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This is such a weird thread. There are a bunch of people arguing about why their favorite math thing is more a miracle than someone else's favorite math thing.
by kerkeslager 7y ago
This is such a weird thread. There are a bunch of people arguing about why their favorite math thing is more a miracle than someone else's favorite math thing.
Nothing is surprising if you've seen it before. Let's just let each other be excited about our favorite math, okay?
Euler's identity is the one that gets me:
e^(i * pi) + 1 = 0
How can this be? The five fundamental constants are related!
- matt-noonan 7y agoYou mean the seven fundamental constants? e^(i * pi) + 1 = 0 * 17 * 42
- tus88 7y agoStill not sure what i and 17 are...
- AstralStorm 7y agoNumber i is the complex unit, but I've no idea about 17 either.
- deleted 7y ago[deleted]
- magicseth 7y ago17 is the favorite number of David Kelly, a mathematician and professor of Hampshire College. He loves it for a very large number of reasons. He was instrumental in creating a summer camp at Hampshire which has trained many young brilliant mathematical minds. (Including Vi Hart). Thus it has spread. http://www.vinc17.net/yp17/index.en.html http://www.vinc17.net/yp17/index.en.html
- oarabbus_ 7y agoYou can find such "collections" for numbers like 7, 8, or 13, even ignoring the integers 0-4, and 42 is obviously a reference to Hitchhikers guide, a work of fiction. Am I missing something?
- lucas_membrane 7y agoFollowing the links from that link, I find wikipedia for 17 says, "Carl Friedrich Gauss (1777–1855) showed that two regular "heptadecagons" (17-sided polygons) could be constructed with ruler and compasses." (I learned it as straightedge and compass.) Wikipedia did not mention the story that Gauss has a regular heptadecagon inscribed on his tombstone.
- lonelappde 7y agoSurely he loves it for 17 reasons.
- yesbabyyes 7y agoIt's the first arbitrary number. 4711 is the second arbitrary number. They were used as the numeric foo and bar when I studied programming in high school and college during the 90s. I always thought it was universal hackerdom, but I actually rarely see them referenced. Perhaps it's a European thing?
- tus88 7y agoDo you have a source on "arbitrary number"? There were no obvious explanations online.
- perl4ever 7y agoI was thinking, what is the smallest number not found in any sequence on https://oeis.org/ https://oeis.org/?
- yesbabyyes 7y agoIt's not a mathematical or in any way strict thing. It's just a number chosen to exemplify any number. Just like "foo", "bar", "baz", "qux", etc, a sequence of numbers could be e.g. 17, 42, 4711. I grew up with it, and googling it seems that at least 4711 is a German thing.
- boyobo 7y agoi= sqrt(-1) https://en.wikipedia.org/wiki/Euler%27s_identity https://en.wikipedia.org/wiki/Euler%27s_identity
- deleted 7y ago[deleted]
- throwaway2048 7y agoits a joke about how Euler's identity isn't normally simplified and written as: E^(iπ) = -1 because people wanna be "cute" about it containing 0 and 1, so why not a bunch of other arbitrary numbers too?
- lonelappde 7y agoSetting an expression equal to 0 is quite standard form.
- avmich 7y agoRoger Hui: In the year 2033 Earth was discovered by Survey Fleet MCXII of the Galactic Empire. The Emperor ordered Earth to send a representative to the court, with strict instructions to “bring something beautiful”. Emperor: What beautiful things does Earth have? Earth Representative: Your excellency, our mathematician Euler proved in our year 1737 that +/(1+⍳∞)-s ←→ ×/÷1-(⍭⍳∞)-s Emperor: What is the ⍭ symbol? Earth Rep.: ⍭i is the i-th prime. Emperor: And what is ∞? Does it do anything useful? Earth Rep.: It denotes infinity, your excellency. Emperor: (ponders equation for a minute or two) Respect! Emperor: Neat notation you have there. Tell me more. Earth Rep.: Your excellency, it’s called APL. It was invented by the Canadian Kenneth E. Iverson …
- boyobo 7y agoLet's just let each other be excited about our favorite math, okay? I don't think it's malicious, we're just trying to narrow down exactly what makes something interesting. In the process, we get to understand our favorite math more deeply. Nothing is surprising if you've seen it before. On the other hand, if you can find a theory that makes a miracle look mundane, then the theory is probably interesting. So finding reasons why something is `not surprising' is a good guide to finding interesting math.
