I think there's definitely something to your suggestion here. For instance, there is no result to sqrt(-1) on the real line, but it becomes possible in the complex plane because you have an additional degree of freedom.

One qualifier I'd add though is that they're 'asymmetric' 2D numbers. The real vs imaginary axes have different behaviors. If I multiply by the positive unitary value on the imaginary axis (i.e. 'i'), I get CCW rotation by 90 degrees; with the negative imaginary unit I get CW rotation; positive real unit, no change occurs; negative real unit, rotation by 180 degrees.

Actually that's something I wonder about—could there be a 'symmetric' version of that? Maybe something where the real axis behaves more like the imaginary axis, but maybe some signs are flipped or something.

I guess an important aspect of how using the complex plane is beneficial is the fact that it combines these disparate elements though—we can start with something real, do certain transformations to it in our more broadly capable complex plane, then bring it back into the real line.

You want geometric algebra. In 2D GA, you would have two unitary vectors, i and j, such that i * i = 1 and j * j = 1. The (non-commutative) product of between them (or their division) would be a bivector: i * j = ij = - j * i. You rotate 90 degrees the i and j vectors using the bivector.

The good thing about GA is that the same concept can be easily extended to 3D (quaternions), and in fact to 4D and nD.

That does sound pretty close to what I had in mind. Are i and j from your example associated with orthogonal axes in a plane?

Yes, they are. i would be the unitary vector in the horizontal direction, x, and j in the vertical one, y.

In fact, geometric algebras are very general, and the one I briefly sketched is not the only possible interpretation. If you are interested, you can find many good introductions online, directed at different audiences. You can also search for the term "Clifford algebras" if you are interested in a more formal approach.

Reduce these to the integral powers of i: i^0, i^1, i^2, i^3. The symmetries occur two apart (0, 2; 1, 3). Higher integral powers are equivalent to the 4 listed, i^n = i^(n mod 4).

Really, each power corresponds to an additional 90 degree rotation. 0th = 0 degree rotation (just scaling), 1st = 90 degree rotation, 2nd = 180 degree rotation, 3rd = 270 degree rotation (equivalently -90 degree). So that seeming CW rotation is also 3 CCW rotations.

Sure, I guess that's a consequence of general rotational symmetry.

The main thing I was pointing out were the different geometric behaviors of multiplication by real vs imaginary units—not so much the difference between negative and positive units on either axis.

Separate out the components, rotation and reflection and scaling. -1 is reflection. i is rotation by 90 degrees CCW. -i is both rotation by 90 degrees and reflection. Positive reals are scaling. So -ai is really rotation by 90 degrees, reflection, and then scaling by a.

But none of that is related to my key point: multiplication by unitary values is significantly different depending on which axis the unit comes from (real vs imaginary), hence my usage of 'asymmetric'.

In the last part of my first comment I point out the benefit (that I perceive) arising from the asymmetry. I also still wonder if a symmetric version (like I describe in the first comment) would be possible.

Splitting the geometric behavior into rotation, reflection, and scaling is interesting—so thanks for the description—I just don't see how it relates to my comment.

It would be great to get this level of explanation extrapolated all the way out to describe the behavior of octonions. Kidding, but not really ;)

I don't know about math, but at least in physics, "quantum teleportation" is a terrible name; it made way more sense to me when somebody called it "quantum telegraphy" instead.

There is "random variable" a.k.a function.

random variable as a shortening of "measurable function from a probability space" doesn't seem too bad to me

Don't see how this works as the extended version doesn't mention random or variable.

My point is that is isn't just a function, but a measurable one (with weak restriction on the measure space it has as domain) so we can't call it just "function" as you were suggesting.

The actual name might be debatable, but a shortened name rather than the full definition definitely makes sense here. Random variable seems like a good choice of name to me, thinking about the intuition we're formalizing with it.

But you propose to simply relabel as function, which doesn't work in general because random variable corresponds to a specific type of function. You could compromise by calling it a probability function, but then you start to collide with other uses of that word.

I agree random variable is awkward, though. I always avoided stats courses because it's full of so much jargon that collides with nomenclature used by mathematicians.

Yes, it's not just a function - there are strings attached but we deserve a better name - probability function is x100 better. "random variable" is exceptionally bad in that it leads the mind in irrelevant directions.

