[HN Gopher] Euler's number pops up in situations that involve op...
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Euler's number pops up in situations that involve optimality
Author : elsewhen
Score : 108 points
Date : 2021-11-25 11:58 UTC (11 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| quantum_state wrote:
| That's why it is physicists' best friend :-).
| woopwoop wrote:
| This 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 wrote:
| Unfortunately, it is trivial... A joke. The constants there
| could amount to almost anything.
| nick__m wrote:
| And my favorite equation is ^(ip)+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 wrote:
| Also p and the three most important operations: addition,
| multiplication, exponentiation.
| 323 wrote:
| > 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
| Scarblac wrote:
| The 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/E_(mathematical_constant)
| poizan42 wrote:
| But that is the Euler-Mascheroni constant which is normally
| denoted by gamma.
|
| There is a footnote on
| 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
| 0xdeadb00f wrote:
| I 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.
| adunk wrote:
| One 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
| janto wrote:
| Indeed. It shows that the pi formulation is actually somewhat
| ugly because it lacks symmetry.
| f00zz wrote:
| This 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
| sidpatil wrote:
| And _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
| Rompect wrote:
| Also an amazing way are Taylor polynomials, this article
| explains the process of thought really well:
|
| https://betterexplained.com/articles/taylor-series/
| f00zz wrote:
| Yeah, 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 wrote:
| Which only slightly obfuscates the fact that e^(ip) = -1.
| Bamboozles the rubes!
| 323 wrote:
| e^(ip) = -1 is basically the unit circle in the complex
| plane:
|
| https://en.wikipedia.org/wiki/Circle_group
| _Microft wrote:
| I wonder how many mathematicians and physicists were harmed by
| the submitted title ;)
|
| (I would like to increase the count by e^0 btw)
| ReleaseCandidat wrote:
| :D
|
| Mathematician. Thought about a new (at least to me)
| transcendental number ...
| agumonkey wrote:
| Isn't there another fix point like value for hyperoperations ?
| westcort wrote:
| The 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.
| hinkley wrote:
| The 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.
| [deleted]
| ReleaseCandidat wrote:
| I prefer the Champernowne constant.
| vmilner wrote:
| Tim 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-d...
| scythe wrote:
| I was expecting this to happen because the proof that the limit
| at zero of (e^h - 1)/h = 1 is tricky -- nope, the student
| doesn't recognize the derivative formula in the first place.
| Aardwolf wrote:
| > The particular topics he wanted me to cover were integrating
| log x, or ln x as he called it
|
| What's wrong with calling it ln x? The way this is written in
| the article implies there's something weird about calling it
| that. The name 'log' can mean log2, log10 or natural logarithm
| depending on the field.
|
| Removing ambiguities from math notation should be considered a
| good thing.
|
| The author expressed a worry about math education. Consider
| that a clear non ambiguous notation would help.
| dan-robertson wrote:
| Most mathematicians use log to mean either natural log, or
| sometimes log in the relevant base (e.g. 2 if you are talking
| about information theory).
|
| In school (and engineering or physics I guess) you often are
| made to use ln for natural log and you are taught a way to
| pronounce that name (somewhere between lun and l'n)
|
| It feels like the point is "this person had not been exposed
| to university style mathematics".
| Aardwolf wrote:
| > It feels like the point is "this person had not been
| exposed to university style mathematics".
|
| Imho math is about logic and reasoning, not about what
| group you're part of
| jfengel wrote:
| The group is a bigger deal than you might expect. There
| are an infinite number of true theorems, almost all of
| which are boring.
|
| Mathematicians decide what is interesting, and that's not
| a matter of logic. A computer can bang out new theorems
| at light speed but nobody cares. Mathematics, like
| science and programming, is as much about humans as about
| the raw logic and data.
|
| You're welcome to be a group of one and please only
| yourself. But then you wouldn't care if it were
| published, and it wouldn't be unless you showed it to
| someone and they took an interest.
| CornCobs wrote:
| For me it's the opposite - in secondary school education
| the math teachers made the distinction between "log" and
| "lon" (how they pronounced it) probably because that's
| what's written on our Casio calculators!
|
| Whereas in uni log is generally assumed to be the natural
| log, or else it's specified, or else the base is
| unimportant (like in big O notation)
| SilasX wrote:
| "Yes, how dare someone have a different context than me [in
| which ln x is correct and log x is not]."
|
| Similar to those who mock people for saying a word
| incorrectly that they only learned from reading.
| filmor wrote:
| Actually, another possible convention is "I don't care
| about the base", as in O(n log n) or in general in most of
| Asymptotic Analysis. This becomes fun when people start
| talking about O(2^(log n)) where the chosen base becomes
| relevant again :)
| yellowcake0 wrote:
| The ln notation has gone out of fashion with mathematicians.
