6 ms·

Euler's Formula: A Complete Guide

134 points by ColinWright 6y ago

sgdpk 6y ago
I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f'=f (with suitable normalization). In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc. With this definition, you can define the constants e and pi in a natural way. In particular, 2 * pi * i is the period of e^x. It then makes sense to analyse the function e^ix, with period 2*pi, and separate its real and imaginary parts. These are sure to be periodic real functions, and we can name them cos(x) and sin(x). This Quora answer talks about this approach pretty well [1]. [1] https://qr.ae/pNy7qp https://qr.ae/pNy7qp
- throwaway77384 6y ago
  Here's a fun perspective from someone who knows very little about maths, but has a huge interest in logic. I have a question for you: At what 'level' of maths or understanding do you reckon someone has to be in order to make sense of your introductory paragraph? Let me map out what happens in my head when I read that first paragraph: > "I find that the most elegant way to understand exponential-related things"... Okay, I know what exponential increase means. Something grows by an exponent of something. 2416... > "is by defining the exponential as the function satisfying f'=f (with suitable normalization)"... I don't know what an exponential is. A function, to me, is something I call as part of code, so I am not sure if I might be confusing the programmer and mathematician-definition of 'function'. I don't really know how to interpret f'=f in any way. (F-apostrophe equals f? Why?). I also don't really know how normalization plays into it. For me, normalization is something related to homogenising a series of things. Sometimes it can be related to normalising the volume of a bunch of mp3 files, or in other cases it can be turning a bunch of badly formatted phone numbers into a common format for a database. > "In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc." I have often heard terms like "eigen"-something, but this always causes me trouble, because I end up parsing it in German ("eigen" means "self" in German). So now it is the "self"-function of something? Okay, of the 'derivative'. Is a derivative the mathematician's way of saying "figuring out something you don't know by using a bunch of other things you do know"? I do not know what a linear operator is at all. I also do not know what ODEs are. You see, I love logic and science and have been a programmer for so long, yet when I hear something like the statement above, I almost become despondent over the fact that seemingly I know nothing of the underpinnings or logical foundations of the things I spend my life with (aka computers and programming being born from maths...). Anyway, just a random drive-by thought.
  - gxs 6y ago
    Tangentially related to your point - math is an extremely dense subject. Small little statements with just a few symbols can be LOADED with information - things like complex, meaty definitions for even the most benign terms (e.g., a function), and stuff that can be deduced from it from a million different theorems. This is part of the reason why it's so easy to fall behind in math. Once you are missing a vital part of information, it can be hard to identify where that gap is and the train just keeps moving and moving. This was my experience with math, and something that I noticed when I tutored people in some low level math as well.
    - hansvm 6y ago
      I've thought about this on occasion. Anecdotally (and subject to selection and confirmation biases) I've seen students who struggled excessively in a given course, dropped, and taken the previous course to excel not just in the subject they originally dropped but also in subsequent classes. I've wondered if my alma mater might be better off recommending students to start a level or two below where they currently are to fill in some of those vital gaps.
  - sgdpk 6y ago
    Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university. So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers. Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator. Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication. The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?
- gxs 6y ago
  That was an EXCELLENT read, thank you. I was a math major in college, and somethings you learn how to work out mechanically and understand intellectually. I dont do anything remotely close to academic, undergraduate mathematics, but it's interesting how now that I'm older i seem to understand things better, to internalize them beyond something mechanical or simply accepting them because they make logical sense. Wish I'd had that type of understanding in school. This explanation is very intuitive - and the author makes a good point. There might be better ways to introduce pi other than the ratio of the circumference of a circle to its diameter.
  - jonsen 6y ago
    > There might be better ways to introduce pi other than the ration of the circumference of a circle to it's radius. Sure, the ratio of the circumference of a circle to it's diameter ;-)
    - gxs 6y ago
      Ha! I thought I corrected it before anyone saw :)
  - qsort 6y ago
    > There might be better ways to introduce pi To be fair that's the elementary school way of defining pi. I'm somewhat rusty myself, but I'm pretty sure even undergraduate real analysis courses define pi in terms of the complex exponential. "Ratio of circumference to diameter" is insanely hard to make rigorous.
    - Tainnor 6y ago
      My undergraduate real analysis course didn't, probably because here, I don't think complex numbers are necessarily part of the high school curriculum. Instead, we defined sin and cos via power series, and then IIRC, pi as the first positive zero of the sin function or something similar.
  - sgdpk 6y ago
    Indeed! I highly recommend following the author.
deleted 6y ago
[deleted]
adamnemecek 6y ago