- lacker 7y agoI will take the contrary viewpoint. It matters what mathematics we consider to be beautiful and intuitive, because better mathematical intuition will make you better at solving mathematical problems. On that note, I think e^i pi = -1 is drastically overrated. That is typically the definition of exponentiating to a complex power, that e^ix = sin x + i cos x. Of course something can be beautiful if you take it to be an axiom. It’s like saying x=x is beautiful. Also it’s really just sticking -1 in the more-intuitive, more-useful formula so that it looks cool. A really beautiful formula is that x^p - x is divisible by p when x is an integer and p is prime....
- kd5bjo 7y agoThough it’s probably the most commonly used definition for complex exponentiation, e^i pi=-1 is not an axiom— it is derived from the basic rules of integer arithmetic expanded in scope until they make a closed system. And from this, we can derive all of the trigonometric functions using only algebra and no geometry. (See http://www.feynmanlectures.caltech.edu/I_22.html http://www.feynmanlectures.caltech.edu/I_22.html , particularly section 22-5)
- thaumasiotes 7y ago> I think e^i pi = -1 is drastically overrated. That is typically the definition of exponentiating to a complex power, that e^ix = sin x + i cos x. Yes, but it's the definition of exponentiating to a complex power because it is easily proven by reference to the real Taylor series of those functions. The complex definition is motivated by the real result, not -- as you imply -- the other way around. (Also, you have it backwards; e^ix is cos x + i sin x.)
- username90 7y agoYou can easily derive e^ix = i cos x + sin x. Just look at the Taylor series for e^x and compare with the Taylor series for sin x + cos x. You'd see that they are basically the same, just that the trigonometric functions alternate their signs every 2 steps. How do we alternate the signs in e^x? We just add a factor of i, that way we get alternate signs every two steps but get an additional i on every odd step. Odd steps are sin, so just add i to sin and we are done: e^ix = i sin x + cos x. Now we also see that you remembered the wrong formula, it the sine part is imaginary since it corresponds to the odd factors.
- jimbo1qaz 7y agoI prefer e^(i * 2pi) = 1 (or any multiple of i 2pi). 2pi (or tau) is simply a more fundamental constant for nearly all formulas related to trigonometry and DSP (excluding the sinc function).
- jacobolus 7y ago> exp(iπ) = –1 ... How can this be? The “plain English” expression of this formula is: a half turn rotation in a plane is equivalent to a reflection across the axis of rotation. YMMV, but I have found that small children can understand this statement.
- deleted 7y ago[deleted]
- ColanR 7y agoThat is the best explanation I've ever seen.
- creato 7y agoTalking about rotation as if that is obviously relevant completely skips the surprising and interesting part of this identity. edit: Said another way, what you said is an explanation for what (cos(pi), sin(pi)) is. That exp(i*pi) has anything to do with sine or cosine is what makes the identity interesting, and your explanation says nothing about that.
- dnautics 7y agoYeah and also that an exponential is a turning function, or that you can even take an number to an imaginary power in the first place, or that turning the right amount just happens to take the power of e (and not, say, 2)
- username90 7y agoIt's a lot less surprising if you look at the functions as Taylor series, then you'd see that exponentiation is basically the same thing as cos + sin. This also explains "why e and not 2?", because the Taylor series for e, cos and sin are very nice by definition (e is defined to have trivial derivative, and radians are defined to make sin and cos have simple derivatives) while the Taylor series for exponentiation by 2 is not.
- 7y ago
- nabla9 7y agoFor me the most mysterious equation is: ⌊π⌋ - ⌈e⌉ = 0 This equation shows connection between pi, e, zero and number theory.
- soulofmischief 7y agoI mean, you could arbitrarily chop the precision of all sorts of things to make them "equal". It hardly seems significant or illuminating.
- lern_too_spel 7y agoIt's a joke.
- j7ake 7y agoThere’s nothing magical here, the first term describes a 180 degree rotation around a circle of radius 1, landing you at -1. Then you add 1 and you land at the Center of your circle, (0,0).
- ColinWright 7y agoSo why does the first term describe a 180 degree rotation? People learn that 2^3 means to take three copies of 2 and multiply them together, and that i^2 is -1, so what does it mean to take (i pi) copies of e and multiply them together? The part you are describing is not where the magic resides ...
- Koshkin 7y agoWell, there is nothing magical in a magic after you explain it.
- lonelappde 7y agoBut why does rotation work as a model of complex numbers? It's (mathe)magical!
