Why does e to pi i equal -1? (2015) [video]
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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]
#2Is 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 that will help you in college"
Re: Why does e to pi i equal -1? (2015) [video]
#31) 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.
Re: Why does e to pi i equal -1? (2015) [video]
#4This seemed like voodoo until i took signals and systems. Then it's just a normal observation about vector sums on the complex unit circle.
Re: Why does e to pi i equal -1? (2015) [video]
#51) 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 rotation). One thing to draw from this: for the real line, negation doesn't reflect the line, but instead it's a rotation through another dimension of numbers!
When we went on to differential equations, I tried to convince them that e^(ix) is meaningful using your #2: it is a path around the unit circle since it's the solution to z'=iz, and we already agreed multiplication by i is a 90 degree rotation, and circles are what you get when velocity is orthogonal to vector position. It's a bit tougher to convince someone that x is arclength, though, especially when they aren't too comfortable with complex numbers yet.
Re: Why does e to pi i equal -1? (2015) [video]
#6For anyone that finds it beautiful isn't there a bit of humanisation and definition involved (for example the Sine function used in deriviation uses 'pi' instead of 90 degrees), not to mention Sine is a human created function. You could have e to pi (90 * -1) too.. or a different method to define angles instead of having 360 degrees (base 60)
Re: Why does e to pi i equal -1? (2015) [video]
#7For anyone that finds it beautiful isn't there a bit of humanisation and definition involved (for example the Sine function used in deriviation uses 'pi' instead of 90 degrees), not to mention Sine is a human created function. You could have e to pi (90 * -1) too.. or a different method to define angles instead of having 360 degrees (base 60)
Yeah, you can have the same thing with degrees if you use a different exponent base than e:
e^((pi/180)ix) = (e^(pi/180))^ix = E^ix
E^180i = -1
The particular base e has some nice properties, though, like that its rate of change d/dx e^ix at a given point x is just i e^ix. The rage of change for E^ix, on the other hand, is d/dx E^ix = (pi/180) i E^ix, which is a little less "natural". This strange fact -- that using radians makes the expression e^ix = cos x + i sin x have a nice derivative -- is one of the reasons why mathematicians like to define these functions in terms of radians instead of degrees.
Re: Why does e to pi i equal -1? (2015) [video]
#8For anyone that finds it beautiful isn't there a bit of humanisation and definition involved (for example the Sine function used in deriviation uses 'pi' instead of 90 degrees), not to mention Sine is a human created function. You could have e to pi (90 * -1) too.. or a different method to define angles instead of having 360 degrees (base 60)
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Re: Why does e to pi i equal -1? (2015) [video]
#9This seemed like voodoo until i took signals and systems. Then it's just a normal observation about vector sums on the complex unit circle.
It seems like voodoo when you have high-school mathematics, because you've been taught about "to the power of" in terms of multiplying something by itself a certain number of times, and multiplying something by itself an imaginary number of times is nonsensical.
Once you've done enough maths you think of it in a different way and e^ipi seems obvious.
It's just a different conception of what the symbols mean, I suppose.
Re: Why does e to pi i equal -1? (2015) [video]
#10The most elementary explanation I ever got for why e^(i pi) = -1 is from the definition
e = lim_{n -> infinity} (1 + 1/n)^n
First, we can generalize this to
e^z = lim_{n -> infinity} (1 + 1/n)^(z n) = lim_{m -> infinity} (1 + z/m)^m
using the substitution m = zn. Therefore,
e^(i pi) = lim_{m -> infinity} (1 + i pi/m)^m
Now, converting the complex number (1 + i pi/m) in terms of polar coordinates (r, theta) yields
r = (1 + pi^2/m^2) ~ 1
theta = sin^{-1}(pi/m) ~ pi/m
Since the product of two complex numbers is
(r, theta) (r', theta') = (r r', theta + theta'),
we have
(1 + i pi/m)^m ~ (1^m, m * pi/m) = (1, pi) = -1.