I 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:
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:
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.
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…