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Euler’s number pops up in situations that involve optimality
- deleted 5y ago[deleted]
- nick__m 5y agoAnd my favorite equation is ℇ^(ⅈπ)+1=0 ! It contains Euler, the imaginary unit, the unit, the zero and some hidden trigonometry. P.S. does anyone know why the unicode symbol for the Euler constant render as a weird E when it is usually represented as a slightly italicized e ?
- mkl 5y agoAlso π and the three most important operations: addition, multiplication, exponentiation.
- 323 5y ago> addition, multiplication, exponentiation. Which are the hyperoperations of rank 1, 2 and 3: > In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called hyperoperations in this context) that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3). https://en.wikipedia.org/wiki/Hyperoperation https://en.wikipedia.org/wiki/Hyperoperation
- adunk 5y agoOne of the things I really like about the tau manifesto (the proposal to use tau == 2 * pi instead of pi in many situations) was their explanation of how tau made this equation all the more interesting (IMHO) by making it "almost like a tautology": https://tauday.com/tau-manifesto#sec-euler_s_identity https://tauday.com/tau-manifesto#sec-euler_s_identity
- janto 5y agoIndeed. It shows that the pi formulation is actually somewhat ugly because it lacks symmetry.
- Scarblac 5y agoThe weird E is Euler's _constant_, and the slightly italicized e is Euler's _number_. https://en.wikipedia.org/wiki/Euler%27s_constant https://en.wikipedia.org/wiki/Euler%27s_constant https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant)
- poizan42 5y agoBut that is the Euler–Mascheroni constant which is normally denoted by gamma. There is a footnote on https://en.wikipedia.org/wiki/Letterlike_Symbols https://en.wikipedia.org/wiki/Letterlike_Symbols that says: > It's unknown which constant this is supposed to be. Xerox standard XCCS 353/046 just says 'Euler's'. See also this discussion on math stackexchange: https://math.stackexchange.com/a/3123704 https://math.stackexchange.com/a/3123704
- 0xdeadb00f 5y agoI think they're aware, but replying to when the parent asked "anyone know the symbol for Euler's constant" when they really neeeded the symbol for Euler's number.
- f00zz 5y agoThis follows from e^(ix) = cos(x) + i sin(x)! I'm currently reading the Qiskit quantum computing textbook, and the appendix on linear algebra has a demonstration: https://qiskit.org/textbook/ch-appendix/linear_algebra.html https://qiskit.org/textbook/ch-appendix/linear_algebra.html
- sidpatil 5y agoAnd that follows from De Moivre's formula, (cos(x)+ i sin(x))^n = cos(nx) + i sin(nx). https://en.wikipedia.org/wiki/De_Moivre%27s_formula https://en.wikipedia.org/wiki/De_Moivre%27s_formula
- Rompect 5y agoAlso an amazing way are Taylor polynomials, this article explains the process of thought really well: https://betterexplained.com/articles/taylor-series/ https://betterexplained.com/articles/taylor-series/
- f00zz 5y agoYeah, in the link I posted the formula is derived via Taylor (or Maclaurin) series, but the explanation in your link is great. Thanks for sharing!
- wrycoder 5y agoWhich only slightly obfuscates the fact that e^(iπ) = -1. Bamboozles the rubes!
- 323 5y agoe^(iπ) = -1 is basically the unit circle in the complex plane: https://en.wikipedia.org/wiki/Circle_group https://en.wikipedia.org/wiki/Circle_group
- lunchladydoris 5y agoIf you want to go deeper, Eli Maor's "e: The Story of a Number" [0] is a great read that doesn't shy away from showing a few equations. [0]: https://press.princeton.edu/books/paperback/9780691168487/e-the-story-of-a-number https://press.princeton.edu/books/paperback/9780691168487/e-...
- vesinisa 5y agoTitle should be edited to lowercase e for Euler's number.
- mromanuk 5y agoYes: Why Euler's number (e), the Transcendental Math Constant, Is Just the Best
- adunk 5y agoFor everyone that, like me, like to read only the headline and then proceed directly into the comments: The title of the link currently is "Why E, the Transcendental Math Constant, Is Just the Best". But the article really is about Euler's constant - the lower case e - and not about any of the capital E:s out there (like the capital E sometimes used in scientific notation, or the expected value in probability theory).
