And we'll laugh about it some day.
Like the "hand held shot" that became popular a decade or so back — in every serious drama the director seemingly gave a GoPro to someone with tremors. A number of TV series were comically unwatchable (well, for me at least).
Perhaps this persisted as habit even after the development of colloidal photography?
Yeah, I am even now a little suspicious of the photos authenticity for this reason.
Probably when I started my slow deviation away from math (certainly a part of why I never majored in it).
It's too bad though. Over the years since I have come to trust mathematicians more and more.
A co-worker explained to me how the imaginary component of complex numbers represents the phase information when performing an FFT. I think he was even trying to explain to me how it is why an FFT is not reversible, why you lose the phase information from the original (but I was already lost).
(Odd too that the human ear cannot distinguish between two audio sources for which every thing is the same but for the phase. Related?)
I am assuming now, from a position of ignorance if that is not obvious, that imaginary numbers are quite clever after all, perhaps neither a hack nor "imaginary".
It’s often useful because you can use complex numbers as an intermediary step in many calculations (such as solving cubic equations) while still ending up with a non imaginary number at the end.
The history is actually fairly interesting: https://www.youtube.com/watch?v=cUzklzVXJwo
I wish it'd been explained like: counting numbers like 1, 2, 3 obey certain laws, e.g. adding in different ways gets the same result. If you relax just a few of the laws in the right way, you can find a broader class of things obeying the shared logic. In the case of complex numbers, we're dropping ordering, and finding that takes us from 1-d sliding and stretching (adding and multiplying) to 2-d, where the stretching becomes stretching and turning.
Approaching imaginary numbers as:
> "Eventually, mathematicians got sick of not being able to achieve a negative result through multiplication, so Rafael Bombelli finally made up imaginary numbers using the work of other Italians and even some Greeks. Unfortunately for him, most people thought his idea and rule set was stupid until about two hundred years later. We will now learn what sorts of nonsense he made up--those of you who are interested in electricity had better listen close--"
will inevitably lead to a different mindset about math than:
> "Today's lesson is that the square root of negative one is i. Write that down, because it WILL be on the test."
-1 x -1 = -1 * (0 - 1)
= (-1 * 0) - (-1 * 1)
= 0 - (-1)
They feature fairly prominently in quantum mechanics.
Maybe just teacher quality though. I can’t forget the frustration of both the class and our calc teacher when she realized our trig teacher from the prior year never conveyed the relationship between various trigonometric identities and the Pythagorean theorem and instantly made it vastly more intuitive.
// BOGUS: assuming 'x' will never be greater than 1024.
Sort of tells future engineers, yeah, I know it's shit.
A comment I left that I am sometimes reminded of by ex-coworkers still at that company.
Yes, I can also recommend Futility Closet for HN readers.
(Or if a 3rd party had marketed an inexpensive replacement power-brick for the author's 2001 Apple iBook perhaps he would still be using it.)