Multiplying Even and Odd Numbers

Odd NumbersI hope you will forgive me for writing about math today. Last night, I was lying in bed thinking about the numbers 7 and 11. I had been listening to a podcast with Ezra Klein and Molly Ball. Ball had mentioned that the number of white Christians in the United States had gone from (I think!) 54 percent when Obama came into office and that it was 47 percent now. Klein must have misheard her, because he later referred to it being an 11 percentage point drop. But that was why I was thinking about the two numbers — 7 percentage points is the actual number.

These numbers are interesting in that they are consecutive primes. And being so, they hold a certain fascination for me. But it got me thinking about the number 9. Nine is not a prime, since it reduces to 3×3. And then something occurred to me that I’d never thought about before: two odd numbers multiplied always create an odd number.

I know this is obvious, but since when has that ever stopped me? Why is it that odd numbers multiplied are always odd?

Multiplying Even Numbers

Let’s start with an easier question: why are even numbers multiplied always even? That’s almost definitional. An even number is any whole number divisible by 2. So if you have two even numbers x and y, you know that both x/2 and y/2 must be whole numbers. Thus, for example:

2×(x/2)×y

Note that it doesn’t matter if y is even. Thus: an even number times any number will be even.

Multiplying Odd Numbers

Looking at two odd numbers is more interesting. Or I think it is. Let’s stick with our variables above. Now we have two odd numbers: x+1 and y+1. If we multiply them, we get the following:

(x+1)×(y+1) = x×y + x + y + 1

Given that x and y are even, we know that x×y is even. So we have: even plus even plus even plus one. The whole thing doubles back on itself: we defined our odd numbers as evens plus one. And that’s what we get here.

Using Addition

Another way to think about it is via addition. This is the way that ought to come more naturally. Multiplication is, after all, just addition. Four times three is just 3+3+3+3. Sadly, math is usually taught so badly that people don’t think in this way. So people end up thinking that addition, subtraction, multiplication, and division are four different things when they are all just one really simple thing: addition.

Thinking in this way, (x+1)×(y+1) would be the number y+1 added x+1 times. I would show you how this all works with a series, but doing so requires more typesetting ability than I have here. But think about it. If you add an odd number an even number of times, you will get an even number. So when you add that odd number one more time, it makes the even number odd.

The beautiful thing about math is that this is all intuitive. I didn’t have to work out the steps in my mind. It all looks awful on the page. In the mind, it’s comforting. Of course, I did have to get out of bed. I figured if I didn’t write down the idea, I would forget to write this article. Then wouldn’t you all be sorry…

8 thoughts on “Multiplying Even and Odd Numbers

    • Have you ever seen Beneath the Planet of the Apes? In it, a group worships an atomic bomb. In a fundamental sense, I worship math. It is more real to me than the outside world, which I have great skepticism of. I don’t know that the outside world exists. But, as the great man said, I think therefore I am. Not that I agree with him! But math is the closest to an absolute reality that I can find.

      • I just don’t understand this obsession of yours. I also don’t know how math is more real then reality.

        • It is probably just that I think “reality” is less real than you do. Since math is something that is only inside me, I don’t question it the way that I do things that I observe. Those things could just be hallucinations. The ideas that “observe” are just as real, I suppose. But there is a lack of purity in them.

  1. Maybe a little simpler, using your (x+1) times some number, where x is even: The x’s will always add up to an even number, so the addition of the 1s is what matters. If you add up an even number of 1s, the total result will be even. Add up an odd number of 1s and that “extra” 1 (over the preceding even # of 1s) makes the total result odd. If you get what I mean–if you don’t, that probably means my way of looking at it is not simpler!

    • Oh, that’s good. So: x+x+x+1+1+1. That’s one of the things I love about math: there are so many ways to approach a problem. I remember that when I tutored math. I would see people solving equations where nothing they did was wrong, but they weren’t moving in a useful direction. It’s a lot like musical improvisation. Once you “see” it, you just play. It isn’t about rules. People who are good at math understand how intuitive it is. People who are bad at math think it is rigid. I find that sad. And I blame the way it is taught.

      • “People who are good at math understand how intuitive it is. People who are bad at math think it is rigid.”

        Incompleteness theorem notwithstanding, it IS utterly rigid. That’s what makes it so great. No hand-waving, not a lot of “everybody is right, in a way”. It’s beauty is inherent.

        For me, what makes doing math so rewarding is that it is bloody hard. I studied math and linguistics at university and because math was that much more difficult, I was that much more passionate about it, even though I loved working on both subjects.

        • I know what you mean, but I don’t see math as rigid. I mean, it is rigid in the way that climbing a mountain is rigid. There is somewhere you’re going, but endless ways to get there — each one different in terms of beauty, efficiency, creativity, and so on. And that’s largely true of literature too. A tragedy is rigid, for example.

          Is math hard? To me it’s a lot easier than, say, oil painting. Regardless, even if hard, I think people find it harder than it is because they’ve been trained to hate it. I’d really like to see improvements in teaching math.

          The incompleteness theorems are, of course, where the action really is. And I don’t think that’s hard at all. But it is amazingly creative. I remember when I first read the Nagel and Newman’s article on it, I thought, “Who would imagine approaching the problem in this way?!” It was so exciting.

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