@Dex Stewart
The repeated addition thing is just wrong.
I mean the teacher’s wrong. The kid’s right.
There’s something called “ordinal arithmetic” that extends natural number arithmetic to
transfinite numbers. The idea is that each number corresponds to the set of its predecessors. So 0 is the empty set, 1 is the set {0}, 2 is {0,1}, and so on. So what’s {0,1,2…}? What happens when you “count to infinity”? Well, we call that ω. That’s a lowercase Greek omega, to be clear. And you can keep going. {0,1,2…,ω} is ω+1. {0,1,2,…,ω,ω+1} is ω+2. Yada yada. All the way up to {0,1,2,…,ω,ω+1,ω+2,…}, a.k.a. ω+ω, a.k.a. ω·2.
Now where the arithmetic part comes in is that the plus and times I’ve been using can be defined in a rigorous way. (Exponentiation too.) For example, ω+2 corresponds to a set of ω
followed by a set of 2: {0,1,2,…,0’,1’}. And ω·2 corresponds to 2
copies of ω, back to back: {0,1,2,…,0’,1’,2’,…}. But here’s the thing. As defined, these operations are not commutative. ω+2 is not the same as 2+ω. 2+ω is 2 followed by ω: {0,1,0’,1’,2’,…}. That’s just ω. Count to infinity once and you’re done. Likewise, 2·ω is ω copies of 2: {0,1,0’,1’,0’’,1’’,…}. Again, just ω.
This has been a part of mathematics for nearly a century and a half. It’s established practice. And according to that practice, 5·3 means 3 copies of 5, not the other way around. 5+5+5.
And it’s not just ordinals. You crack open any book on mathematical logic and look at the axiomatizations of natural number arithmetic. Peano, Robinson, Heyting if you’re some kinda weirdo. You look at how they define multiplication. It’s inductive, two parts. x·0=x, x·Sy=x·y+x. (S is the successor function. It just means “+1”.) Now you take all that and you run back to your little elementary school problem. 5·3 = 5·SSS0 = 5·SS0+5 = 5·S0+5+5 = 5·0+5+5+5 = 0+5+5+5 = 5+5+5.
![Thread Starter - Hazbin Hotel Thread [Possible NSFW/Dark]](https://derpicdn.net/badges/2017/7/2/0bd59d4eff9c3aeaad.svg "Thread Starter - Hazbin Hotel Thread [Possible NSFW/Dark]")
