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I can't recommend this site enough for explaining imaginary numbers and rotation through animations: http://acko.net/blog/how-to-fold-a-julia-fractal/

> But if you stop this half way then everything has just settled on zero, and doing that twice is obviously not going to get to the whole reflection.

I just wanted to say, this is probably the best intuitive explanation I've ever heard, so thanks.

Complex numbers being associated with rotation -- well the math and geometry always worked out no problem, but it always felt sort of... random or arbitrary to me.

But framing it that there ultimately needs to be a total reflection in the real numbers when multiplying by -1, and that rotation is the simplest way to achieve a smooth path to that which preserves all the necessary... that just clicks.

So thanks again!

> The key idea of imaginary numbers is to consider multiplication -1 to be a rotation by half a turn rather than a reflection. That is a lot easier to do half of! As soon as you have multiplication by -1 as a rotation by half a turn, it is obvious to identify i as rotation by a quarter turn.

It gets even more interesting when you add additional degrees of freedom so that such rotations can happen via more than one path. For example, quaternions add two extra degrees of freedom, and this lets you have an infinite number of square roots of -1. Any imaginary unit quaternion (i.e. ip+jq+kr where p^2+q^2+r^2=1) is a square root of -1.

Even in the complex numbers, there is more than 1 path, i^2 and (-i)^2.

On a non-geometric note, i and -i are algebraicly equivelent over the reals.

> it's because that's the whole point of defining imaginary numbers in the first place!

People used imaginary numbers for a long time before Cartesian coordinates even existed.

Indeed. There were 18th century mathematicians like Gauss who realized the importance of the geometric nature of complex numbers, but it didn't become central to the subject until the 19th century. The appearance of imaginary numbers as formal square roots of negative numbers goes back to the mid 16th century. As for Cartesian coordinates, i^2 = -1 has an intrinsic, coordinate-free interpretation in terms that would be instantly recognizable to the ancient Greeks, but it's certainly true that this way of thinking wasn't at the basis of the discovery and initial development of complex numbers, and thinking of geometric operators as generalized numbers would have seemed pretty alien for most of the 19th century as well.

> and thinking of geometric operators as generalized numbers would have seemed pretty alien for most of the 19th century as well.

Do you know of any resource treating that subject explicitly? I assume you mean the same kind of operator as in 'differential operator'—is that right? I can kinda see it maybe... but would definitely be interested in hearing the idea expanded on :)

By an operator I just mean a transformation. Certainly linear operators like differential operators qualify. The fact that these have an algebra in their own right goes back to work in the 19th century by Felix Klein and Sophus Lie on transformation groups and to later 20th work on linear algebra and functional analysis. It's stuff pretty much everyone learns as an undergrad nowadays, but the fact that you can do algebra on operators hardly without thinking is a relatively modern perspective.

I'll be honest, I've never understood how humanity didn't invent Cartesian coordinates until 1637, with all the other engineering we had.

Once we had linear equations, for example with the ancient Greeks, not one person ever thought to plot a line with it? Or to use it to calculate the necessary building materials for something like a pediment or cathedral?

Nicholas Oresme essentially thought of them in the 14th century. I'd be surprised if as you suggest there weren't a lot of sporadic particular uses of the idea. But Descartes's big accomplishment was to abstract the coordinates away from any specific problem. No matter how many people prior to his work used coordinate ideas in problem solutions that's a major accomplishment.

I think one of the keys to understanding math, and to a lesser but still significant extent, physics history is to remember what you believed as a child, and how you struggled with the concepts taught to you. And that's even with a math curriculum designed to lead you to modern math. (One can debate how effective it is at that, but that's a separate topic.) Those misconceptions we had as children are pretty fundamental to the human wetware. Even today, with centuries of refinement and educational advancement, really only a small fraction of people come away from school with the ability to think truly mathematically.

For instance, even "just" negative numbers is a fairly counterintuitive concept. Despite how they may seem universal today, they actually didn't pop up in all that many cultures historically before their current line. And that story gets repeated over and over for all sorts of developments. It takes time for fields of study to process and abstract these things, because they weren't just handed it on a silver platter in school.

(To the extent that that doesn't seem to be the case today, I'd say that as we have become more and more mathematically sophisticated and the area of mathematical inquiry exponentially increases, the dominant factor 'holding back' math today is our inability to cover territory. Today nobody can completely cover a major discipline before the next generation is already coming in with fresh brains. That's a relatively recent development.)

> It takes time for fields of study to process and abstract these things, because they weren't just handed it on a silver platter in school.

A sense of the phrase "knowledge is power" aligns with this.

I read Alan Kay's "User Interface - A Personal View" [1] recently, wherein he discusses Seymour Papert's [2] ideas on learning, specifically the 3 stages of learning. I found Papert's conception (with only mild exaggeration) to be illuminating. For example, I now have a explanatory model as to why certain inventions that did not require the du jour technology of the industrial age were developed so late in the game.

[1]: http://www.vpri.org/pdf/hc_user_interface.pdf [2]: https://en.wikipedia.org/wiki/Seymour_Papert

Apollonius used what basically amounts to Descartes’s coordinate method in the 1st century for studying conic sections, and European mathematicians of the 16th–17th centuries were all quite familiar with his work. But the formulation was somewhat cumbersome and context-specific. https://en.wikipedia.org/wiki/Apollonius_of_Perga#The_coordi...

The world also had a long cartographic tradition based on coordinates.

Moreover, Descartes’s book only used one coordinate axis at a time, only used positive numbers, and used it as a tool for setting up geometry problems to be solved algebraically, not as a general tool for what we now think of as graphing functions/equations. The way we think of the “Cartesian plane” is not the way Descartes thought about it.

The history of these conceptual developments is richer and more complicated than popularly imagined today.

Always replace "intuitive" with "familiar" and you get an insight into what the person is talking about. So the task is to explain Euler's equation in terms of what you're already familiar with. Hmmm.

[1] Theorem by Jeff Raskin - https://www.asktog.com/papers/raskinintuit.html