It makes more sense when you use tau as tau is a full rotation, not half rotation. e^tau=1. Also I think that i is not interesting because i * i=-1, but because i * conjugate(i)=1. I never saw the -1 in the context of complex numbers but if you say it's 1, it's easy to see the unit circle. Similarly, with split-complex numbers, you have j * conjugate(j)=-1. Split complex numbers are about mirror reflection so the -1 makes sense.
mellosouls 6y ago
Famously evaluates when x = π (pi) to Euler's Identity (listed halfway down the page), which is one of those beautiful "Wow!" formulas that brings seemingly unconnected things together. e^(i x π) = -1 https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity
- Twisol 6y ago
  Apropos of nothing, that page suggests that the identity (with the 1 shifted across the equals sign) is beautiful because it relates e, pi, i, 1, and 0. I always felt 1 and 0 were shoehorned in, because they're clearly nonessential when one writes it the way you did. It honestly detracts from the pure relationship between e, pi, and i.
scythe 6y ago
Oddly this leaves out what I would consider the most "elementary" proof, starting from Bernoulli's definition of e and requiring only the squeeze theorem. You begin by writing exp(z) = lim_{N->\infty} (1 + z/N)^N. Note that in this limit we do not need the analytic continuation of powers: N is an integer, so the function y^N is single-valued on C. This gives a definition of the exponential function that is immediately entire and does not rely on calculus. Then we can write exp(ix) = lim_{N->\infty} (cos(arctan(x/N)) + i sin(arctan(x/N)))^N ⋅ (1 + x^2/N^2)^N. (note that sec(arctan(t)) = 1 + t^2). It is easy to show that lim_{N->\infty} (1 + x^2/N^2)^N = 1 for all finite x, so we find exp(ix) = lim_{N->\infty} (cos(arctan(x/N)) + i sin(arctan(x/N)))^N. It is not necessary to apply L'Hopital's rule since neither of the factors approaches zero or infinity -- we can simply split the limit. By induction we can then show that (cos(x) + i sin(x))^A = cos(Ax) + i sin(Ax) for all integer A. We apply this to the previous expression to conclude that exp(ix) = lim_{N->\infty} = cos(N ⋅ arctan(x/N)) + i sin(N ⋅ arctan(x/N)). Therefore we only need to find lim_{N->\infty} N ⋅ arctan(x/N). Consider the triangle with vertices (0,0), (1,0), (1, x/N): its hypotenuse intersects the unit circle at (N / sqrt(N^2 + x^2), x / sqrt(N^2 + x^2)), subtending an arc of exactly arctan(x/N). A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length. We therefore have Nx / sqrt(N^2 + x^2) < N*arctan(x/N) < x. The left and (trivially) right limits both equal x, so by squeezing lim_{N->\infty} N arctan(x/N) = x and therefore e^(ix) = cos(x) + i sin(x).
- codethief 6y ago
  Be careful with your last paragraph. It builds upon results about the relationship between trigonometric functions (usually defined in terms of power series) and the unit circle's geometry, as well as intuition from 2D geometry in general. These things can actually be quite hard to prove. (Prime example in a slightly different context: https://en.wikipedia.org/wiki/Jordan_curve_theorem https://en.wikipedia.org/wiki/Jordan_curve_theorem ) For instance, > A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length. already uses the fact that straight lines are the shortest lines in the class of C¹-differentiable curves due to the Euler-Langrange equations. (Good luck with proving this statement in the class of C⁰ curves.) Then, to use this fact, you need to prove differentiability of a curve parametrizing the unit circle, i.e. differentiability of the trigonometric functions. Now, at this point you might as well use the standard definition of e^x in terms of a power series because it simplifies the differentiability proof tremendously.
sevensor 6y ago
Euler's formula got me through a bunch of EE classes back when I was a student. I was pathologically unable to commit trig identities to memory, possibly due to laziness. So instead I learned Euler's formula and worked everything through in complex numbers until I was ready to give an answer. I never got marked down for it.
- qsort 6y ago
  I did this exact same thing for math competitions. You often need some stupid exact formula that you are supposed to have memorized, but I never did. De Moivre also yields the same results for this, and it's somewhat easier/faster to use.
calebm 6y ago
e^ix is useful for radio stuff since it generates a wave that oscillates in 2 dimensions (like a spiral). Here's it being used in code: https://github.com/calebmadrigal/radio-hacking-scripts/blob/master/ask_modulate_radio_signal.py#L53 https://github.com/calebmadrigal/radio-hacking-scripts/blob/...
plaidfuji 6y ago
Gauss was known to have said “if you do not immediately recognize Euler’s formula as an intuitive result, you will never be a first-class mathematician.” (Paraphrased) Good thing I didn’t go into pure math.
markisus 6y ago
I really like the animations and graphs. Anyone know what library if any was used for those?
- SamBam 6y ago
  It looks like they both have links to Desmos.com on them.
  - andrepd 6y ago
    Desmos is absolutely incredible.
nayuki 6y ago
https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
umvi 6y ago
I remember discovering this formula by accident when I was a junior in high school. I was messing around on my Ti-83, when I discovered that `e^(i*pi)` resulted in `-1`. That blew my mind, so I showed my pre-calc teacher and he thought it was just a calculator glitch because it didn't know how to calculate imaginary powers. I kind of forgot about it until the full formula surfaced in one of my electrical engineering classes in college and I realized I had (partially) stumbled across the formula by accident a few years earlier!
mhartl 6y ago
The Tau Manifesto shows how using tau = C/r makes the geometric meaning of Euler’s formula (and hence of Euler’s identity) even clearer. Disclaimer: I am the author of The Tau Manifesto. https://tauday.com/tau-manifesto#sec-euler_s_identity https://tauday.com/tau-manifesto#sec-euler_s_identity
- sgdpk 6y ago
  My favorite hill to die on! Thanks for writing!

