The Four-Operations Conspiracy, or How I Learned To Stop Worrying and Love the Commutative Property

We all know that addition is commutative--

Hold up. What's that? You don't even know what "commutative" means? I'm not surprised - I didn't even learn that this was a thing until my Calculus III professor used the word to explain that dot products are commutative while cross products are not (wait please don't go I promise those don't show up in this post, honest!)

Okay, so what the fuck is the commutative property? It's actually just a name for something you already know is true, because you've been doing it since your very first math class ever without even realizing it.

Basically, the commutative property means that 3 + 5 = 5 + 3. Addition is commutative. Multiplication is also commutative. 2 × 6 = 6 × 2

But those are examples of it. To define it, just remember that if something's commutative then that means you can switch things around as much as you want without changing the answer.

WHY IS THIS IMPORTANT?

I'm getting there, goddammit.

Now that you know addition and multiplication are commutative, you can extrapolate that subtraction and division are not, because we all know that 3 - 5 ≠ 5 - 3 and 8 ÷ 2 ≠ 2 ÷ 8. Pretty fuckin' basic, you guys.

But I'm here to tell you something that will blow your fucking mind.

This is a lie.

Subtraction and division don't even EXIST. They are not even REAL. They are nothing but illusions given to students to make things seem simpler. Subtraction and division are nothing but a vast mathematical conspiracy!

That's a pretty bold claim there, Guindo! This is a thing you are thinking to yourself, obviously, because I do not think about myself in the third person - usually.

I know I just turned your world upside down but quit freakin' out for a second and I'll explain. You see, numbers can be positive or negative (but Guindo what about imaginary numbers SHUT UP I'LL GET THERE LATER this post isn't about that). When you subtract, what you're really doing is adding a negative number.

Oh man, are you confused yet? Let me make it worse.

Check out multiplication. Multiply two positive numbers and your answer is positive. Multiply two negative numbers and your answer is positive. Multiply a negative and a positive, and your answer is negative.

In handy visual format:

+ × + = +
- × - = +
+ × - = -

Think of a minus sign as a negative one being multiplied into the number.

3 - 5 = 3 + (-1)(5)

(Also you can signify multiplication with parentheses NOW YOU KNOW)

Now it's addition! Instead of subtracting five, you're adding negative five. Mindblowing shit, I know.

Here's the cool thing: now that you're adding, you can use the commutative property to switch things around.

3 - 5 = 3 + (-1)(5) = (-1)(5) + 3 = -5 + 3

GOD DAMN! THAT'S AWESOME!

Why is that, uh, important? At all? Fuck you, that's why. It's important because it means you can rearrange subtraction and that's vital when you start working with polynomial functions (HOLY SHIT WHAT ARE THOSE look I'm just setting up a foundation you guys quit freakin' out on me).

BUT WHAT ABOUT DIVISION? YOU SAID THAT WAS A LIE TOO!

Yes, it is a lie! It is a lie in the same way that subtraction is a lie: it's multiplication written differently. But to understand this, you need to understand fractions.

OH GOD NOT FRACTIONS. I HATE FRACTIONS. Yeah, yeah, so does everyone. We'll cover that in more detail later, but right now all you need to know is this: multiplying and dividing by 1 doesn't change your number, right? So 1 × 3 = 3 and 3 ÷ 1 = 3.

Well, all division can be written as a fraction. So let's rewrite 3 ÷ 1 = 3 as That also means that any time you see a 3, you can write it as (because they are equal and you can do this with things that are equal to each other).

When you divide, what you are really doing is multiplying by the inverse. Inverses are a complex thing that I don't feel like talking about now because this post is long enough as it is, but for now just keep in mind that an inverse of a number is that number as a fraction, flipped upside-down. For example, the inverse of is

Let me show you what I mean.

(fractions are multiplied straight across, top times top and bottom times bottom)

So there you have it, division is multiplication by the inverse. That means that you can rewrite division as multiplication by 1 over whatever, and just like with subtraction, you can re-organize shit now and move that little fraction wherever the fuck you want.

Now that you've had the illusions of subtraction and division shattered for you and your entire world has been torn asunder by the revelation, NEXT TIME WE'LL TALK ABOUT ~*FRACTIONS*~!!

---

* Equations provided by codecogs.com, sorry I took the links out of the nice html you provided!