I remember learning about this on the last day of Calculus class. It was probably the most dramatic classroom moment of any math class I took in college or high school
Lots of high schoolers who thought they knew math saw the first little hint that all these separate areas of the subject might be more related than they had imagined
I think this really hits at why the identity is so famous: people tend to have been introduced to pieces in entirely separate ways, and from that perspective it’s quite surprising to see them all come together to create something nice.
I think that e^(i*pi) + 1 = 0 is massively overrated. Because the beauty that people find is largely based on the symbols that are present and because it doesn't really say anything meaningful beyond itself. It's not even a formula or an equation - it's a result. It's like being impressed by cos(pi) = -1.
The more general formula e^(ix) = cosx + i*sinx is a lot more impressive. Both because of it's implications for the rest of mathematics and aesthetically. If you look at the way the vectors at 38:00-39:20[1] cancel out to always land on the unit circle - I can get behind calling that beautiful.
If you already know the power series expansions of cos and sin then yes. But then you put the burden of proof into showing that the power series of cos and sin indeed define the same functions as the trigonometric definition.
I do not think this is true: you can introduce the complex numbers (including the imaginary unit `i`) without defining exponentiation or introducing `e`. As a consequence, no properties of `e` can be deduced from the definition of the imaginary unit.
The first line of the article is "Beauty, they say, is in the eye of the beholder" and here your comment seems to relate directly to this line and the rest of the article and yet someone still found reason to downvote you for having your own point of view. Come on HN, seriously...
> "Beauty, they say, is in the eye of the beholder"
Incidentally, I reject this claim. I take beauty to be objectively real and not a subjective reaction to something. The latter I would call taste. (A materialist would of course object because for him, the world is just a bunch of boring atoms bumping into each other, though he will never explain how his mind, just another boring bunch of jostling atoms, is capable of entertaining such subjective delusions and why they can exist in his mind-as-jostling atoms but not in the world "out there", more jostling atoms.)
That some variation in opinion exists about what is beautiful (and in what way) does not disprove the claim. Not everyone is equally discerning, some have perverse tastes, etc.
The lecture in the sequences and series portion of calculus where the prof took the Taylor series of e^x and then substituted x=i*theta into the resulting polynomial and all of the terms magically divided themselves into the Taylor series’s for sin and cos was a jaw dropping moment for me.
I was in my second year of EE, and it completely changed my relationship with a ton of practical material for me, eg the Fourier transform.
On the other hand, I also took vector calculus that year but it wasn’t until the following year when I took the E&M course (electric and magnetic fields) where I truly appreciated surface integrals, path integrals, etc. One of the biggest Aha! moments there was really appreciating how properly structuring the problem could dramatically simplify the computations. Eg if you can make it symmetric you can often cause integrals to disappear implicitly without having to compute them at all, or you can take a dramatically simpler path through a vector field and end up at the same place without a ton of complex computation. Absolutely beautiful :D
> Because the beauty that people find is largely based on the symbols that are present
The identity involves the five most fundamental constants and the four most fundamental operations - and nothing else. That is why a lot of people find it satisfying.
>“Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.”
Well that's the thing. They're not unconnected. The hype depends on the people who aren't aware of the connection.
It's most certainly not obvious how the relation df(x)/dx = f(x), the ratio between the diameter and circumference of a circle , and the square root of -1 -- a "trick" to extend the number line with a second dimension -- are related, let alone be used to create an useful construct.
Add in the fact that the reason they are related appears to involve infinity (power series). It is tough to argue against it being an impressive result.
I wouldn't say power series are the 'reason' they are related. The more intuitive reason for Euler's identity is the combination of e^x being its own derivative, and the relationship between complex multiplication and rotation.
"its own derivative" is defined in terms of things happening at infinity though. Calculus is very much a study of what happens to things at the infinite and infinitesimal; and e has its heft because of calculus.
Aren't they historically unconnected, as the quote alludes to?
1's origin is in counting.
pi's origin is in the study of the circle.
0's origin is in accounting.
i's origin is polynomial algebra.
e's origin is in the study of exponentials and derivatives.
The relationships between all of these fields is a discovery in itself, and a revelation that was not apparent at all throughout history, and it is still not apparent to most people who haven't studied math in-depth.
