I think it's because of the name. If you called them double numbers or paired numbers nobody would say that.
Imaginary numbers are not pairs. Complex numbers are. Continuing Gauss, I'd rename "complex numbers" as "planar numbers".
Also vector numbers could be somewhat of a useful name, since they behave like a 2D vector (or even higher dimensional vectors for e.g. quaternions)
Don't know how relevant this is, but I've been thinking about a better naming scheme for hypercomplex number systems. I came to it after seeing a paper about the "dual-complex numbers", which are not a straightforward complexification of the dual numbers as one might expect. Hopefully, the scheme should be pronounceable, and without the possibility of confusing it for something else. This town needs law! I'm thinking of asking for suggestions for what it should be exactly.
indeed. the name is terrible; causes a lot of folks consternation, or tricks them into thinking strangely about the complex numbers.
anyone happen to know of any languages where they're not referred to as "imaginary", or anything that implies they're less "real"?
Almost all the Latin alphabet languages use a variant of 'imaginary'. Icelandic appears to be an exception, where it means cross-number. https://en.wiktionary.org/wiki/þvertala#Etymology
(I don't know non-Latin alphabets I can't tell you anything about them.)
I'm not sure that's true.
Just think of solving x^2+k=0. It's clear that for k<0 you get two solutions and k=0 you get one solution. But when k>0 the graph doesn't touch the x-axis... so why should I expect a "double number" or "paired number" to be the solution?
I'm teaching college algebra right now and introduce 'i' algebraically as a solution to x^2+1=0... but then we talk about graphing quadratics and there's no simple connection between the geometry/graph and the algebra.
Even if I had the time to talk about the geometry of multiplication and such, it's still a big leap to the graph of z^2+1 and its roots in the complex plane.
And it's this leap which, IMO, makes them seem "not real".
Take an equation like x^2 -2x + c
When C is some large negative number, the roots are symmetric about x = 1. As you increase C, roots are real until C = 1, basically the two roots meet in the middle.
When you increase it beyond 1, the roots become complex numbers, but they stay symmetric, they "lift off" from being real into being complex, but still symmetric (conjugates) where 1 is always the real part but the imaginary parts become sqrt(C - 1). You can conveniently imagine the complex plane on top of the real XY plane and the roots go orthonogal to the direction they were going when they were real, from the point where they met.
That's kind of how I visualize it for quadratics. For higher orders you cut a complex circle into n equal pieces. Haven't quite figured it out in my head yet.
There's a big conceptual leap here.
Again why would anyone posit that a pair of numbers suddenly appears when trying to solve x^2+k=0 as k goes from negative to positive?
And why is two dimensions enough for third and higher degree equations? Were there a simpler geometric connection, you’d probably have a nicer proof of the Jordan curve theorem… but you don’t.
Not sure what you mean by pair of pairs… the OP said that calling imaginary numbers “double numbers” would clear up a lot of issues but that is not clear at all to me.
By pair of pairs I just mean that you're looking for two solutions, and they turn out to be computed numbers, both of them.