I can't identify with zero

Another case of forgetting to write a post beforehand. Whoops. Spring Break is bad for productivity. Gonna have to put off graphing again (see, it's a post that will require work). Let's talk about something interesting and conceptual instead.

Identities! Also, zero. I'm going to explain why you can't divide by zero.

An identity is a number that, when processed through an operation with another number, gives you that other number. That sounds confusing. It isn't, really. You already know this, you just don't have a name for it.

The multiplicative identity is 1. x*1 = x, you see. It doesn't change the value of what you're multiplying it with. This is important because remember, you can write 1 as anything over itself. So 1 being an identity allows you to re-write equations and put them into a different form. Pretty cool!

The additive identity is 0, because x+0 = x. This one also lets you re-write equations. You use it constantly in simple algebra while solving for x. x+3 = 7, subtract 3 from both sides, x + 3 - 3 = 7-3; x + 0 = 4. You're using the fact that adding zero doesn't change the answer to figure out the solution to the problem.

Yeah, yeah, this is stuff you already know, but nobody sat down and bothered to explain what it meant, I'm certain. I didn't find out about identities as a THING until linear algebra, when you start using this thing called an identity MATRIX. You multiply it with a matrix and get the original matrix. UNSURPRISINGLY it's an idea that comes from more basic math concepts which are never adequately explained!

I mean, you can say it's obvious that x+0 = x, but why? Why is that obvious? Math operates on the methods we use to define it. Did you know that x⁰ is 1? Yeah, something times itself 0 times is equal to 1. Why? Because if it wasn't, the rest of our math wouldn't make sense. (Speaking of, the exponential identity is also 1.)

So we created this thing called an identity and defined it the way we did because it helps us solve problems.

This is only tangentially related to identities, but it's a good place to talk about zero. Why can't you divide by zero? We know from doing limits that division by zero goes to infinity. It's not finite, but the limit exists. Well, the fact that it's not finite is important. That means that x/0 doesn't actually have a value. You can't say it equals infinity, because infinity is a concept, not a number.

You can also explain it algebraically. I'll even bust out the code for this one.

Divide both sides by x, where x ≠ 0


This is true. It works.

But if you try it the other way around:



Dividing by a number is actually multiplying by the inverse, so we try multiplying by 1/0.


Unfortunately, all you get is a mess.

we know is zero. we know is infinite.

So what's if the top makes it 0 and the bottom makes it infinite? We don't know. There's no way to reconcile that. It's undefined. The answer there doesn't exist.

Unlike all other numbers, 0 does not have an inverse that turns it into 1. x*1/x = 1, but 0*1/0 ≠ 1 because 0/0 is not defined and it's definitely not equal to 1. And that's why you can't divide by zero unless you're taking limits! Because 0 does not have an inverse function!

Maybe I'll actually do graphing next time. Or maybe I'll forget until Monday morning again and write another filler post. START A BETTING POOL.