134 points by ColinWright 6y ago

I find that the most elegant way to understand exponential-related things is by defining the exponential as the function satisfying f'=f (with suitable normalization). In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc. With this definition, you can define the constants e and pi in a natural way. In particular, 2 * pi * i is the period of e^x. It then makes sense to analyse the function e^ix, with period 2*pi, and separate its real and imaginary parts. These are sure to be periodic real functions, and we can name them cos(x) and sin(x). This Quora answer talks about this approach pretty well [1]. [1] https://qr.ae/pNy7qp https://qr.ae/pNy7qp

throwaway77384 6y ago
Here's a fun perspective from someone who knows very little about maths, but has a huge interest in logic. I have a question for you: At what 'level' of maths or understanding do you reckon someone has to be in order to make sense of your introductory paragraph? Let me map out what happens in my head when I read that first paragraph: > "I find that the most elegant way to understand exponential-related things"... Okay, I know what exponential increase means. Something grows by an exponent of something. 2416... > "is by defining the exponential as the function satisfying f'=f (with suitable normalization)"... I don't know what an exponential is. A function, to me, is something I call as part of code, so I am not sure if I might be confusing the programmer and mathematician-definition of 'function'. I don't really know how to interpret f'=f in any way. (F-apostrophe equals f? Why?). I also don't really know how normalization plays into it. For me, normalization is something related to homogenising a series of things. Sometimes it can be related to normalising the volume of a bunch of mp3 files, or in other cases it can be turning a bunch of badly formatted phone numbers into a common format for a database. > "In other words, it is the "eigenfunction" of the derivative, which is a linear operator. This is a very natural thing to ask when studying derivatives, ODEs, etc." I have often heard terms like "eigen"-something, but this always causes me trouble, because I end up parsing it in German ("eigen" means "self" in German). So now it is the "self"-function of something? Okay, of the 'derivative'. Is a derivative the mathematician's way of saying "figuring out something you don't know by using a bunch of other things you do know"? I do not know what a linear operator is at all. I also do not know what ODEs are. You see, I love logic and science and have been a programmer for so long, yet when I hear something like the statement above, I almost become despondent over the fact that seemingly I know nothing of the underpinnings or logical foundations of the things I spend my life with (aka computers and programming being born from maths...). Anyway, just a random drive-by thought.
- gxs 6y ago
  Tangentially related to your point - math is an extremely dense subject. Small little statements with just a few symbols can be LOADED with information - things like complex, meaty definitions for even the most benign terms (e.g., a function), and stuff that can be deduced from it from a million different theorems. This is part of the reason why it's so easy to fall behind in math. Once you are missing a vital part of information, it can be hard to identify where that gap is and the train just keeps moving and moving. This was my experience with math, and something that I noticed when I tutored people in some low level math as well.
  - hansvm 6y ago
    I've thought about this on occasion. Anecdotally (and subject to selection and confirmation biases) I've seen students who struggled excessively in a given course, dropped, and taken the previous course to excel not just in the subject they originally dropped but also in subsequent classes. I've wondered if my alma mater might be better off recommending students to start a level or two below where they currently are to fill in some of those vital gaps.
- sgdpk 6y ago