Basically, e^ipi + 1 = 0 is proof of the connection, one that is not at all immediately apparent, and has not been known for the majority of humanity's history.
In my opinion, the truly beautiful concept involved here is Euler's formula:
e^(ix) = cos x + i sin x.
It unifies algebra, trigonometry, complex numbers, and calculus. Euler's identity is only a special case of Euler's formula, i.e., Euler's formula with x = π gives us Euler's identity:
e^(iπ) = -1.
This is cute but Euler's formula is truly beautiful. In fact with x = τ = 2π, we get another cute result:
e^(iτ) = 1.
From Chapter 22 of The Feynman Lectures on Physics, Volume I:
"We summarize with this, the most remarkable formula in mathematics: e^(iθ) = cos θ + i sin θ. This is our jewel. We may relate the geometry to the algebra by representing complex numbers in a plane; the horizontal position of a point is x, the vertical position of a point is y. We represent every complex number, x + iy. Then if the radial distance to this point is called r and the angle is called θ, the algebraic law is that x + iy is written in the form re^(iθ), where the geometrical relationships between x, y, r, and θ are as shown. This, then, is the unification of algebra and geometry."
The one involving pi looks more elegant when written in the first manner. The one involving tau looks more elegant when written in the fourth manner. I screwed up this one while going back and forth between these options. I have fixed it now. Thanks for noticing and commenting!
Agreed, the full formula is much more intersting and deep: it shows there is a mapping between the reals and the circle group that preserves the structure (turns addition into multiplication). This concept leads to the very general exponential map of a Lie algebra to a Lie group and Pontryagin duality, which is the essence of Fourier transform.
> It unifies algebra, trigonometry, complex numbers, and calculus.
I vividly remember my maths teacher impressing us upon what you stated above. By far the best math teacher I had. He didn't work out the proof on the board but made us work it out, took us on a journey, him as a guide, so to speak.
In a way it's analogous to how Einstein's E = mc^2 is a special case of how the norm of the four-momentum is defined in special relativity, which is mc = √((E/c)^2-p^2). For the special case of a stationary object we have p = 0, so E = mc^2 follows automatically. But the special case is somehow more memorable and more famous, and I believe something similar has happened with Euler's identity.
Visual Complex Analysis is the text that introduced me to this concept. The functions cos and sin suddenly look a lot like accessor functions for a two member object.
All of complex analysis is filled with magical results like this. For example, Cauchy's formula and its generalizations. Max/Min results, Loiville's theorem, Nevanlinna theory, Picard's Great and Little theorems. It's all absolutely bonkers. Just think:
"Every complex function with an essential singularity achieves all but one value infinitely often in every neighborhood of the singularity"
or
"Every holomorphic function can be approximated arbitrarily well by a region of the Riemann Zeta function"
There is this joke about how proofs start in different branches of math. So in real analysis you start with "Let ε > 0". And in Abstract Algebra, it's "Suppose not." But in Complex Analysis: "Nothing up my left sleeve, nothing up my right sleeve.."
comparing the pi and tau formulas, though, the advantage of using the pi formula is you can pose the question (to someone who has never seen it) as such:
Given the formula, e^(ix) = -1, what would you suppose x is?
If you tried to use the tau formula, it falls flat:
Given the formula, e^(ix) = 1, what would you suppose x is?
It's a unification of geometry and algebra in 2 dimensions, but generalizations beyond that are scarce. Effectively we're left with a solution where we can represent any space in dimensions modulo 2.
While quaternions and some higher dimensional complex numbers exist, is there a unified formula expressing Euler's formula for arbitrary numbers of dimensions? is there one for an infinite dimension space?
I second your nomination of Euler's formula as the most beautiful mathematical statement!
Another candidate in my mind is the generalized variant of Stokes' theorem (there's nothing wrong with the basic calculus variant, but the general variant is just so succinct): https://en.wikipedia.org/wiki/Generalized_Stokes_theorem
In addition to being pleasingly succinct and symmetric in its very statement, it couples shapes and functions on them in a beautiful way.
A favorite of mine is the Gauss-Bonnet theorem which relates the Euler characteristic of a surface (i.e. its topology) to its Gaussian curvature (its differential geometry).
