Since n, m, and 2 are all integers, when multiplied together, it gives you an integer. So we can say:


Since that is an even number, we have proven it.

After learning these facts in third or fourth grade, it is cool to be able to understand the reasoning behind them. They seem almost so simple that there isn't any need for a proof, but I think the proofs are cool nonetheless.

Our problem of the week is back! If you were not following Cool Math Stuff last summer, let me explain. In the summer, each month, I will create two very long problems: an easy problem and a hard problem. Problems so long that it takes five days to figure out the answer. Each day, I will post the next part of the problems for you to solve. After the five days, you will have a month to determine the answers. I will post the answers on the Saturday's post following the next month's problem of the week.

Last year, there was guidance with the problem. This year, I am going to make it a little more challenging. For June, there will be the same level of guidance as last year. July, I will be taking it down a notch. You will have some hints (maybe the important formulas, or some calculator buttons to look for), but no in-depth instruction like before. In August, it's all you. There will be a problem, and an answer the next September.

June's problem of the week will be given from 6/18 - 6/22.

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