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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]

#141

Post by askvictor »

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…

In every maths subject at uni, I'd do quite well in the exams, but not fully 'getting it'. Then the next semester, in the next subject along the, I suddenly understood it, as if the semester break had given things a chance to arrange themselves, and the context of the more advanced subject made things clear.

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

#142

Post by ouid »

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.

This isn't a proof, and while that may be obvious to you, it's not obvious to everyone who sees it.

There exists no logical path to the statement from the previous definition of e^x because, up to this point, e^x is a function on real numbers. e^ix is nonsense until you define e^x in complex coordinates, at which point the proof needs to rely on properties of that definition.

what you have posted misses the nature of the insight.

If two analytic functions on the complex numbers agree on uncountably infinitely many points, then they agree everywhere they are defined. This means that there is a unique analytic function that maps (ix)[x \in R] to the circle and (x)[x \in R] to the natural exponential, and they are the same complex function. Without the knowledge that analytic continuations are unique, the statement is entirely (pun intended) meaningless.

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

#143

Post by cousin_it »

ouid wrote: post #142
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.

This isn't a proof, and while that may be obvious to you, it's not obvious to everyone who sees it. There exists no logical path to the statement from the previous definition of e^x because, up to this point, e^x is a function on real numbers. e^ix is nonsense until you define e^x in complex coordinates, at which point the proof needs to rely on properties of that definition. what you have posted misses the nature of…

Sure, step 1 relies on the existence of e^z which is not obvious. Step 2 relies on the chain rule over C, step 4 relies on having a unique solution to a differential equation, etc. My comment can be expanded to a proof, but it was more aimed to explain why Euler's identity is natural.

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

#144

Post by ouid »

cousin_it wrote: post #143
ouid wrote: post #142

Earlier quoted context omitted.

This isn't a proof, and while that may be obvious to you, it's not obvious to everyone who sees it. There exists no logical path to the statement from the previous definition of e^x because, up to this point, e^x is a function on real numbers. e^ix is nonsense until you define e^x in complex coordinates, at which point the proof needs to rely on properties of that definition. what you have posted misses the nature of…

Sure, step 1 relies on the existence of e^z which is not obvious. Step 2 relies on the chain rule over C, step 4 relies on having a unique solution to a differential equation, etc. My comment can be expanded to a proof, but it was more aimed to explain why Euler's identity is natural.

step 1 isn't "not obvious"; it's nonsense. What, exactly, are you trying to prove exists? e^z? e^z is undefined at this point. If you're trying to prove that e^i\pi=-1, you need to start with a definition of e^z.

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

#145

Post by AndrewOMartin »

hueving wrote: post #134
AndrewOMartin wrote: post #95

Earlier quoted context omitted.

I'm refuting that difference.

Not very well. I'm not talking about pilots. I fly with many frequent fliers that are completely comfortable with flying while having no technical understanding of the mechanics.

My mistake. I misread the phrase "many people that fly" as "many people who pilot planes".

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

#146

Post by AndrewOMartin »

nitrogen wrote: post #132
AndrewOMartin wrote: post #79

Earlier quoted context omitted.

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

If you are unsure why you were downvoted, I strongly suspect it's because politely giving and receiving due credit are important, knowing that one's interlocutor was involved in specific research is very useful information (to know what questions to ask), and it's probably kind of rude to call someone to account in this manner. Hope this helps :)

I was being sarcastic on a whim, fully aware that it would attract downvotes.

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

#147

Post by ouid »

E6300 wrote: post #111
jameshart wrote: post #99

Earlier quoted context omitted.

The problem with that is that complex numbers initially emerge as roots of polynomials with real coefficients. Getting to affine transformations from there seems a much bigger leap than asserting there is a square root of -1.

The video linked in this post will only make sense if you accept that x e^(theta i) and x k are respectively the rotation part and the scaling part of a linear transformation of x. I'm not aware of any other way to intuitively grasp an expression like e^(pi i).

this is an old comment, but e^z (over C) can be defined as the analytic continuation of e^x (over R). This is a much more interesting definition, since it depends on the fact that that function is unique.

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

#148

Post by mtourne »

jacobolus wrote: post #138
mtourne wrote: post #121

Earlier quoted context omitted.

Quaternions are a useful tool for manipulating rotations in a lot of common applications. But that wasn't my point here. Also quaternions are hard to grasp by humans. I find axis-angle much more palatable in general. Imaginary numbers can be represented with a 2x2 skew symmetric matrix with no stretch of the imagination at all. And 3x3 skew symmetric matrixes represent rotations most compactly with only 3 actual vari…

From what I understand that would be (sorta; I’m not an expert in Lie theory) the logarithm of a rotation. I find the stereographic projection to be a more useful way to compress an arbitrary rotation down to 3 dimensions, for most purposes.

Yes, precisely. exp() is the mapping from so(3) to SO(3), so you can use log to go the other way around.

Here is the relationship with the other representations [1]

Where it becomes interesting for our rigid-body transform application (or recovery of it, in the case of computer vision), is with Twist coordinates (6 element vector) which will map to a 4x4 Transform, again using the matrix exp() operator [2].

[1] https://en.wikipedia.org/wiki/Axis%E2%80%93angle_representat... [2] https://en.wikipedia.org/wiki/Screw_theory#Twists_as_element...

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