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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]

#41

Post by gizmo686 »

snogglethorpe wrote: post #39
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.

Is this just the contents of the video in text form? 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!

I can't tell if you are being sarcastic, but no, parent's comment is not the contents of the video. The video provides a geometric construction, and makes no use of calculus.

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

#42

Post by gizmo686 »

dvt wrote: post #37

I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane): > ... 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,…

>It's entirely non-obvious WHY we should be okay with rotating all of a sudden.

Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural; and it defiantly seems to be well defined. Additionally, it appears obvious that the presented generalization is equivalent to the original definition if considered only along the x axis.

From the perspective of the video, the fact that e^x is written as an exponential is "a vestige of its relationship with repeated multiplication". In the construction presented in this video, we should think of e^x as the "natural" homomorphism from adders to multipliers.

The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.

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

#43

Post by snogglethorpe »

gizmo686 wrote: post #41
snogglethorpe wrote: post #39

Earlier quoted context omitted.

Is this just the contents of the video in text form? 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!

I can't tell if you are being sarcastic, but no, parent's comment is not the contents of the video. The video provides a geometric construction, and makes no use of calculus.

Not being sarcastic, I just didn't watch the video.

I was wondering if it was basically the same thing with nice animations...

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

#44

Post by recursive »

snogglethorpe wrote: post #39
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.

Is this just the contents of the video in text form? 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!

It's not really any clearer to me. I cannot generalize my understanding of exponents to anything that deals with imaginary numbers.

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

#45

Post by jacobolus »

recursive wrote: post #44
snogglethorpe wrote: post #39

Earlier quoted context omitted.

Is this just the contents of the video in text form? 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!

It's not really any clearer to me. I cannot generalize my understanding of exponents to anything that deals with imaginary numbers.

Here, http://www.shapeoperator.com/2016/12/12/sunset-geometry/

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf

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

#46

Post by hueving »

snogglethorpe wrote: post #35
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.

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.

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

#47

Post by moresilenter »

dvt wrote: post #37

I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane): > ... 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,…

as a masters in mathematics, this is exactly what i came here to write. thanks!

To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. I honestly think the best way to get a visual sense for why multiplication by rexp(itheta) is to look at the first few terms of the taylor series added together and see that the adders combine into a spiral that converges on rcos(theta) + isin(theta). that is still plenty visual for me.

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

#48

Post by godmodus »

kmill wrote: post #5
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.

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.

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

#49

Post by dvt »

gizmo686 wrote: post #42
dvt wrote: post #37

I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane): > ... 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,…

>It's entirely non-obvious WHY we should be okay with rotating all of a sudden. Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural; and it defiantly seems to be well defined. Additionally, it appears obvious that the presented generaliza…

> Why shouldn't we? We have a set of object called multipliers that we want to generalize to the 2 dimensional plane. The presented generalization (eg, the multiplier identified by x maps the point at 1 to the point at x) seems natural

Rotating all of a sudden does not feel natural to me at all. Why not start cutting up the 2D plane? Why not fold it? In analytic terms, we would cause discontinuities, whereas rotating is "smooth". But this is definitely nontrivial. Rotating is just so arbitrary. I think that skewing or flipping, for example is just as "natural" as rotating.

> The fact that this function has anything to do with exponential, trigonometry, or analysis is neat, but not important in this context.

It's actually at the heart of why rotation (and not some other geometric operation) is key to e^iπ.

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

#50

Post by dvt »

moresilenter wrote: post #47
dvt wrote: post #37

I've seen this a while ago, and while it's pretty instructive, it's actually also pretty confusing. The magic happens in a seemingly-innocuous throwaway sentence at around 4:20 (after being introduced to the 2D plane): > ... 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,…

as a masters in mathematics, this is exactly what i came here to write. thanks! To have a notion that multiplication by imaginaries causes rotation, you'd need Euler's formula. I honestly think the best way to get a visual sense for why multiplication by r exp(i theta) is to look at the first few terms of the taylor series added together and see that the adders combine into a spiral that converges on r cos(theta) + i…

Sure thing!

> .. see that the adders combine into a spiral that converges on rcos(theta) + isin(theta)

Incidentally, that's how Mathologer explains e^iπ and I like that explanation a lot more.

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