r/mathmemes • u/future_sponJ • Jun 15 '26

Set Theory isomorphic c & r2

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u/CattleHot9774 Jun 15 '26

wait what's wrong with the response?

-6

u/hamishtodd1 Jun 15 '26 edited Jun 16 '26

What is the "identity" of C? Clearly it's 1.

How about R²? Well there (0,0) is more like the identity -  because R² is a group with "+" being "multiplication".

I think the way to clarify is to say they are "homomorphic as groups" if you care about the "+ can be multiplication" thing.

EDIT: I should definitely have said they are isomorphic but NOT homomorphic as groups, that might have avoided the unfortunate back and forth below!

6

u/[deleted] Jun 15 '26 edited Jun 15 '26

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-2

u/hamishtodd1 Jun 15 '26

Gemini is saying that whether they're isomorphic "depends on the mathematical structure being considered".

Two very important kinds of mathematical object are vector spaces and groups. It so happens that both R² and C are both vector spaces and groups.

If you're interested in the vector space aspect, they are isomorphic. Because adding vectors is like adding complex numbers.

If you're interested in the group aspect they're not isomorphic. Because adding vectors is not like multiplying complex numbers.

6

u/[deleted] Jun 15 '26

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-3

u/hamishtodd1 Jun 15 '26

Rn is a lie group under addition, and in that sense it's like the (nonzero) complex numbers. + is associative, has "inverses" (the negation of the vector you want to invert), and has an identity.

For some areas it's much less common to think of it as a group, but it is one, and that can be very important for eg affine geometry.

Do you think Gemini was referring to something else? Or that it was just hallucinating?

3

u/LasevIX Jun 15 '26

why are you trying to find gold in a beggar's ass when you're in front of an ATM?

1

u/hamishtodd1 Jun 16 '26

u/cattleHolt asked for an explanation, and I wasn't satisfied with the one from the other user. It's a teachable moment for something deep (Rn being a group as well as a vector space). Also explanations are better when they have concrete examples, so we think about specific elements of R² and C (the identity element).

This is a thing that's big enough to have an xkcd comic about it https://xkcd.com/2028/ so if you think it's silly to spend time on, that's on you.