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In fact the first time students see this is in Primary School, when they can do it for themselves with Cuisenaire Rods. In the picture we can see an orange rod of length 10, and next to it a blue rod of length 9, and a white cube for 1, and we can see that 9+1=10. Now just reverse the order of the rods and we have 1+9=10, thus proving the Commutative Property.

Next up the same thing with algebra in high school. If a=2, b=3, then
a+b
=2+3
=(1+1)+(1+1+1)
=1+1+1+1+1
=(1+1+1)+(1+1)
=3+2
=b+a...

3/3
And obviously that holds true for all values of a and b

After that we teach students how to do inductive proofs, which can also be used...

Prove it's true for n=1

1+1=2=1+1 (obviously you need to imagine I swapped the order, as we did with the Cuisenaire Rods)

Assume true for n
n+n=n+n (ditto)

Show it's true for n+1
n+1+n+1
=n+n+1+1
=(n+n)+(1+1)
Thus true for all values of n

These are so trivially easy it astounds me that people say it can't be proven! Need to go back to high school! 🙄

@cardinal_reinhardt
"So what axioms are you using? " - oh, you're one of those types.

"in abstract algebra" - which you don't need when you have an arithmetic proof 🙄

"induction only includes one (which you call n)" - that's how inductive proofs work. Prove it's true for 1, assume it's true for n, show it's true for n+1

"Maybe there's a use for mathematicians after all!" - says someone who doesn't know how inductive proofs work 🙄

@SmartmanApps

“which you don't need when you have an arithmetic proof 🙄”

No offense but have you ever studied abstract algebra, group theory, rings, fields, etc, because this. does not make sense

“that's how inductive proofs work. Prove it's true for 1, assume it's true for n, show it's true for n+1”

You're right, that is basically how they work. And "it" is a+b=b+a, so I think you're saying "assume it's true for n" means you substitute both a and b by n? That right?

@cardinal_reinhardt
"because this. does not make sense" - I guess you don't understand arithmetic proofs then if it doesn't make sense 🙄

"You're right, that is basically how they work" - that's exactly how they work. We teach it to high school students

"you substitute both a and b by n?" - you know a and b are allowed to be the same value, right? I have 10 apples and 10 bananas, a=10, b=10

@SmartmanApps
- "have you ever studied abstract algebra"
-- "..."

OK, if you don’t want to answer I won’t make you, but owning what you don't know is much better than dodging qs, someone asks me about cohomology I be like ¯\_(ツ)_/¯
Here’s a good textbook if instead you want to learn it: https://www.amazon.com/Contemporary-Abstract-Algebra-Textbooks-Mathematics/dp/1032778911

"a and b are allowed to be the same value"

So in the same way, to assume a = b is true for n, you would assume n = n?