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It's a nice video but it goes so fast I don't think people who don't already understand this stuff will get it.
"Young man, in mathematics you don't understand things. You just get used to them."
I've been watching 3Blue1Brown a lot recently, and he's been linked to many times here too. His work is absolutely amazing, and he even wrote a python library to make the animations!
I do have 1 gripe with this video though, and that is his handwaving around "natural".
More specifically, as far as this particular case goes, there is nothing natural about the choice of e^x, or the significance of pi.
As he identifies, we are interested in some function that maps adders into multipliers with the property f(x+y)=f(x)f(x).
To uniquely identify such a function, we need to add an additional constraint. He chose to add f(pi) = -1. He justifies this by arguing that pi is the length you would travel along the unit circle to arrive at -1. This is true (and the underlying reason why pi and e end up being natural), however using this argument seems to break the abstraction for me.
Under this construction, in the equation f(i * pi)=-1, "i * pi" is an object in the set of multipliers, and "-1" is an object in the set of adders. Specifically, "i * pi" is a function which takes a plane (or perhaps a point) and rotates it, while "-1" is a function which takes a plane (or point) and slides it.
He then invokes an unstated mapping, g, to convert the multiplier [0] "i * pi" into the real number "pi". He than insists that g(x) gives the distance a point would travel along the unit circle when the multiplier x is applied to it.
At this point, because arriving at the point "-1" [1] from the point "1" through rotation requires traveling pi distance, it makes sense that f(i * pi) = f(g^-1(pi)) = -1
I am still not convinced that (within this construction alone), this is a more natural choice that saying that f(i) = 1, but would agree that it is one of the two natural choices. To get to f(x) = e^x being the natural choice requires showing that it comes up in all sorts of unrelated parts of math, so it is probably more natural.
[0] It might be better to speak of "rotational multipliers" here, as I am not sure how to natural define g(x) for multipliers that stretch the plane, instead of or in addition to simply rotating it.
[1] This "1" is again distinct from the adder "1" and multiplier "1", but plays a central role in defining them, so I do not object to its usage.
Earlier quoted context omitted.
"Young man, in mathematics you don't understand things. You just get used to them."
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.
Earlier quoted context omitted.
"Young man, in mathematics you don't understand things. You just get used to them."
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.
"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...
Earlier quoted context omitted.
"Young man, in mathematics you don't understand things. You just get used to them."
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.
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.
> ... This can now include rotating along with some stretching and shrinking ...
It's entirely non-obvious WHY we should be okay with rotating all of a sudden. The real answer is not super complicated, but it deals with a couple of amazing relationships between exponential functions and trigonometric identities[1]. So really, we don't have to accept "rotating" as some weird new action we can do when moving into the complex plane, we just have to accept that trigonometry is weirdly related to analysis due to some very cool properties.
[1] https://www.phy.duke.edu/~rgb/Class/phy51/phy51/node15.html
1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.
In a linear algebra course I helped teach last semester, I had the students go through the exercise of writing down the matrix for "multiplication by i," where the complex numbers are thought of as a two-dimensional vector space with {1,i} as a basis. Then I asked them to recognize the matrix (it's a 90-degree rotation of the plane), and then I asked them for an interpretation of that matrix squared (a 180-degree rot…
1) e^x is a function whose derivative is equal to its value. 2) e^ix is a function whose derivative is equal to its value rotated by 90 degrees (ie^ix). 3) As x goes from 0 to pi, the trajectory of e^ix always has a velocity vector perpendicular to its current position. For example, when x = 0, the current position is 1 and the velocity vector is i. 4) So the trajectory a circle arc of length pi, which ends at -1.
In any case, thanks! I don't need to watch the video now, as you've very clearly explained it in only four lines of text!
Is it pure chance that this was posted only two or three days after I watched it along with a few other e to pi = -1 videos? Randomness aside. 3blue1brown makes some wonderful math videos that I find really explain the intuitiveness of some of the ideas. I was unfortunately cursed with a math teacher who for whatever reason required us to memorize until we passed the test. Imaginary numbers were taught as "something…
Richard Feynman used to go up to people all the time and he'd say "You won't believe what happened to me today... you won't believe what happened to me" and people would say "What?" and he'd say "Absolutely nothing". I agree wholeheartedly with the quality assessment of 3blue1brown videos. All mathematics should be clear and intuitive, by definition :)
I disagree with this completely, and so does all of higher mathematics. It's neither clear nor intuitive. In fact, as soon as you start learning about infinities (Calc I), intuition becomes hit and miss.