7 ms·

Is Euler’s Identity Beautiful? and If So, How?

49 points by vo2maxer 5y ago

ian-g 5y ago
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
- joppy 5y ago
  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.
- Yajirobe 5y ago
  > if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician. -- Carl Friedrich Gauss
beforeolives 5y ago
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. [1] https://youtu.be/ZxYOEwM6Wbk?t=2280 https://youtu.be/ZxYOEwM6Wbk?t=2280
- jleahy 5y ago
  Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.
  - sobriquet9 5y ago
    In hindsight, and if you know Mclaurin series. Complex numbers were known and used for many years before Euler found this obvious result.
  - red_trumpet 5y ago
    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.
  - japanuspus 5y ago
    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. In particular, the result is not "obvious".
- spuz 5y ago
  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...
  - bobthechef 5y ago
    > "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.
sobriquet9 5y ago
I don't find identity itself to be beautiful. It's just a special case of (much more useful) Euler formula. That I do find beautiful.
thrwyoilarticle 5y ago
>“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.
- SimplyUnknown 5y ago
  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.
  - roenxi 5y ago
    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.
    - eutectic 5y ago
      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.
      - roenxi 5y ago
        "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.
        5y ago
susam 5y ago
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." See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.
- SuchAnonMuchWow 5y ago
  e^(iτ) = 1, not 0
  - susam 5y ago
    Oops! I meant to write e^(iτ) - 1 = 0. I was going back and forth between the various formulations of the identities: e^(iπ) + 1 = 0, e^(iτ) - 1 = 0, e^(iπ) = -1, e^(iτ) = 1. 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!
    - BlueTemplar 5y ago
      I don't "get" the first two. 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...)
js8 5y ago
I thought of the other Euler's identity: https://en.wikipedia.org/wiki/Euler_characteristic https://en.wikipedia.org/wiki/Euler_characteristic ..which I think is beautiful too.
- gspr 5y ago
  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 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_theorem https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_th...
- MiguelVieira 5y ago
  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). https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem
vishnugupta 5y ago
Obligatory mentioning of the Better Explained article[1]. [1] https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...
teucris 5y ago
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 [...]
KirillPanov 5y ago
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.
- ThrustVectoring 5y ago
  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.
  - KirillPanov 5y ago
    > 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? [1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... [2] https://en.wikipedia.org/wiki/Central_simple_algebra https://en.wikipedia.org/wiki/Central_simple_algebra
deleted 5y ago
[deleted]
jqgatsby 5y ago
I think even prettier is the formula in hyperbolic form. Then at least it adds up properly!

49 points by vo2maxer 5y ago

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

joppy 5y ago
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.
Yajirobe 5y ago
> if this formula was not immediately apparent to a student upon being told it, that student would never become a first-class mathematician. -- Carl Friedrich Gauss

beforeolives 5y ago

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. [1] https://youtu.be/ZxYOEwM6Wbk?t=2280 https://youtu.be/ZxYOEwM6Wbk?t=2280

jleahy 5y ago
Even e^(ix) = cosx + i*sinx is an obvious result of the definition i^2= - 1.
- sobriquet9 5y ago
  In hindsight, and if you know Mclaurin series. Complex numbers were known and used for many years before Euler found this obvious result.
- red_trumpet 5y ago
  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.
- japanuspus 5y ago
  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. In particular, the result is not "obvious".
spuz 5y ago
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...
- bobthechef 5y ago
  > "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.

sobriquet9 5y ago

I don't find identity itself to be beautiful. It's just a special case of (much more useful) Euler formula. That I do find beautiful.

thrwyoilarticle 5y ago

>“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.

SimplyUnknown 5y ago
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.
- roenxi 5y ago
  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.
  - eutectic 5y ago
    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.
    - roenxi 5y ago
      "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.
      - 5y ago

susam 5y ago

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." See the bottom of the page at https://www.feynmanlectures.caltech.edu/I_22.html https://www.feynmanlectures.caltech.edu/I_22.html for the above excerpt.

SuchAnonMuchWow 5y ago
e^(iτ) = 1, not 0
- susam 5y ago
  Oops! I meant to write e^(iτ) - 1 = 0. I was going back and forth between the various formulations of the identities: e^(iπ) + 1 = 0, e^(iτ) - 1 = 0, e^(iπ) = -1, e^(iτ) = 1. 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!
  - BlueTemplar 5y ago
    I don't "get" the first two. 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...)

js8 5y ago

I thought of the other Euler's identity: https://en.wikipedia.org/wiki/Euler_characteristic https://en.wikipedia.org/wiki/Euler_characteristic ..which I think is beautiful too.

gspr 5y ago
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 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_theorem https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_th...
MiguelVieira 5y ago
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). https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem https://en.wikipedia.org/wiki/Gauss–Bonnet_theorem

vishnugupta 5y ago

Obligatory mentioning of the Better Explained article[1]. [1] https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding...

teucris 5y ago

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 [...]

KirillPanov 5y ago

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.

ThrustVectoring 5y ago
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.
- KirillPanov 5y ago
  > 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? [1] https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_constru... [2] https://en.wikipedia.org/wiki/Central_simple_algebra https://en.wikipedia.org/wiki/Central_simple_algebra

deleted 5y ago

[deleted]

jqgatsby 5y ago

I think even prettier is the formula in hyperbolic form. Then at least it adds up properly!