Negative Numbers

Ancient people rejected the idea of negative numbers for a long time. It made no sense to them.

To you, negative numbers seem perfectly reasonable. But could you empathize with these ancient people at all? Why on earth would they consider negative numbers to be weird.

Try to understand their perspective. It’s not easy to do. It took me a while to really grok it.

They’re Weird

Imagine this conversation:

You: 3+5=8.
Ancient Person: Huh?
You: You have three cows right? Here’s five more. Now how many do you have.
Ancient Person: Oh ok. I get it.
You: 3–5=-2
Ancient Person: Huh?
You: Imagine you have three cows and I take away fi-
Ancient Person: BUT HOW COULD TAKE AWAY FIVE, I ONLY HAVE THREE! THAT’S PHYSICALLY IMPOSSIBLE!!!

No They’re Not

Here’s how I would respond:

You’re right — it is physically impossible to take five cows away from a herd of three. BUT that doesn’t mean 3–5 is senseless.
Imagine that you commit a crime and your punishment is to pay the victim 5 cows. But you only have 3. So, you pay the victim 3 cows now, and you owe him 2 more cows. How many cows do you have?
Well, there are 0 cows physically in front of you. But if you acquired 1 cow, you’d have to give it to the victim, and you’d again have 0 cows. If you acquired another cow, you’d still have to give it to the victim, and you’d still end up with 0 cows in front of you.
Now, the next time you acquire a cow, you get to keep it because your debt is payed off. This is a very different state than the one you were in earlier, right? Before, you had to hand your hard earned cows off to the victim, now you get to keep them!
How can we distinguish between these two states mathematically? With negative numbers!

Abstract

Numbers don’t have to represent physical things, they could represent concepts. The fact that you’re in debt and have -2 cows isn’t a physical thing you could see with your eyes — it’s a concept. And we have a number for it.

But numbers don’t always fit with every concept. For example, physically taking away 5 cows from a herd of 3 cows makes no sense and is indeed physically impossible. This type of subtraction does not fit with the concept of physically acting on actual cows.

Consider another example — how would you divide 9 orphans equally amongst 5 sets of parents? You can’t do it. There’s no such thing as 1.8 orphans. Children aren’t fractions. This type of division doesn’t fit with the concept of orphans.

Numbers are what we call abstract. They aren’t referring to a specific concept (physical cows, debt, orphans…). They’re an idea that may or may not be applicable to a given concept.

Open Minded

Please be open minded. Sometimes a number may seem weird, just like negative numbers seemed weird to ancient people. But remember, a number is just an abstract idea. Don’t be mad at people for having abstract ideas.

And don’t immediately conclude that the abstract idea is useless. It may not be applicable to the concepts that you have in mind, but perhaps there are other concepts they could be applied to.

For example, you may think it’s stupid to allow numbers to be eight dimensional (wat), but super smart physicists who can perfectly describe the electron wave equation don’t. There are deep and fundamental aspects of our physical world (aka our reality) that can only be described with mathematical weirdness.

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