| Generally the base is clear from the context, or it's
| irrelevant.
|
| As someone who has done a lot of mathematics in their life,
| I've never found this perceived ambiguity to be an issue.
| Aerroon wrote:
| I think the reason for this is that derivation from 'first
| principles' isn't really done. You'll do it once or twice in
| the intro to derivatives and that's it. The other 40 hours you
| spend on derivatives won't even touch it.
|
| The issue with being able to derive the formulas for derivation
| yourself is that it's not very useful. You simply don't have
| time to make those derivations during a test. It's like trying
| to use grammar rules in a conversation - conversations happen
| at a pace where you cannot apply grammar rules. You'll just
| have to know the patterns.
|
| You learn things in school to do a test. The usefulness of the
| vast majority of the knowledge they attain is purely to help
| them do the test. Later in life you might wish you knew more
| about this or that, but that's not at all apparent to the
| student.
| tomrod wrote:
| This 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 wrote:
| Yes exactly! Deriving calculus from set axioms truly opened
| my mind to math, and more generally critical thinking.
| londons_explore wrote:
| It'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?"
| dan-robertson wrote:
| I think the problem is they want calculus in the curriculum
| and it is too late to be able to put it in context. There are
| some great uses for calculus that are accessible to many high
| school students. In particular, with physics you usually
| learn about capacitors and nuclear decay. Both of these cases
| are basically solving the differential equation y' = ky but:
|
| - the physics course can't depend on the concurrent maths
| course because you are allowed to take physics without taking
| maths, so you just learn weird equations full of exponential
| a instead of the ODE
|
| - I _think_ the maths course doesn't even teach differential
| equations. They are in FP1 (from a separate 'further maths'
| course) but definitely not in AS (penultimate year of school)
| maths. Possibly a few turn up in A2 (final year) but then
| they can't have any good examples from physics because not
| everyone doing maths will be able to depend on knowledge
| about what a capacitor is or how nuclear decay works. But I
| guess population models might work.
|
| - there can be some better stuff in the further maths course
| (e.g. I think they might even have the 'exponentiate a
| matrix' solution to systems of first order linear ODEs)
| PeterisP wrote:
| I recall in my highschool the math and physics (both were
| mandatory) teachers explicitly coordinated so that the
| derivatives and other relations were taught right before
| they got applied in physics. There are much simpler
| examples than capacitors and nuclear decay, you can explain
| all aspects of physics (starting with basic mechanics,
| position/speed/acceleration) simpler if you can rely on
| calculus.
| eigenket wrote:
| I've seen pretty bright seeming UK university applicants able
| to do whatever you ask them but then completely shit the bed
| when you ask them to differentiate e^y with respect to y
| rather than e^x with respect to x.
| Rompect wrote:
| I genuinely don't know whether the trick of that question
| is swapping the `e^x` with `e^y`, so just renaming a
| variable - or is `y` a function?
| ithinkso wrote:
| Even worse, I've seen a lot of people that where convinced
| the derivative of f(x)=e^7 is e^7
| londons_explore wrote:
| That would trick me too... Exam questions never ask
| 'trick' questions like that where the answer is
| zero/infinity/undefined.
| DeathArrow wrote:
| I find this beautiful:
|
| e^ip = -1
| montroser wrote:
| Yeah! Which of course also means that _e_ and p can be defined
| in terms of one another.
| dotancohen wrote:
| That is probably the most insightful thing I've read all
| year. I wonder if there are any subtle implications.
| tsimionescu wrote:
| One interesting thing is that it means it's not impossible
| to think that p+e or pe or p^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 wrote:
| The 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 wrote:
| I 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.
| jstx1 wrote:
| I 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 wrote:
| The fact that e = 2.718... is a fundamental property of the
| exponential function, though. It's not an arbitrary choice.
| ianai wrote:
| Indeed. 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 wrote:
| I 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 wrote:
| Which 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 wrote:
| The 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.
| ianai wrote:
| Idk feels pretty arbitrary to say the Fourier expansion
| of a function matters more than any other expression of
| the function when the whole point of the Fourier
| transformation is precisely its ability to express any
| function in terms of an orthonormal set of functions.
| naasking wrote:
| > the fact that exp(1) = 2.718.. and we call that e is pretty
| insignificant and boring.
|
| The constant itself is still pretty interesting. Using e as a
| base for all numbers yields optimal information density IIRC.
| Binary (base 2) is close to e so it's information density is
| not bad, but this also tells us that trinary (base 3) would be
| even better on this metric since it's closer.
|
| There are lots of interesting properties like this that end up
| linked to e.
| Qem wrote:
| Related: https://web.williams.edu/Mathematics/sjmiller/public
| _html/10...