Like art, discounting the human experience in mathematics strips away so much of its meaning.
> It took me an entire three-year education as a mathematics undergraduate before I could look at Euler’s identity and say, “Yeah, it’s no big deal. I just did not know enough five years ago. I see it all now.”
The mystery had gone, replaced by deeper understanding. And there, with that very personal, emotionally rich, intellectual journey of discovery that we all must follow if we wish to understand mathematics, we find the deeper beauty of mathematics [...]
I am still struggling to convince myself that complex numbers are more than just clever notation for polar coordinates.
I know that the real numbers aren't "just notation" because things like $\pi$, e, $\sqrt{2}$ etc can't be expressed using pairs of natural numbers (i.e. as rational numbers). I'm having trouble finding an equally simple example of "can't do this using pairs of reals" justification for complex numbers. Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? Trigonometry arises whenever you are able to take limits over any kind of cyclical/repeating process, so the fact that sin and cos appear in Euler's Identity does not seem terribly profound to me.
Also, why stop at complex numbers, since it's "turtles all the way"? Quarternions, octonions are neat too. Why doesn't Feynman think octonions are "our jewel" too? At each step (real -> complex -> quaternion -> octonion) you get a larger class of formulae having roots, but sacrifice more properties of the ring structure. For example, the reals have a total order but the complex numbers do not.
The concept you are looking for here is algebraic closure. If you want "ax^2 + bx + c" to have numeric roots and be factorable for all coefficients, you have to expand your definition of number from real to complex.
Probably the most blatant physical example of this is modeling mass-spring-damper systems. The differential equations will map the physical coefficients to such a polynomial, and the roots of that polynomial determine the system's behavior. Without complex numbers, you wind up only being able to describe real exponential responses and don't have the vocabulary to describe the underdamped (and oscillating) system responses with complex exponentials in the results.
> The concept you are looking for here is algebraic closure.
Yes, I know that the complex numbers are an ACF.
Why apply the Cayley-Dickson construction [1] only once? The complex numbers may be an algebraically closed field, but they aren't a Central Simple Algebra over the Reals [2]. Apply the Cayley-Dickson construction again and you get the quaternions, which are real central simple. Apply it a third time and you get the octonions. You can keep doing this over and over and over getting more and more closure properties...
> Without complex numbers, you wind up only being able to describe real exponential responses
Why can't you describe them using pairs of real numbers in polar coordinates?
Complex numbers are pairs of real numbers. As a vector space they are isomorphic to R^2 and one construction of the complex numbers is, in effect, just as pairs of real numbers. (There are of course other ways of constructing the complex numbers, too, such as via matrices. These are also all isomorphic to R^2 as a vector space.)
Of course, what makes complex numbers special is that you can multiply them, and that gives you a field.
Following that train of thought,
> Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)?
You can of course develop all of complex analysis without introducing the traditional a+bi notation and just referring to vectors in R^2. That's notation. However, complex differentiability (being holomorphic) is a stronger definition than regular differentiability in C(R^2, R^2), in particular, the Cauchy-Riemann equations need to be satisfied. From that stronger definition, you get all sorts of other theorems such that all holomorphic functions are analytic and so on.
Thank you for bringing up complex differentiability.
Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1].
If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel".
I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful".
[1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev
, former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-P...
54 comments
[ 2.1 ms ] story [ 135 ms ] threadLots of high schoolers who thought they knew math saw the first little hint that all these separate areas of the subject might be more related than they had imagined
-- Carl Friedrich Gauss
The more general formula e^(ix) = cosx + i*sinx is a lot more impressive. Both because of it's implications for the rest of mathematics and aesthetically. If you look at the way the vectors at 38:00-39:20[1] cancel out to always land on the unit circle - I can get behind calling that beautiful.
[1] https://youtu.be/ZxYOEwM6Wbk?t=2280
Complex numbers were known and used for many years before Euler found this obvious result.
In particular, the result is not "obvious".
