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felipec  unbelief  2yr ago (felipec.substack.com)   3599 thread views

2+2 = not what you think

felipec.substack.com

Changing someone's mind is very difficult, that's why I like puzzles most people get wrong: to try to open their mind. Challenging the claim that 2+2 is unequivocally 4 is one of my favorites to get people to reconsider what they think is true with 100% certainty.

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https://math.stackexchange.com/questions/1556009/quotient-ring-mathbbz-4-mathbbz

Somebody was confused when defining Z/4Z and not getting integers; every response notes that Z/4Z is strictly not a set of integers, but a set of sets of integers.

I could go look for a (presumably pirate) online textbook if you really want (I learned this from lectures in uni, not from a textbook), but it'd be a pain.

(The elements of the underlying rings of the quotient ring - Z and 4Z - are of course integers, but the elements of Z/4Z aren't.)

OK. But in the answers it's claimed that this defines a new way to say what elements equals to what else, so 3=7. Therefore 4=0, and 2+2=0.

Yes. It is true that 2 + 2 = 0 in Z4; I've not disputed that. It's just also true that 2 + 2 = 4.

Yes. But the whole point of my post is to get people to reconsider what basic notions like 2+2 are.

And if I understand correctly in ℤ/4ℤ there is no 2 in the main set, it's {..,-6,-2,2,6,...}, so it's actually {...,-6,-2,2,6,...}+{...,-6,-2,2,6,...}={...,-8,-4,0,4,8,...}, or something like that. 2 is just a simplification of the coset.

Second part is right, yes.

OK. But then I do get it: 2+2 = 0 (mod 4).

Yes, without any other context 2+2 is assumed to be 4, but 2+2 (mod 4) is a different thing, because 2 and 2 (mod 4) are different (the latter is actually {..,-6,-2,2,6,...}). Correct?

I have updated the article to be more correct.

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