To follow one of the more common patterns for identifying a class of functions, it could have been named after one of the early pioneers in the field. But yeah, it would be hard to do worse than random variable, which is illogical and misleading.

I found this doc on the origins of the name (author also agrees it's terrible): http://www.glennshafer.com/assets/downloads/talks164_The-inv...

Sounds like it got mangled as work was being translated back and forth between Russian, English, French, and German.

Another approach is to use language not commonly spoken ( quickly - what does topology means) and consequently less burdened with cultural baggage.

They're not simply 2d numbers, though. ℝ^2 is also a 2d number, but has different multiplication rules, for example.

Intuitively elements of R2 are not numbers, they're vectors, or points in a 2 dimensional vector space. You can also treat complex numbers as vectors in the 2 dimensional complex plane.

"Intrinsic curvature" should be called something else, because people frequently drop the "intrinsic". This implies the existence of a higher dimensional space which usually doesn't exist. "Curvinture" would be a better name.

Hmm. I only see 'intrinsic curvature' used when there is a potential confusion about whether it's intrinsic or extrinsic (otherwise it's just 'curvature'). And if that is the case, so that there is an embedding in a higher dimensional space, then intrinsic/extrinsic curvature are excellent terms (imo). They were some of the first math terms I came across where I really appreciated the selection of terminology.

A great explanation about imaginary numbers: https://www.quora.com/How-do-you-explain-the-imaginary-numbe...

But... they aren't just 2D numbers. That would be vectors. Yes, they behave like vectors under addition, but they also have a ton of other behaviour that '2D numbers' doesn't help with at all, like what it means to raise a number to a complex power.

> Do you know any other examples

https://youtu.be/lFIYKmos3-s

Where "complex numbers" is Chinese, "imaginary numbers" is German, and "2D numbers" is English in this analogy.

Definitely: "determinant" should be called "volume".

Except they can be negative, and relate to area for 2D, not volume, and would require defining volume in dimensions higher than 3, which is not necessarily obvious or intuitive for beginners.

"Signed scaling factor" is more general, and some courses (e.g. ones I teach on) use that to introduce the idea.

I think that 'N-volume' including area, volume, and higher dimensional analogs is pretty understandable to a student. The fact that it's signed is more subtle, but still is not a big problem. The benefits to intuition of calling it 'volume' far outweigh the negatives.

Anyway, 'scaling factor' still leaves the signs mysterious.

Volume of what?

Of the parallelepiped generated by the columns of the matrix. (https://textbooks.math.gatech.edu/ila/determinants-volumes.h...)

The result of the determinant might be negative, which doesn't make sense for a volume, so you need to take the absolute value to interpret it that way.

Another way to think of it is the change in volume caused by the linear transformation determined by the matrix (http://algebra.math.ust.hk/determinant/01_geometry/lecture3....)

This might be one: Instead of rational numbers, call them "[integer-]ratio numbers" -- they're the numbers that can be expressed as a ratio of integers.

When you call them rational numbers, it sounds like you're saying they're the only numbers that are "smart" or that "make sense" -- the connection to ratios is obscured.

It's not a terrible name, it's just that we've forgotten the real meaning of "complex" and use it as a synonym for "complicated".

kowdermeister wrote about imaginary numbers, not complex numbers. "Imaginary" is a terrible name, because all numbers are imaginary. Even integers are imaginary, in the sense that the percentage of integers with actual physical representations is 0%, rounded to however many decimal places you like. "Real" is a terrible name for numbers for the same reason.

"Complex" was originally justifiable but it should be renamed to reflect changes in the English language.

What would happen if we started renaming mathematical objects to reflect changes in the English language? English will continue to evolve, and the vast body of mathematical literature would have to be constantly rewritten.

Wouldn't that cause much, much greater confusion due to mathematical nomenclature being a moving target rather than remaining stable?

Mathematical nomenclature is already a moving target. And the symbols don't have to change, only the English reading of them, e.g. ∫ is from the latin "summa" (sum), but you don't need to know that to read it as "integral"

I can't think of any examples of mathematical objects being renamed, do you have any?

There may be some amount of drift over time, but you can go to the research library of any university math department and find books from pre-WWII that are still totally readable because although style has shifted somewhat, the basic nomenclature hasn't.