Proof of commutative property of multiplication

How can you prove that the multiplication of natural numbers satisfies the commutative property?

[asbarker]

A very nice use of mathematical induction. I suspect that most viewers won't have seen it used in this way before.

[tejedordealas]

Good vibe from Morelia. Now I have the two explanations, discovermaths and shurprofe. Continuing with the channel marathon in English and it is ready. Let's go for it.

[lazaredurand]

How do you define what is the neutral element of an operation and demonstrate that 1 is the neutral element of the product? Is there not a more simple proof? To the use of neutral element concept make it weighty.

[vinayseth]

4:11 I don't get it. You've shown that if the commutative property holds true for numbers x, y such that x+y<m+n, then the commutative property holds true for m, n as well. How does that prove what we had set out to prove?

[chemiflask]

Thank you sir! It kinda itches when you are constantly coming up with shallow assumptions, but you don't really know the proof

[arbitrarilyarbitrary]

There is however one thing that's bugging me about this particular proof. And that is the fact that this proof only holds for natural numbers, as induction only proves a property for natural numbers. However, it is usually given in most undergrad courses that the commutative property holds even for the generalization of real numbers and complex numbers (though properties of complex numbers follow from properties of real numbers).


For example e*π = π*e, here both ''e'' and ''π'' are real numbers. How would you go about proving this for all real numbers? (obviously using a calculator doesn't for proving this case)

[TrojenMonkey]

So much of this video is filled with: "but why.", in particular the last segment.
"The sum of (n-1)*m = n-1+m."
What.

(paraphrasing) "if n-1+m is smaller than n+m, then we've proven that m*n is n*m." What.

The hypothesis itself is "WHAT!?" -- "suppose that the result (n*m?) is true for two factors whose sum is less than n+m". Somehow, that proves the commutative property. What?? Wtf does that even mean? If I had enough capacity to know what that meant, or why that's important, you'd think I'd be here? There are so many assumptions here. :/

That was somehow worse than reading my uni book... :|