| hinkley wrote:
| I wonder sometimes if when Dennard scaling finally grinds to
| a halt, some desperate and clever individuals will switch us
| to trinary circuitry, for that last 37% theoretical limit.
| lunchladydoris wrote:
| If 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-...
| vesinisa wrote:
| Title should be edited to lowercase e for Euler's number.
| mromanuk wrote:
| Yes:
|
| Why Euler's number (e), the Transcendental Math Constant, Is
| Just the Best
| [deleted]
| jhncls wrote:
| In a unique Numberphile video featuring Grant Sanderson
| (3blue1brown), this weird number pops up in a game of darts.
|
| [0] https://youtu.be/6_yU9eJ0NxA
| adunk wrote:
| For 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 wrote:
| Are either of the latter transcendental constants
| throwaway81523 wrote:
| Yeah the title almost seemed like clickbait since "Euler's
| constant" usually means g=0.5772... provoking a reaction of
| "where does _that_ show up in optimization? ". That the
| constant turned out to be e=2.718... which shows up all over
| the place was a big disappointment
| lordnacho wrote:
| See 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
|
| This one is:
| https://en.wikipedia.org/wiki/E_(mathematical_constant)
|
| Worth coming up with some better way to talk about this.
| [deleted]
| poizan42 wrote:
| The first one is usually (at least from what I've seen)
| called the Euler-Mascheroni constant which is denoted by g,
| so I don't think there is much confusion.
| ianai wrote:
| Euler 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.
| Qem wrote:
| To any person interested in understanding calculus through
| infinitesimal and infinite numbers, see:
| https://people.math.wisc.edu/~keisler/calc.html
| lordnacho wrote:
| > 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.
|
| My math teacher from high school, who I still keep in touch
| with, sends out a reminder every April 15th.
| adunk wrote:
| You should come to Stockholm, Sweden, during the week at
| the beginning of December when the Nobel prizes are
| awarded. While it is not an entire month of celebrations in
| the name of science, at least it is one full week:
| https://www.nobelprize.org/ceremonies/nobel-week-2021/
| st_goliath wrote:
| > See what happens when you're so smart you get multiple
| things named after you?
|
| Yes, this problem is touched on in the Wikipedia "List of
| Things named after Leonhard Euler" (https://en.wikipedia.org/
| wiki/List_of_things_named_after_Leo...)
|
| I particularly like the remark in the introduction:
|
| > In an effort to avoid naming everything after Euler, some
| discoveries and theorems are attributed to the first person
| to have proved them _after_ Euler.
| jamespwilliams wrote:
| This was probably the result of HN's autocapitalisation of post
| titles. The title of the article itself uses lowercase e. In
| any case, the title has been changed now.
| mensetmanusman wrote:
| I 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 wrote:
| sounds 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
| thomasahle wrote:
| > 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 wrote:
| "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.
| kwhitefoot wrote:
| > hated when my instructors put "important" results that we
| have never seen before on an exam.
|
| You would have hated my 1977 quantum mechanics final; not a
| single question that had been directly covered in the
| course. Really sorted out those who had been paying
| attention from those who thought that memorization was
| enough.
| iratewizard wrote:
| It doesn't seem like many universities do it like that at
| all anymore. MIT's old comp sci curriculum used to be
| great. I've since seen them replace teaching fundamentals
| on lisp with python. My guess is that python is taught so
| that they can stuff their curriculum with buzzwords.
| KMag wrote:
| If I recall correctly, the stated reasoning is that most
| software construction today relies heavily on combining
| libraries, and the batteries-included nature of Python
| allows them to get to this point earlier in the semester.
| Also, using Python makes more of the material directly
| transferable to upper-level machine learning and big data
| subjects.
|
| As far as buzzwords, I think the weight of "MIT" is much
| heavier than any buzzwords that could be attached.
| (Though, I'm biased.)
| tfigment wrote:
| My intro to Physics prof did this. First exam, average was
| like 35%. I was in top 3 at like 70%. He got into trouble
| because he also said no grading on curve and most of class
| complained to his dept head.
| hinkley wrote:
| We had a physics professor who did the same, except he
| did grade on a curve. My 38% was a B-. I don't think we
| had three people above 70%. #1 was an outlier and might
| have hit 70.
|
| The guy I studied with sat behind me, and at one point
| one of us started stress laughing. Then it was two of us
| in the middle of a lecture laughing like our gun just
| jammed while the horror movie cereal killer was almost
| within striking distance.
|
| There were a lot of pissed off people in class for the
| next couple of weeks.
| ianai wrote:
| Oddly 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.
| dtgriscom wrote:
| > Was surprised to have that learning moment in the middle of
| the exam and not prior...
|
| Better than at the end of the exam...
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