  Sure, there is certainly a lot of jargon here which I could have explained better, but in the end one has to be familiar with some calculus and complex numbers to make sense of this topic. Depending on where you're from, this could be either late high-school to second/third year of university. So by exponential I mean the exponential function, in the mathematician's sense: something which takes numbers as inputs and produces other numbers as outputs in a unique way. Already a complication: these can be complex numbers. Then, the derivative of a function is its "instantaneous rate of change". How much does the function increase for small (infinitesimal, really) changes in input? Since this rate may be different for every point, we are defining a new function f': for every input of f, f' gives you the local rate of change. So there you go, the derivative takes a function f and produces a function f'. We call this an operator. Because the derivative has very nice properties, we call this a "linear" operator. Finally, the "eigen"-things. These are really fancy words for the simple concept of a fixed-point: when you apply an operation to a certain thing X and just get that X back. So, the function x^2 has two fixed-points: 0 and 1. These would be "eigenpoints", but we don't call them that because we reserve the term "eigen" for linear operations, such as the derivative operator or matrix-multiplication. The point of my answer is that the exponential function is elegantly understood as the eigenfunction of the derivative. In other words: which function is its own derivative? Which one tells you how it changes just by looking at it?
gxs 6y ago
That was an EXCELLENT read, thank you. I was a math major in college, and somethings you learn how to work out mechanically and understand intellectually. I dont do anything remotely close to academic, undergraduate mathematics, but it's interesting how now that I'm older i seem to understand things better, to internalize them beyond something mechanical or simply accepting them because they make logical sense. Wish I'd had that type of understanding in school. This explanation is very intuitive - and the author makes a good point. There might be better ways to introduce pi other than the ratio of the circumference of a circle to its diameter.
- jonsen 6y ago
  > There might be better ways to introduce pi other than the ration of the circumference of a circle to it's radius. Sure, the ratio of the circumference of a circle to it's diameter ;-)
  - gxs 6y ago
    Ha! I thought I corrected it before anyone saw :)
- qsort 6y ago
  > There might be better ways to introduce pi To be fair that's the elementary school way of defining pi. I'm somewhat rusty myself, but I'm pretty sure even undergraduate real analysis courses define pi in terms of the complex exponential. "Ratio of circumference to diameter" is insanely hard to make rigorous.
  - Tainnor 6y ago
    My undergraduate real analysis course didn't, probably because here, I don't think complex numbers are necessarily part of the high school curriculum. Instead, we defined sin and cos via power series, and then IIRC, pi as the first positive zero of the sin function or something similar.
- sgdpk 6y ago
  Indeed! I highly recommend following the author.

deleted 6y ago

[deleted]

adamnemecek 6y ago

It makes more sense when you use tau as tau is a full rotation, not half rotation. e^tau=1. Also I think that i is not interesting because i * i=-1, but because i * conjugate(i)=1. I never saw the -1 in the context of complex numbers but if you say it's 1, it's easy to see the unit circle. Similarly, with split-complex numbers, you have j * conjugate(j)=-1. Split complex numbers are about mirror reflection so the -1 makes sense.

mellosouls 6y ago

Famously evaluates when x = π (pi) to Euler's Identity (listed halfway down the page), which is one of those beautiful "Wow!" formulas that brings seemingly unconnected things together. e^(i x π) = -1 https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity

Twisol 6y ago
Apropos of nothing, that page suggests that the identity (with the 1 shifted across the equals sign) is beautiful because it relates e, pi, i, 1, and 0. I always felt 1 and 0 were shoehorned in, because they're clearly nonessential when one writes it the way you did. It honestly detracts from the pure relationship between e, pi, and i.

