I think Mathologer's take is much more intuitive:
Why does e to pi i equal -1? (2015) [video]
101–110 of 148 posts
Re: Why does e to pi i equal -1? (2015) [video]
#102Earlier quoted context omitted.
I love this answer. For some reason, I've been dreaming about negative numbers lately. I think they deserve their own number set notation.
Speaking of number sets. I never understood why complex numbers are considered one. I mean, yes they "are" a set, but besides that they are completely different. All number sets I learned about did fill some gaps in one dimension, but complex numbers somehow added a new dimension. Like real numbers stood in an entirely different context to rational numbers than complex numbers stood to real numbers.
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Re: Why does e to pi i equal -1? (2015) [video]
#1031) 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…
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Re: Why does e to pi i equal -1? (2015) [video]
#104Earlier quoted context omitted.
Honestly, after spending months studying the subject, I don't think it's really possible to "get" complex numbers. I just view them as affine transformations written in an unusual notation. I don't think they make sense as anything but a recontextualization of R^2.
So what don't you get? Are you implying there is something more to complex numbers?
What I don't get is why someone would bother with complex numbers and their silly notation when linear algebra would work perfectly fine. I know that in certain contexts they are useful. E.g. to avoid losing information when solving polynomials. But that's a minuscule fraction of their range of application. Nearly every practical use I've seen of complex numbers just use them as a vector representation.
Re: Why does e to pi i equal -1? (2015) [video]
#105Earlier quoted context omitted.
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…
Posits look too good to be true! Is there a reason why regime bits do not include the sign bit? Then both "0 0001" and "1 1110" could be interpreted as 4 regime bits. Even better we could include the last flipped bit as well and we would have 5 bits. Edit: Well. I see it would result in losing the values 0 and 1. Another question: Since it is fixed length of 4 bits (for N=32) why don't we just extract the 4 bit value…
You need to be a lot more explicit about what you’re asking.
Look at the slides, searching down for “At nbits = 5, fraction bits appear”. Notice that every possible bit pattern is used and meaningful.
http://web.stanford.edu/class/ee380/Abstracts/170201-slides....
Re: Why does e to pi i equal -1? (2015) [video]
#106Earlier quoted context omitted.
I love this answer. For some reason, I've been dreaming about negative numbers lately. I think they deserve their own number set notation.
Speaking of number sets. I never understood why complex numbers are considered one. I mean, yes they "are" a set, but besides that they are completely different. All number sets I learned about did fill some gaps in one dimension, but complex numbers somehow added a new dimension. Like real numbers stood in an entirely different context to rational numbers than complex numbers stood to real numbers.
C (complex numbers) is a field, just like R (real numbers). It's an algebraically closed field, unlike R, so in some sense it's actually the most natural set to call "numbers".
Re: Why does e to pi i equal -1? (2015) [video]
#107This 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)…
Your (1) is just one of many possible (technically equivalent) definitions for the exponential function. And arguably not the most basic/natural/intuitive one. The primary motivation for the exponential function is to be the inverse of the logarithm function. And the motivation of logarithms is to convert multiplication problems to addition problems, so they could be solved with table lookups (later performed on a sl…
Of course! I think there's a good reason (1) is the first definition of e^x we get in school, though. I don't think it's the most "natural" definition, but I think it's the most grok-able. It's easy for high-schoolers to learn precisely _because_ each part of it is so intimately familiar that it's clear without hand-waving precisely what everyone's talking about.
When we then step into viewing real exponentials as "The thing that gets bigger at a rate proportional to where we're at", it's un-abstract, which is honestly important for intuition. Of course you're going to want to then view it with the calculus definition, or as a matrix, or as an operator... But I wouldn't start there.
So I feel the same about the complex version. There's another nice proof in this thread that starts with the calculus definition and follows a similar track. It's a much more beautiful proof! But I worry some will lose intuition when we start the conversation with "a complex exponential's rate of change is perpendicular to its position"...
That said, if I were to ever teach a math class on complex numbers, I think fleshed-out versions of these two 4-step proofs are the ones that together best build intuition for what "rotation in complex exponentials" really means. That rotational aspect is the foundation of much of the handwaving done in control theory, DSP, ASP, signals and systems, and really anything involving fourier and laplace transforms. So, it's important to grok it well.
Re: Why does e to pi i equal -1? (2015) [video]
#108I had never understood how imaginary numbers had any bearing on physics (probably because I'd never been taught). But lately I've thinking about how all mathematics must have some natural physical equivalent or relation. E.g. how does multiplication happen in nature.. what does it mean really to multiply something?...how does this happen in nature? For the most part, we take these things as facts and learn the mechanics, but in physics, it seems to me, you really need to understand how these algebraic interpretations translate into physical realities (also, perhaps sort of obvious for everyone reads HN). For someone who enjoys discovering these perhaps obvious things later in life, it's clear how mathematicians and physicists could clearly come to the same mathematical conclusions from completely opposite vectors.. I guess this probably happens all the time.
Re: Why does e to pi i equal -1? (2015) [video]
#109In Lie theory, given a Lie group, there's a general notion of an "exponential function", which maps elements in the tangent space to the identity to "full" elements of the group.
In this case, our group is the unit circle in the complex plane. This is not circular logic, by the way. If a, b are complex numbers on this circle, then |a| = |b| = 1 and so |ab| = |a||b| = 1.
The identity of the group is the complex number 1 (with plane coordinates (1,0)). So the tangent space to the identity is the vertical line 1 + it, for t a real-valued parameter. In two-dimensional coordinates, that expression looks like (1,0) + t * (0,1).
(To be completely precise, the tangent space is a vector space, not a line displaced from the origin. In particular, the tangent space must contain a zero vector.)
If v is an element of this tangent space, then in Lie theory the exponential of v, exp(v), is defined to be g(1), where g is the unique geodesic on the circle (passing through the identity element 1) whose velocity at time 0 is v.
Visually, this is sort of like taking the tangent vector v = ti, placing its base at 1, then wrapping it around the circle and marking where its endpoint ends up at. If the vector is v = pi*i, then it has length pi, so it will end up demarcating an arc length of pi on the circle. Since we're working with the unit circle, this takes us straight to (-1,0).
I'm still leaving a lot out, of course -- most importantly why this notion of exponential has anything to do with the ordinary one.
Re: Why does e to pi i equal -1? (2015) [video]
#110This 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 agree, the suggestion that anyone could/would be able to just throw out everything they know about arithmetic is sketchy at best, and even it's sketchier to make it seem as though a result like Euler's formula could purely be derived from geometric intuition. Sure, it "makes sense" in the video, but I don't think it's possible to truly grok and appreciate the result without having done any calculus. The real joy of…
The Maclaurin trick is what I'd call the "standard proof", and I think it's the source of a lot of the lack of intuition surrounding e^i... It's exactly what you call it, a "trick", where you prove two rabbits are identical by transforming them both into an infinite number of hats.
On the one hand, it's all sound logic and algebra, but on the other hand, I certainly don't blame Randall Munroe for seeing that proof in high school and going on to say "I have never been totally satisfied by the explanations for why e to the ix gives a sinusoidal wave."[0]