- corndoge 5y agoAre either of the latter transcendental constants
- lordnacho 5y agoSee what happens when you're so smart you get multiple things named after you? This is not the e you know and love from school: https://en.wikipedia.org/wiki/Euler%27s_constant https://en.wikipedia.org/wiki/Euler%27s_constant This one is: https://en.wikipedia.org/wiki/E_(mathematical_constant) https://en.wikipedia.org/wiki/E_(mathematical_constant) Worth coming up with some better way to talk about this.
- deleted 5y ago[deleted]
- poizan42 5y agoThe first one is usually (at least from what I've seen) called the Euler–Mascheroni constant which is denoted by γ, so I don't think there is much confusion.
- ianai 5y agoEuler was much more than smart. The man went home during the Black Plague and studied math so hard he went blind in one eye - presumably so his brain could use those neurons for math instead of sight. He was also discredited in his time and for centuries after for an intuitive understanding of calculus through infinitesimal and infinite numbers - which was only relatively recently put into rigor akin to epsilon-delta calculus. Also considered the last person to be able to know all of the known world of mathematics at his point in time. I kind of wish we had a holiday of some kind to appreciate either Euler himself or even a month to discuss the historical contributions to knowledge by philosophers and scientists alike.
- DeathArrow 5y agoI find this beautiful: e^iπ = −1
- montroser 5y agoYeah! Which of course also means that e and π can be defined in terms of one another.
- dotancohen 5y agoThat is probably the most insightful thing I've read all year. I wonder if there are any subtle implications.
- tsimionescu 5y agoOne interesting thing is that it means it's not impossible to think that π+e or πe or π^e or some other combination of the two could be a simpler number (right now most of these numbers have no known/proven properties - they could even be rational for all we know).
- jstx1 5y agoThe more general formula (e^ix = cosx + i*sinx) looks better to me because it defines exponentiation of a complex number as a rotation around a unit circle. It has a nice proof, some cool visualisations and a lot of implications to a bunch of other things in mathematics - I can get behind calling that beautiful. The special case of x=pi... it's like being excited that sin(pi)=0 or cos(pi)=-1. It doesn't really say anything meaningful or consequential, people like it only because of the symbols it includes. It feels kind of like a math meme that people like to repeat and I can't get behind it. Maybe it's just not for me and I should just let other people like what they like.
- qq4 5y agoI feel this way as well. In fact every time I have tried to remember the "most beautiful equation" I had to think of it in the context of the unit circle and work it out by assigning pi to x. Otherwise I don't get any wow out of it.
- _Microft 5y agoI wonder how many mathematicians and physicists were harmed by the submitted title ;) (I would like to increase the count by e^0 btw)
- ReleaseCandidat 5y ago:D Mathematician. Thought about a new (at least to me) transcendental number ...
- ReleaseCandidat 5y agoI prefer the Champernowne constant.
- vmilner 5y agoTim Gowers uses the differentiation of e^x as an example of something bright UK maths A-level students often don't understand fully: https://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-necessarily-give-you/ https://gowers.wordpress.com/2012/11/20/what-maths-a-level-d...
- tomrod 5y agoThis article reminded me of my maths journey. I was mechanically dutiful as a student, and would make lateral connections but had a lot of patchwork understanding. It wasn't until I understood the derivations in Real Analysis that things started to click.
- alexilliamson 5y agoYes exactly! Deriving calculus from set axioms truly opened my mind to math, and more generally critical thinking.
- londons_explore 5y agoIt's because most exams and curriculums in the UK are so strictly defined that all questions are almost guaranteed to follow one of a small set of structures. And schools have figured out that rather than teaching the subject from first principles, it's easier to get students to get high grades by teaching them each of the structures. Eg. "Whenever there is a question about differentiating x^7, just put 7x^6 as the answer." They then get the students to try a few examples (x^3 becomes 3x^2, x^77 becomes 77x^76, etc), and thats the way every science-y subject is taught. I often think it leads to students who do well in exams, but can't solve many real world problems. It could be solved by having a part of every exam paper be never-seen-before applied problems. For example, for differentiation, one might ask "A road's height in meters as a function of the horizontal distance along the road in kilometers is defined as sin(x)cos(x)tan(x). At what points are the steepest uphills? Would you describe the slope of the road as 'very hilly', and why?"