scythe 6y ago

Oddly this leaves out what I would consider the most "elementary" proof, starting from Bernoulli's definition of e and requiring only the squeeze theorem. You begin by writing exp(z) = lim_{N->\infty} (1 + z/N)^N. Note that in this limit we do not need the analytic continuation of powers: N is an integer, so the function y^N is single-valued on C. This gives a definition of the exponential function that is immediately entire and does not rely on calculus. Then we can write exp(ix) = lim_{N->\infty} (cos(arctan(x/N)) + i sin(arctan(x/N)))^N ⋅ (1 + x^2/N^2)^N. (note that sec(arctan(t)) = 1 + t^2). It is easy to show that lim_{N->\infty} (1 + x^2/N^2)^N = 1 for all finite x, so we find exp(ix) = lim_{N->\infty} (cos(arctan(x/N)) + i sin(arctan(x/N)))^N. It is not necessary to apply L'Hopital's rule since neither of the factors approaches zero or infinity -- we can simply split the limit. By induction we can then show that (cos(x) + i sin(x))^A = cos(Ax) + i sin(Ax) for all integer A. We apply this to the previous expression to conclude that exp(ix) = lim_{N->\infty} = cos(N ⋅ arctan(x/N)) + i sin(N ⋅ arctan(x/N)). Therefore we only need to find lim_{N->\infty} N ⋅ arctan(x/N). Consider the triangle with vertices (0,0), (1,0), (1, x/N): its hypotenuse intersects the unit circle at (N / sqrt(N^2 + x^2), x / sqrt(N^2 + x^2)), subtending an arc of exactly arctan(x/N). A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length. We therefore have Nx / sqrt(N^2 + x^2) < N*arctan(x/N) < x. The left and (trivially) right limits both equal x, so by squeezing lim_{N->\infty} N arctan(x/N) = x and therefore e^(ix) = cos(x) + i sin(x).

codethief 6y ago
Be careful with your last paragraph. It builds upon results about the relationship between trigonometric functions (usually defined in terms of power series) and the unit circle's geometry, as well as intuition from 2D geometry in general. These things can actually be quite hard to prove. (Prime example in a slightly different context: https://en.wikipedia.org/wiki/Jordan_curve_theorem https://en.wikipedia.org/wiki/Jordan_curve_theorem ) For instance, > A similar triangle inscribed at this vertex has a short side which is necessarily less than the arc length. already uses the fact that straight lines are the shortest lines in the class of C¹-differentiable curves due to the Euler-Langrange equations. (Good luck with proving this statement in the class of C⁰ curves.) Then, to use this fact, you need to prove differentiability of a curve parametrizing the unit circle, i.e. differentiability of the trigonometric functions. Now, at this point you might as well use the standard definition of e^x in terms of a power series because it simplifies the differentiability proof tremendously.

sevensor 6y ago

Euler's formula got me through a bunch of EE classes back when I was a student. I was pathologically unable to commit trig identities to memory, possibly due to laziness. So instead I learned Euler's formula and worked everything through in complex numbers until I was ready to give an answer. I never got marked down for it.

qsort 6y ago
I did this exact same thing for math competitions. You often need some stupid exact formula that you are supposed to have memorized, but I never did. De Moivre also yields the same results for this, and it's somewhat easier/faster to use.

calebm 6y ago

e^ix is useful for radio stuff since it generates a wave that oscillates in 2 dimensions (like a spiral). Here's it being used in code: https://github.com/calebmadrigal/radio-hacking-scripts/blob/master/ask_modulate_radio_signal.py#L53 https://github.com/calebmadrigal/radio-hacking-scripts/blob/...

plaidfuji 6y ago

Gauss was known to have said “if you do not immediately recognize Euler’s formula as an intuitive result, you will never be a first-class mathematician.” (Paraphrased) Good thing I didn’t go into pure math.

markisus 6y ago

I really like the animations and graphs. Anyone know what library if any was used for those?

SamBam 6y ago
It looks like they both have links to Desmos.com on them.
- andrepd 6y ago
  Desmos is absolutely incredible.

nayuki 6y ago

https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...

umvi 6y ago

I remember discovering this formula by accident when I was a junior in high school. I was messing around on my Ti-83, when I discovered that `e^(i*pi)` resulted in `-1`. That blew my mind, so I showed my pre-calc teacher and he thought it was just a calculator glitch because it didn't know how to calculate imaginary powers. I kind of forgot about it until the full formula surfaced in one of my electrical engineering classes in college and I realized I had (partially) stumbled across the formula by accident a few years earlier!

mhartl 6y ago

The Tau Manifesto shows how using tau = C/r makes the geometric meaning of Euler’s formula (and hence of Euler’s identity) even clearer. Disclaimer: I am the author of The Tau Manifesto. https://tauday.com/tau-manifesto#sec-euler_s_identity https://tauday.com/tau-manifesto#sec-euler_s_identity

sgdpk 6y ago
My favorite hill to die on! Thanks for writing!