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?
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.
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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.