Evens and Odds

Did you know that even numbers are considered female and odd numbers are considered male? The Chinese and the Pythagoreans alike often imbued numbers with personality and human qualities, forming a mystical mathematics alongside the practical and logical mathematics. But there are some interesting properties of odd and even numbers that seldom get discussed. But knowing these properties can help you determine if an answer you arrived at is correct.

Even or Odd?
If I asked you to list some even numbers, you'd probably say, "2, 4, 6, 8, ... do you want me to keep going?"
"No, no, " I'd say. "That's fine."
But what do all of those numbers have in common? Yes, I know. They're even. But what makes them even?

Of course! When you divide them by 2, there's no remainder. In other words, 2 goes "evenly" into each number. Another way to say this is that each number is equal to 2 times some other number. For instance, 8 is even because it is 2 x 4. 12 is even because it is 2 x 6. 2 is even because it is 2 x 1. And even zero is even because it is 2 x 0.

In Algebra, we often represent an unknown even number as 2n. That's because no matter what we choose for n, multiplying it by 2 makes the result even.

So if an even number is "evenly" divided by 2, what's an odd number? It's a number that is NOT "evenly" divided by 2, of course. In other words, there is a remainder when we divide an odd number by 2. But one of the other things that we notice about odd numbers is that they come immediately after the even numbers. In other words, if we just add 1 to an even number, we get an odd number. 9, for instance, comes right after the even number 8. So if 8 is 2 x 4, then 9 must be 2 x 4 +1. We can write all of our odd numbers this way.

In Algebra, we often represent an unknown odd number as 2n + 1. That's because an odd number is always one more than an even number.

Addition Properties:
If you take any two even numbers, say 24 and 8, and add them together, you get another even number, in this case, 32. Try it. Find a bunch of different pairs of even numbers and have at it. Satisified? Ok.

Now how about trying to add any two odd numbers together. Let's take 9 and 17 as an example. We get 26 when we add them, an even number. Does this always happen? Try a bunch of odd number pairs. Go ahead. I'll wait.

Ok, now try adding one even and one odd number together. Yup. Say 8 and 17. That's right, we get 25, an odd number. Once again, test this out with several even/odd pairs to see if it ALWAYS works.


Algebraic Proofs:
Now, you might have convinced yourself that you've discovered some properties of addition:

even + even = even
odd + odd = even
even + odd = odd

But could there possibly be a case for which one of these properties doesn't work? In other words, is there a pair of even numbers that doesn't give you back an even result? In order to know for sure, mathematicians like to prove things, in this case using Algebra.

Let's use our Algebraic representations of even and odd numbers to help us out. We said before that an even number can be represented as 2n and an odd number can be represented as 2n + 1. So, let's uses these two facts to try out our addition facts. Let's start with adding two even numbers.

Conjecture: The sum of two even numbers is even.
Proof:
Let's represent each even number Algebraically. One of them will be 2n and the other will be 2m. (We don't want to make them both be 2n, because that would mean that we are adding an even number to itself. We want to be sure that we can pick two unique even numbers if we want to.)

2n + 2m

We can simplify this a bit by factoring out (or un-distributing, if you remember your distributive property) a 2.

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

Now, we need to ask ourselves if 2 ( n + m ) is even. If we look back at how we described our even numbers, we said that an even number is a number that we can form by multiplying some other number by 2. Well, regardless of what n and m are, if we add them, we definitely get "some other number". And that "some other number" is definitely being multiplied by 2. So, 2 (n + m ) is definitely even.

So, we've just proved, Algebraically, that the sum of two even numbers is ALWAYS another even number.

Conjecture: An odd plus and even is an odd.
Proof:

We need to add an odd and an even. The algebraic representation of an odd number is 2n + 1, and the even number is 2m. Let's add them.

2n + 1 + 2m

A bit of rearranging gives us:

2n + 2m + 1

Now, we just saw, in the previous proof, that 2n + 2m can be turned into 2(n + m), which is even.

2 ( n + m ) + 1

So this little expression is an even number plus 1. Well, that's the description of an odd number, for sure. So by adding an even and an odd, we've managed to get another odd.

Because we don't ever care what someone decides to pick for n and m, this is a proof that ALWAYS works.

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Suggested Activities:

1. Prove, using Algebra, that the sum of two odd numbers is even.
2. Prove, using Algebra, that the sum of an odd number and an even number is odd.
3. Develop some rules for multiplying even and odd numbers.
4. Prove, using Algebra, your multiplication rules.