Incidentally, I reject this claim. I take beauty to be objectively real and not a subjective reaction to something. The latter I would call taste. (A materialist would of course object because for him, the world is just a bunch of boring atoms bumping into each other, though he will never explain how his mind, just another boring bunch of jostling atoms, is capable of entertaining such subjective delusions and why they can exist in his mind-as-jostling atoms but not in the world "out there", more jostling atoms.)
That some variation in opinion exists about what is beautiful (and in what way) does not disprove the claim. Not everyone is equally discerning, some have perverse tastes, etc.
I was in my second year of EE, and it completely changed my relationship with a ton of practical material for me, eg the Fourier transform.
On the other hand, I also took vector calculus that year but it wasn’t until the following year when I took the E&M course (electric and magnetic fields) where I truly appreciated surface integrals, path integrals, etc. One of the biggest Aha! moments there was really appreciating how properly structuring the problem could dramatically simplify the computations. Eg if you can make it symmetric you can often cause integrals to disappear implicitly without having to compute them at all, or you can take a dramatically simpler path through a vector field and end up at the same place without a ton of complex computation. Absolutely beautiful :D
The identity involves the five most fundamental constants and the four most fundamental operations - and nothing else. That is why a lot of people find it satisfying.
Well that's the thing. They're not unconnected. The hype depends on the people who aren't aware of the connection.
Let D_n be a difference operator like D_n(F) = F(x+n) - F(x)
Solving by induction for the equation D_n(F)/D_n(x) = F(x), F(0)=1, it gives (1+n)^(x/n).
So now you have a "family" of functions that when "derived" by its difference, gives the same constant ^ x. Euler is the case when n→0.
This aplies for every n, including complex numbers.
1's origin is in counting.
pi's origin is in the study of the circle.
0's origin is in accounting.
i's origin is polynomial algebra.
e's origin is in the study of exponentials and derivatives.
The relationships between all of these fields is a discovery in itself, and a revelation that was not apparent at all throughout history, and it is still not apparent to most people who haven't studied math in-depth.
Basically, e^ipi + 1 = 0 is proof of the connection, one that is not at all immediately apparent, and has not been known for the majority of humanity's history.
"We summarize with this, the most remarkable formula in mathematics: e^(iθ) = cos θ + i sin θ. This is our jewel. We may relate the geometry to the algebra by representing complex numbers in a plane; the horizontal position of a point is x, the vertical position of a point is y. We represent every complex number, x + iy. Then if the radial distance to this point is called r and the angle is called θ, the algebraic law is that x + iy is written in the form re^(iθ), where the geometrical relationships between x, y, r, and θ are as shown. This, then, is the unification of algebra and geometry."
See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.
e^(iτ/2) = -1 can be translated as "half a full rotation is equivalent to a symmetry"
(And a full rotation to identity (for usual objects), but this is kind of obvious...)
I vividly remember my maths teacher impressing us upon what you stated above. By far the best math teacher I had. He didn't work out the proof on the board but made us work it out, took us on a journey, him as a guide, so to speak.
"Every complex function with an essential singularity achieves all but one value infinitely often in every neighborhood of the singularity"
or
"Every holomorphic function can be approximated arbitrarily well by a region of the Riemann Zeta function"
There is this joke about how proofs start in different branches of math. So in real analysis you start with "Let ε > 0". And in Abstract Algebra, it's "Suppose not." But in Complex Analysis: "Nothing up my left sleeve, nothing up my right sleeve.."
Given the formula, e^(ix) = -1, what would you suppose x is?
If you tried to use the tau formula, it falls flat:
Given the formula, e^(ix) = 1, what would you suppose x is?
While quaternions and some higher dimensional complex numbers exist, is there a unified formula expressing Euler's formula for arbitrary numbers of dimensions? is there one for an infinite dimension space?
> e^(ix) = cos x + i sin x.
And that right there, is the mathematical basis for a quadrature mixer... And all of software defined radios.
https://hackaday.com/2017/05/16/if-the-i-and-q-of-software-d...
..which I think is beautiful too.
Another candidate in my mind is the generalized variant of Stokes' theorem (there's nothing wrong with the basic calculus variant, but the general variant is just so succinct): https://en.wikipedia.org/wiki/Generalized_Stokes_theorem
In addition to being pleasingly succinct and symmetric in its very statement, it couples shapes and functions on them in a beautiful way.
