4 ms·

What's the deal with Euler's identity?

39 points by surprisetalk 11mo ago

xeonmc 11mo ago
Never liked that form of the Euler's formula. I prefer the following: (-1)ˣ = cos(πx) + i sin(πx)
- badlibrarian 11mo ago
  That's not the point of the Identity. You exponentiated the beauty right out of it.
  - xeonmc 11mo ago
    Beauty is in the eye of the beholder. Instead shoehorning it into an arbitrary symbol salad by gimping its generality, I prefer the one which makes a statement: "What does it mean to apply inversion partially?"
- penteract 11mo ago
  My objection to that is that there isn't a particularly natural reason not to say (-1)ˣ = cos(πx) - i sin(πx) As a formula about e^iπx, there is no such conflict.
rmunn 11mo ago
Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi: e^(i*tau) = 1 I won't reproduce https://www.tauday.com/tau-manifesto https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).
- badlibrarian 11mo ago
  Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
- snthpy 11mo ago
  This! I've been posting the manifesto to friends and colleagues every tau day for the past ten years. Let's keep chipping away at it and eventually we won't obfuscate radians for our kids anymore. Friends don't let friends use pi!
  - avmich 11mo ago
    I wonder how many places we have in modern math symbols which we use for historical reasons, rather than because it's most convenient overall. I guess we are balancing things here.
    - yen223 11mo ago
      Arguably, base-10 counting vs base-12 counting is one such example
      - deleted 11mo ago
        [deleted]
      - snthpy 11mo ago
        Which one of those is preferable? It seems to me that they are both historically based. 10 x 10 is also 100 in base-12 (it's only in base-10 that it looks like 144). IMHO, in a modern setting base-16 would be the most convenient. Then I maybe wouldn't struggle to remember that the CIDR range C0.A8.0.0/18 (192.168.0.0/24) consists of 10 (16) blocks of size 10 (16).
        Sharlin 11mo ago
        There’s nothing particularly convenient about base-ten; for real-world uses base-twelve would be preferable thanks to its large number of divisors (and even larger number of divisors of its multiples like 60). Which is exactly why 12 and 60 historically appear in many contexts. A number theorist would probably want a prime base, so that N (mod 10) would be a field. A power-of-two base wouldn’t be particularly convenient to anyone except a small minority consisting mostly of hardware and software engineers.
zkmon 11mo ago
This is just scratch on the surface. * Enter quaternions; things get more profound. * Investigate why multiplicative inverse of i is same as its additive inverse. * Experiment with (1+i)/(1-i). * Explore why i^i is real number. * Ask why multiplication should become an addition for angles. * Inquire the significance of the unit circle in the complex plane. * Think bout Riemann's sphere. * Understand how all this adds helps wave functions and quantum amplitudes.
- ogogmad 11mo ago
  i^i isn't anything. Please don't write this. Of the two inputs to the function (w, z) -> w^z = exp(z ln(w)), only z is a complex number, so that bit is OK. The problem is that w is NOT a complex number but a point on a particular Riemann surface, namely: The natural domain of the function ln. That particular Riemann surface looks like an endless spiral staircase. The more grown-up term might be "a helix". When you write informally "w=i", that could mean any of ln(w) = i pi/2, i (2pi + pi/2), i(4pi + pi/2), etc. Incidentally, w^z is then always a real number. However, there's an infinite sequence of those numbers that it could equal. I suppose that by pure convention, "w=e" is understood as denoting a single unique point on the helix. But extending that convention to w=i starts to look like a recipe for confusion.
  - JohnKemeny 11mo ago
    Is your argument that complex powers isn't anything?
    - xinu2020 11mo ago
      Their argument is that ln(z) where z is a complex number is a multi-valued function, so the statement "Explore why i^i is real number" could be misinterpreted as i^i = a single well-defined real value.
      - JohnKemeny 11mo ago
        Yes, but it seems strange to claim that i^i isn't anything. That just completely ignores what's interesting, namely that i(π/2 + 2πk) is real for all k ∈ Z.
stared 11mo ago
Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler https://p.migdal.pl/equations-explained-colorfully/#euler.
sevensor 11mo ago
I do like this explanation better. I learned the Maclaurin series explanation in school, where you can show that the series approximations line up, but I never felt that explained why it worked. The idea of starting with -1 as a half rotation and then taking fractions of that really appeals to the intuition.
aap_ 11mo ago
Nobody ever considers the spinorial version. e^iπ is a 360° rotation on a spinor, and + is averaging spinors rotationally. so e^iπ + 1 = 0 means there is no way to interpolate between the identity and a twist in the spinor, because the axis of a 360° rotation is undefined. Things get so much more fun once you embrace spinors.
ccppurcell 11mo ago
It's obviously of more interest as a piece of outreach than as a piece of mathematics. Nevertheless I've always wondered about the e^ipi + 1 = 0 formulation. It seems ugly and ad hoc, and the connection between the "5 constants" is not all that meaningful. That e^ipi = -1 is related to the much more profound observation that the complex numbers represent a sort of rotation into a previously unknown dimension of numbers.
constantcrying 11mo ago
The real magic is that complex exponentials are describing rotations in R^2. To be honest, this equation completely fails to represent this.
ginkoleaf 11mo ago
I do not think I appreciated the formula until I had been exposed to the exponential map in its generalized forms: The so-called 'matrix exponential' and the exponential map of Lie algebra. They place Euler's humble formula into a grander and rather beautiful setting. Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).

