[francasso]

There's no disagreement, the algebraic one is the correct one, obviously. Anyone that says differently is wrong. :)

[srean]

Being an engineer by training, I never got exposed to much algebra in my courses (beyond the usual high school stuff in high school). In fact did not miss it much either. Tried to learn some algebraic geometry then... oh the horror. For whatever reason, my intuition is very geometric and analytic (in the calculus sense). Even things like counting and combinatorics, they feel weird, like dry flavorless pretzels made of dried husk. Combinatorics is good only when I can use Calculus. Calculus, oh that's different, it's rich savoury umami buttery briskets. Yum.

That's not the interesting part. The interesting part is that I thought everyone is the same, like me.

It was a big and surprising revelation that people love counting or algebra in just the same way I feel about geometry (not the finite kind) and feel awkward in the kind of mathematics that I like.

It's part of the reason I don't at all get the hate that school Calculus gets. It's so intuitive and beautifully geometric, what's not to like. .. that's usually my first reaction. Usually followed by disappointment and sadness -- oh no they are contemplating about throwing such a beautiful part away.

[macromagnon]

School calculus is hated because it's typically taught with epsilon delta proofs which is a formalism that happened later in the history of calculus. It's not that intuitive for beginners, especially students who haven't learn any logic to grok existential/universal quantifiers. Historically, mathematics is usually developed by people with little care for complete rigor, then they erase their tracks to make it look pristine. It's no wonder students are like "who the hell came up with all this". Mathematics definitely has an education problem.

[Sharlin]

"The Axiom of Choice is obviously true, the Well-ordering theorem obviously false, and who can tell about Zorn's lemma?"

(attributed to Jerry Bona)

[ajb]

Hah. This perspective is how you get an embedding of booleans into the reals in which False is 1 and True is -1 :-)

(Yes, mathematicians really use it. It makes parity a simpler polynomial than the normal assignment).

[yaks_hairbrush]

It works if you don't care about magnitudes, distances, or angles of complex numbers. Those properties aren't algebraic.

[cperciva]

The complex numbers are just elements of R[i]/(i^2+1). I don't even understand how people are able to get this wrong.

[FillMaths]

Of course everyone agrees that this is a nice way to construct the complex field. The question is what is the structure you are placing on this construction. Is it just a field? Do you intend to fix R as a distinguished subfield? After all, there are many different copies of R in C, if one has only the field structure. Is i named as a constant, as it seems to be in the construction when you form the polynomials in the symbol i. Do you intend to view this as a topological space? Those further questions is what the discussion is about.

[emil-lp]

Obviously.