One more with this pleasing connection is the Atiyah–Singer index theorem. However, it loses some points for requiring a bit too much setup for the statement. https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_th...
https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem
[1] https://betterexplained.com/articles/intuitive-understanding...
> It took me an entire three-year education as a mathematics undergraduate before I could look at Euler’s identity and say, “Yeah, it’s no big deal. I just did not know enough five years ago. I see it all now.”
The mystery had gone, replaced by deeper understanding. And there, with that very personal, emotionally rich, intellectual journey of discovery that we all must follow if we wish to understand mathematics, we find the deeper beauty of mathematics [...]
I know that the real numbers aren't "just notation" because things like $\pi$, e, $\sqrt{2}$ etc can't be expressed using pairs of natural numbers (i.e. as rational numbers). I'm having trouble finding an equally simple example of "can't do this using pairs of reals" justification for complex numbers. Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)? Trigonometry arises whenever you are able to take limits over any kind of cyclical/repeating process, so the fact that sin and cos appear in Euler's Identity does not seem terribly profound to me.
Also, why stop at complex numbers, since it's "turtles all the way"? Quarternions, octonions are neat too. Why doesn't Feynman think octonions are "our jewel" too? At each step (real -> complex -> quaternion -> octonion) you get a larger class of formulae having roots, but sacrifice more properties of the ring structure. For example, the reals have a total order but the complex numbers do not.
Probably the most blatant physical example of this is modeling mass-spring-damper systems. The differential equations will map the physical coefficients to such a polynomial, and the roots of that polynomial determine the system's behavior. Without complex numbers, you wind up only being able to describe real exponential responses and don't have the vocabulary to describe the underdamped (and oscillating) system responses with complex exponentials in the results.
Yes, I know that the complex numbers are an ACF.
Why apply the Cayley-Dickson construction [1] only once? The complex numbers may be an algebraically closed field, but they aren't a Central Simple Algebra over the Reals [2]. Apply the Cayley-Dickson construction again and you get the quaternions, which are real central simple. Apply it a third time and you get the octonions. You can keep doing this over and over and over getting more and more closure properties...
> Without complex numbers, you wind up only being able to describe real exponential responses
Why can't you describe them using pairs of real numbers in polar coordinates?
[1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru...
[2] https://en.wikipedia.org/wiki/Central_simple_algebra
Of course, what makes complex numbers special is that you can multiply them, and that gives you a field.
Following that train of thought,
> Are there any simple and obvious applications that require continuity/calculus or topology on the complex plane that can't be restated in terms of real-pairs (and calculus/topology over those)?
You can of course develop all of complex analysis without introducing the traditional a+bi notation and just referring to vectors in R^2. That's notation. However, complex differentiability (being holomorphic) is a stronger definition than regular differentiability in C(R^2, R^2), in particular, the Cauchy-Riemann equations need to be satisfied. From that stronger definition, you get all sorts of other theorems such that all holomorphic functions are analytic and so on.
Complex analysis is the only field I've found that seems to genuinely require complex numbers rather than just polar coordinate notation. However I get the impression that complex analysis, as a research field, has been a dead end for a long time now with few people working on it [1].
If pure research has come to a halt then it's fair to judge the field based on the practical applications it has achieved. Are there many? The class of holomorphic functions is pretty rigidly constrained -- infinitely differentiable, equivalent to a Taylor series at each point. I once heard that even physicists doubt the legitimacy of assuming that all physical functions are holomorphic (but now I'm speaking beyond my expertise). Continuing to speak beyond my expertise, I don't think Feynman had holomorphic functions in mind when he said "our jewel".
I have to say that the definition of limits for complex numbers always seemed like an artificial straitjacket. I understand why you can't have complex limits defined any other way, but maybe the take-away here isn't "complex limits should work like this" so much as "complex limits, and therefore complex analysis, is not very useful".
[1] "Complex analysis is pretty much dead. There are some remaining problems but most of them are either too specialised or too difficult. This is not a very healthy situation, and there are relatively few people working in complex analysis nowadays.", Dmitry Beliaev , former Assistant Professor (Mathematics) at Princeton University, https://www.quora.com/Are-there-any-mathematics-faculty-at-P...