39 points by surprisetalk 11mo ago

xeonmc 11mo ago

Never liked that form of the Euler's formula. I prefer the following: (-1)ˣ = cos(πx) + i sin(πx)

badlibrarian 11mo ago
That's not the point of the Identity. You exponentiated the beauty right out of it.
- xeonmc 11mo ago
  Beauty is in the eye of the beholder. Instead shoehorning it into an arbitrary symbol salad by gimping its generality, I prefer the one which makes a statement: "What does it mean to apply inversion partially?"
penteract 11mo ago
My objection to that is that there isn't a particularly natural reason not to say (-1)ˣ = cos(πx) - i sin(πx) As a formula about e^iπx, there is no such conflict.

rmunn 11mo ago

Personally, I prefer the version with tau (2 times pi) in it rather than the one with pi: e^(i*tau) = 1 I won't reproduce https://www.tauday.com/tau-manifesto https://www.tauday.com/tau-manifesto here, but I'll just mention one part of it. I very much prefer doing radian math using tau rather than pi: tau/4 radians is just one-fourth of a "turn", one-fourth of the way around the circle, i.e. 90°. Which is a lot easier to remember than pi/2, and would have made high-school trig so much easier for me. (I never had trouble with radians, and even so I would have had a much easier time grasping them had I been taught them using tau rather than pi as the key value).

badlibrarian 11mo ago
Which would be e^(i*tau) - 1 = 0 if you wanted to honor the spirit of the Identity.
snthpy 11mo ago
This! I've been posting the manifesto to friends and colleagues every tau day for the past ten years. Let's keep chipping away at it and eventually we won't obfuscate radians for our kids anymore. Friends don't let friends use pi!
- avmich 11mo ago
  I wonder how many places we have in modern math symbols which we use for historical reasons, rather than because it's most convenient overall. I guess we are balancing things here.
  - yen223 11mo ago
    Arguably, base-10 counting vs base-12 counting is one such example
    - deleted 11mo ago
      [deleted]
    - snthpy 11mo ago
      Which one of those is preferable? It seems to me that they are both historically based. 10 x 10 is also 100 in base-12 (it's only in base-10 that it looks like 144). IMHO, in a modern setting base-16 would be the most convenient. Then I maybe wouldn't struggle to remember that the CIDR range C0.A8.0.0/18 (192.168.0.0/24) consists of 10 (16) blocks of size 10 (16).
      - Sharlin 11mo ago
        There’s nothing particularly convenient about base-ten; for real-world uses base-twelve would be preferable thanks to its large number of divisors (and even larger number of divisors of its multiples like 60). Which is exactly why 12 and 60 historically appear in many contexts. A number theorist would probably want a prime base, so that N (mod 10) would be a field. A power-of-two base wouldn’t be particularly convenient to anyone except a small minority consisting mostly of hardware and software engineers.

