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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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Um, that's what I meant by "Z4" (I couldn't remember and didn't bother with the exact definitional name). The element of Z/4Z that is usually denoted "0" is that set I noted above and can also be correctly denoted "4".

Do you have any source for that? All the sources I've found say the elements of the underlying set of integers modulo 4 are integers.

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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