I believe they are legit. I think my sister pointed me there months ago ... but they were pretty pricey. Seems to be more affordable now — roughly $1 (U.S.) per mask.

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).

I don't know, I felt cheated when I learned about imaginary numbers in high school (wait, you can take the square root of a negative number?). It felt like you had patiently learned the rules of math and then someone just made some shit up.

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".

Imaginary numbers simply represent a different and more powerful system like how integers gain power when you introduce 0 and negative numbers. Rational numbers (fractions) can’t represent irrational numbers like pi or e etc.

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.

School teaching tends to be authoritarian, so that's a reaction I felt too. "Less than no apples? Minus times minus is plus? What the hell is this and who ordered it?"

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.

I think the biggest problem with authoritarian thinking about math is that math is not a discovered set of authoritarian principles, but rather a colossal project to make up weird relationships that satisfy the rules of the weird relationships that were already made up.

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."

They're used in a generalised solution to cubic equations (analogous to the quadratic solution we all learn in high school), where they represent the side length of a square with negative area.

Imaginary numbers seem like something that shouldn’t be taught in high school. Algebra, geometry, trigonometry, and calculus are all useful even if you don’t recall the specifics of the notation and calculation. Imaginary numbers are niche.

They feature fairly prominently in quantum mechanics.

In my humble opinion, imaginary numbers are a core part of algebra, calculus, trigonometry, and physics, and all sufficiently intelligent people should learn about them. The reason exponential curves and sine waves show up all over the place in reality is because they are the solution to some simple differential equations. Imaginary numbers show you how these are fundamentally very similar things. Similarly, imaginary numbers make it clean and simple to solve cubic equations. And the idea that introducing this concept of "imaginary number" makes multiple practical applications much simpler, therefore we added it to mathematics, is core to how mathematics itself evolves.

I’m willing to believe that. But I certainly never learned that. I only briefly got into differential equations by the end of high school. For our program imaginary numbers weren’t much more than learning some rules about how they work exponentially and an awkward introduction to vectors with complex numbers that really should have started with just real numbers.

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.

Why not instead someone just market a replacement keyboard for one of the more popular laptops? It sounds as though modern keyboards are the primary pain point for newer laptops.

(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.)

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