zkmon 11mo ago

This is just scratch on the surface. * Enter quaternions; things get more profound. * Investigate why multiplicative inverse of i is same as its additive inverse. * Experiment with (1+i)/(1-i). * Explore why i^i is real number. * Ask why multiplication should become an addition for angles. * Inquire the significance of the unit circle in the complex plane. * Think bout Riemann's sphere. * Understand how all this adds helps wave functions and quantum amplitudes.

ogogmad 11mo ago
i^i isn't anything. Please don't write this. Of the two inputs to the function (w, z) -> w^z = exp(z ln(w)), only z is a complex number, so that bit is OK. The problem is that w is NOT a complex number but a point on a particular Riemann surface, namely: The natural domain of the function ln. That particular Riemann surface looks like an endless spiral staircase. The more grown-up term might be "a helix". When you write informally "w=i", that could mean any of ln(w) = i pi/2, i (2pi + pi/2), i(4pi + pi/2), etc. Incidentally, w^z is then always a real number. However, there's an infinite sequence of those numbers that it could equal. I suppose that by pure convention, "w=e" is understood as denoting a single unique point on the helix. But extending that convention to w=i starts to look like a recipe for confusion.
- JohnKemeny 11mo ago
  Is your argument that complex powers isn't anything?
  - xinu2020 11mo ago
    Their argument is that ln(z) where z is a complex number is a multi-valued function, so the statement "Explore why i^i is real number" could be misinterpreted as i^i = a single well-defined real value.
    - JohnKemeny 11mo ago
      Yes, but it seems strange to claim that i^i isn't anything. That just completely ignores what's interesting, namely that i(π/2 + 2πk) is real for all k ∈ Z.

stared 11mo ago

Here is the Euler's identity in my recent side project, equations visualised - https://p.migdal.pl/equations-explained-colorfully/#euler https://p.migdal.pl/equations-explained-colorfully/#euler.

sevensor 11mo ago

I do like this explanation better. I learned the Maclaurin series explanation in school, where you can show that the series approximations line up, but I never felt that explained why it worked. The idea of starting with -1 as a half rotation and then taking fractions of that really appeals to the intuition.

aap_ 11mo ago

Nobody ever considers the spinorial version. e^iπ is a 360° rotation on a spinor, and + is averaging spinors rotationally. so e^iπ + 1 = 0 means there is no way to interpolate between the identity and a twist in the spinor, because the axis of a 360° rotation is undefined. Things get so much more fun once you embrace spinors.

ccppurcell 11mo ago

It's obviously of more interest as a piece of outreach than as a piece of mathematics. Nevertheless I've always wondered about the e^ipi + 1 = 0 formulation. It seems ugly and ad hoc, and the connection between the "5 constants" is not all that meaningful. That e^ipi = -1 is related to the much more profound observation that the complex numbers represent a sort of rotation into a previously unknown dimension of numbers.

constantcrying 11mo ago

The real magic is that complex exponentials are describing rotations in R^2. To be honest, this equation completely fails to represent this.

ginkoleaf 11mo ago

I do not think I appreciated the formula until I had been exposed to the exponential map in its generalized forms: The so-called 'matrix exponential' and the exponential map of Lie algebra. They place Euler's humble formula into a grander and rather beautiful setting. Now, I like to think of exponentiation as a kind of integral over infinitesimal generators; and $i$ just happens to be a generator for rotation about a circle in the plane (aka $\mathrm{U}(1)$ aka $x \mapsto e^{it}x$).