- jaster 7y agoIf you're curious about this, betterexplained has a good article on Euler's Formula: https://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/ https://betterexplained.com/articles/intuitive-understanding... As a bonus, understanding how it is related to rotation may give you a new insight on the Fourier Transform: https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/ https://betterexplained.com/articles/an-interactive-guide-to...
- salty_biscuits 7y agoI like your style. Mine is why is the kissing number for circles in the plane 6? Why is it even an integer? Seems totally rigged!
- misil 7y agoFun fact: Gauss is said to have commented on this equation that "if this was not immediately apparent to you upon being told it, you would never be a first-class mathematician".
- yoz-y 7y agoI find it amusing that many replies to your comment illustrate your point very well.
- boyobo 7y agoIf we just throw our hands up in the air and declare everything a "miracle" we never learn anything.
- nadam 7y agoFor me it would be suprising only if the equation would be about real numbers only. Exponentiation of complex numbers is defined in a way that this is trivially true. Or at least this is my perception as a non math-expert. My approach to Euler's equation (as a non-expert) was the following: 1. Try to understand the meaning of the operations in the equation. Search for the definition of exponential on complex numbers, because it was not trivial for me how it is defined on complex numbers. 2. I have read that it is defined by angles and the unit circle: e(alpha * i) is the point on the unit circle at angle alpha. 3: Looking at the equation: this is trivial, it basically says that cos(Pi) is -1. What am I missing? Edit: And why is exponentiation defined this way? Why is the base e? Why isn't it 2, like this: "2^(alpha * i) is the point on the unit circle at angle alpha." Would this definition lead to contradiction or some difficulties?
- ben_w 7y agoThe surprise is that the definition you gave in 2. fully agrees with the definition of multiplication that comes naturally from the cartesian version of complex numbers: z = x + bi -> (x1 + b1i)(x2 + b2i) = x1x2 + x1b2i + x2b1i - b1b2 Vector multiplication has two very different forms (dot and cross), so multiplication (and by extension exponentiation) working out smoothly is neat.
- mikorym 7y agoI suspect it would spiral off the unit circle.
- zests 7y agoThe derivative of e^ix is ie^ix. Multiplication by i is equivalent to rotation by 90 degrees. The derivative of velocity is acceleration. Acceleration perpendicular to velocity is circular motion. With base 2 the motion is not circular. All these pieces (and an initial value problem differential equation) come together to explain the identity.
- obastani 7y agoThat is not actually the way to define e^x for complex numbers. There are a few ways you can do so. The most natural way is as the unique solution to the differential equation df/dx = f The solutions are e^x + const; requiring f(0) = 1 gives you the choice const = 0. Another way to define it is as the infinite series e^x = \sum_{i=0}^inf x^i/i! It is pretty easy to show this definition is equivalent to the previous one. A third way is as lim_{n -> inf} (1 + x/n)^n This definition comes from the intuition that e^x represents the limit of continually compounding interest. As before, it is pretty easy to show that it is equivalent to the previous two. In any case, all these definitions extend directly to complex values of x. The fact that e^{ix} = cos(x) + i sin(x) holds (and hence Euler's identity holds) is a consequence of a more natural definition, not the typical definition.
- robert-wallis 7y agoI recently learned: > It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum. However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it. https://en.m.wikipedia.org/wiki/Euler's_identity https://en.m.wikipedia.org/wiki/Euler's_identity But it is still cool. https://youtu.be/yPl64xi_ZZA https://youtu.be/yPl64xi_ZZA
- throwaway66920 7y agohttps://www.smbc-comics.com/comic/2013-04-02 https://www.smbc-comics.com/comic/2013-04-02 An extension of this I find even more surprising
- nixpulvis 7y agoI too have spent some time pondering this equation. I've written down some of my findings. https://nixpulvis.com/math/00-euler https://nixpulvis.com/math/00-euler
- harry8 7y agoHow do you like: e^(i * tau) = 1 https://tauday.com/tau-manifesto https://tauday.com/tau-manifesto
- kerkeslager 7y agoI actually find this less aesthetically pleasing. It removes the addition identity constant (0). But that's subjective, who cares?
- harry8 7y agoe^(i * tau) - 1 = 0
- graedus 7y ago3blue1brown has an excellent and concise video explaining this: https://www.youtube.com/watch?v=v0YEaeIClKY https://www.youtube.com/watch?v=v0YEaeIClKY