I don't really like this, because it's a self limiting quote that leads to complacency over real understanding. Especially in the case of imaginary numbers and e where intuitive understanding only doesn't exist because mathematics has a history of being poorly taught.
I think there is both a good and bad aspect of this quote, you've done a great job highlighting the bad.
The good, is in essence that you don't always need to be fully comfortable and in fact will not always be comfortable. I find in mathematics if you try to always strive for total comfort you will never progress, as a lot of the comfort comes from advancing past a topic and upon revisiting it you realize you understand the fundamentals better than you thought.
Totally agree. My father would always tell that what's important is to truely understand things and not just be able to do the problem. This had two bad side effects : 1/ i just read the course and didn't do the exercises, and 2/ i "blocked" on things like infinitesimal calculus, because i couldn't get my head on its true meaning.
In every maths subject at uni, I'd do quite well in the exams, but not fully 'getting it'. Then the next semester, in the next subject along the, I suddenly understood it, as if the semester break had given things a chance to arrange themselves, and the context of the more advanced subject made things clear.
Honestly, after spending months studying the subject, I don't think it's really possible to "get" complex numbers. I just view them as affine transformations written in an unusual notation. I don't think they make sense as anything but a recontextualization of R^2.
What I don't get is why someone would bother with complex numbers and their silly notation when linear algebra would work perfectly fine. I know that in certain contexts they are useful. E.g. to avoid losing information when solving polynomials. But that's a minuscule fraction of their range of application. Nearly every practical use I've seen of complex numbers just use them as a vector representation.
Well the two uses I'm most familiar with are AC circuit analysis and quantum mechanics. They can both be reformulated without complex numbers of course, since nothing is special about i.
Yet the complex versions are a lot easier to work with, because even in manifestly real formulations, the complex structure is still there, but in disguise:
The problem with that is that complex numbers initially emerge as roots of polynomials with real coefficients. Getting to affine transformations from there seems a much bigger leap than asserting there is a square root of -1.
The video linked in this post will only make sense if you accept that x e^(theta i) and x k are respectively the rotation part and the scaling part of a linear transformation of x. I'm not aware of any other way to intuitively grasp an expression like e^(pi i).
this is an old comment, but e^z (over C) can be defined as the analytic continuation of e^x (over R). This is a much more interesting definition, since it depends on the fact that that function is unique.
Mild snarkiness aside, is the quote really inaccurate?
"Getting used to" [something] has a pejorative sense, but it also just means "becoming familiar with," and really understanding something is, in a way, simply being so familiar with it that reasoning about it is second nature... at some point, things just sort of start to make sense...
Not quite, you can get used to something without understanding it at all. Many people that fly are used to it but they don't understand the basic physics of flight at all.
To say suggest that you don't understand something without a formal underlying theory is one view, but not mine. I feel like I understand English (as in a deep understanding, not just the ability to interpret sentences) without anything like a set of rules, and everything like a set of experiences akin to a pilot's experience with flight.
