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Why does e to pi i equal -1? (2015) [video]

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Re: Why does e to pi i equal -1? (2015) [video]

#71

Post by bsaul »

pyromine wrote: post #36
johnfn wrote: post #31

Earlier quoted context omitted.

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 unders…

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.

Re: Why does e to pi i equal -1? (2015) [video]

#72

Post by DiThi »

unholiness wrote: post #52

This explanation strikes me as a little too aggressive in throwing out the notation with the bathwater, only reaching its result by redefining the terms we already have intuition for into space-stretching operations that don't work like arithmetic does in my head. In that sense I think I have the same problem with this proof that I do with the standard one, where you add the Maclaurin series of cos(θ) and i * sin(θ)…

I came here to post that video by Mathloger, it's very well explained.

Re: Why does e to pi i equal -1? (2015) [video]

#73

Post by dnautics »

espeed wrote: post #67
dnautics wrote: post #64

Earlier quoted context omitted.

the standard IEEE floating point uses a 'sign and magnitude' system. Negative numbers are basically made by flipping a single bit. I've been playing around with a number system that expresses real numbers as two's complement (like how integers work) - xor all your bits and then add one. In this system, 0 is -0 and the number that is weird for signed integers: 0b100000...0000 is infinity. Now, there's an exponent and…

Have you seen "Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic"? https://www.youtube.com/watch?v=aP0Y1uAA-2Y

Well, I'm the guy in the black shirt who did the demo.

If you liked that lecture, I'm already starting on some verilog implementations.

This is an example multiplication 8 bit * 8 bit -> 16 bit unpacked (20 bits). It differs from standard floating point in that the fractions are stored as two's complement. It takes a little bit of wrapping your head around, but the hidden bit for negative numbers is actually -2 ! Moment of zen.

https://github.com/interplanetary-robot/mullinengine/blob/ma...

Re: Why does e to pi i equal -1? (2015) [video]

#74

Post by jacobolus »

godmodus wrote: post #48
kmill wrote: post #5

Earlier quoted context omitted.

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…

I love this answer. For some reason, I've been dreaming about negative numbers lately. I think they deserve their own number set notation.

What do you mean by “number set notation”?

Re: Why does e to pi i equal -1? (2015) [video]

#75

Post by espeed »

bsaul wrote: post #71
pyromine wrote: post #36

Earlier quoted context omitted.

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 unders…

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.

Re: knowing vs understanding, have you read "Richard Feynman on education in Brazil"?

http://v.cx/2010/04/feynman-brazil-education

Re: Why does e to pi i equal -1? (2015) [video]

#76

Post by [deleted] »

jeffwass wrote: post #62
cousin_it wrote: post #3

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.

When I was a physics TA, I used similar arguments with double derivatives show the connection between imaginary exponentials and sines/cosines. For f(x)=e^(kx), double derivative d^2f/dx^2 = k^2 f(x) Meanwhile for g(x)=sin(kx), d^2g/dx^2 = -k^2 g(x), and similarly for cosine. So if k is imaginary, from a differential equations point of view, the exponential behaves exactly like a sine or cosine. That shows the genera…

[deleted]

Re: Why does e to pi i equal -1? (2015) [video]

#77

Post by mih »

skybrian wrote: post #29

If you prefer slides over a video, here is a slideshow I wrote: https://docs.google.com/presentation/d/1oMNjkDp-LieSGnZEwNpc... Or you might prefer Better Explained's article: https://betterexplained.com/articles/intuitive-understanding...

Upvoted for the Better Explained link. I always liked their explanation of e https://betterexplained.com/articles/an-intuitive-guide-to-e...

Re: Why does e to pi i equal -1? (2015) [video]

#79

Post by AndrewOMartin »

dnautics wrote: post #73
espeed wrote: post #67

Earlier quoted context omitted.

Have you seen "Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic"? https://www.youtube.com/watch?v=aP0Y1uAA-2Y

Well, I'm the guy in the black shirt who did the demo. If you liked that lecture, I'm already starting on some verilog implementations. This is an example multiplication 8 bit * 8 bit -> 16 bit unpacked (20 bits). It differs from standard floating point in that the fractions are stored as two's complement. It takes a little bit of wrapping your head around, but the hidden bit for negative numbers is actually -2 ! Mom…

@espeed didn't ask if you contributed significantly to the lecture, they asked if you'd seen it. Please stick to the question!

Re: Why does e to pi i equal -1? (2015) [video]

#80

Post by AndrewOMartin »

hueving wrote: post #46
snogglethorpe wrote: post #35

Earlier quoted context omitted.

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.

And who would you prefer in the pilot's seat?

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.

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