- eigenket 5y ago
- jstx1 5y agoI really don't like this way of thinking about it. e isn't important, the exponential function is. e shows up so often because we've chosen to write exp(x) as e^x. It's a result of a notational choice - the fact that exp(1) = 2.718.. and we call that e is pretty insignificant and boring.
- Denvercoder9 5y agoThe fact that e = 2.718... is a fundamental property of the exponential function, though. It's not an arbitrary choice.
- ianai 5y agoIndeed. That a member of the real number line has this important relationship to the differential operator, the complex plane and number systems, and thus all of trig, calculus, and quantum mechanics is pretty impressive to put it lightly. (Trig through the many relationships of e^x with cosine and sine functions.) The GP comment reads as either a grab at elite character at best or flat out anti-intellectual at worst. No need to bring it in here.
- jstx1 5y agoI feel like you've missed the point of my comment. I said that the exponential is important and you've repeated that here so we don't disagree about that. My point is to distinguish between the exponential function in general and particular value of the exponential function when evaluated at 1.
- ianai 5y agoWhich just so happens to be the one power of a number that helps the most if you want to do any actual, decimal calculations with a number without any other decimal expansions of it at hand.
- abnry 5y agoThe point being made is that the _function_ is different than the _constant_ producing that function through exponentiation. I think that's kind of fair. Take this headline: The function exp(x) = 1 + x + x^2/2 + x^3/6 + ... is the most beautiful function in mathematics. It is its own derivative, has "product linearity", i.e. exp(x+y) = exp(x) exp(y), and is related to trig functions through complex numbers. The number e isn't doing the heavy lifting, it is the function. The number e comes from the function, not the other way around. Even the famous equation with pi and e is a consequence of the function. And the Taylor series is the easiest way to see the relationship with trig functions. To be fair, there might be a difference in dispensation at play. Those who prefer a more causal or "active" feel to mathematics would prefer the function framing while those who prefer a more platonic or "mystical" feel would prefer the constant framing.
- mensetmanusman 5y agoI remember the ‘aha’ moment I had in my first year of calculus during a test none the less: “Ohhhh when something is growing in proportion to its current size you set up your derivative equality and get an e^x!” The example used was bunnies with unlimited food; then foxes were introduced. Was surprised to have that learning moment in the middle of the exam and not prior…
- annexrichmond 5y agosounds like a well thought out exam question. I always appreciated exams where you actually learn while doing it, instead of being in a mode of regurgitation
- ianai 5y agoOddly it was my calculus 1 final that clicked a lot of things for me. Turned out the authors of the test included a professor who could explain calculus much better than my lecturer for that semester. I remember feeling the most intense and lasting feeling of revelation for several days after that test.
- thomasahle 5y ago> Was surprised to have that learning moment in the middle of the exam and not prior… I sat my first exam for a university course I was teaching last year. I thought I needed to introduce some new ideas, so the students wouldn't be bored doing it. From the evaluations, not all students agreed...
- nerdponx 5y ago"Bored" is the absolute last thing on anyone's minds during an exam! I always hated when my instructors put "important" results that we have never seen before on an exam. It was like adding insult to injury if I didn't know how to solve it. It was different on homework assignments, because usually that you had time to work through the problem in detail and have the "aha" moment, without stress and time pressure.
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- quantum_state 5y agoThat's why it is physicists' best friend :-).
- agumonkey 5y agoIsn't there another fix point like value for hyperoperations ?
- jhncls 5y agoIn a unique Numberphile video featuring Grant Sanderson (3blue1brown), this weird number pops up in a game of darts. [0] https://youtu.be/6_yU9eJ0NxA https://youtu.be/6_yU9eJ0NxA
- woopwoop 5y agoThis is the most beautiful formula in mathematics, because it includes all the most important constants e, i, pi, 0, and 1: (ei)^0 = 1^pi
- mdp2021 5y agoUnfortunately, it is trivial... A joke. The constants there could amount to almost anything.
- westcort 5y agoThe reciprocal of e is about 37% and it pops up in a lot of places. Say, for example, you play a lottery 1000 times and there is a 1 in 1000 chance of winning each time you play. The chances you do not win even once is 37%, or 1/e.
- deleted 5y ago[deleted]
- hinkley 5y agoThe Secretary problem (#2 in the article) is still one of my favorites. Stop playing once you’ve seen at least n/e of the available options and the current one is acceptable.