Euler’s number pops up in situations that involve optimality

mensetmanusman · 10 days ago

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…

ianai · 10 days ago

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.

annexrichmond · 10 days ago

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 · 10 days 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 · 10 days 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.

kwhitefoot · 9 days ago

> 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.

tfigment · 9 days ago

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.

coupdejarnac · 9 days ago

This is why I have nightmares about undergrad engineering. Professors and TAs lose perspective when they teach the same material repeatedly and think they need to make things interesting. No, your job is to communicate abstract ideas clearly, which is apparently an extremely rare skill.

dtgriscom · 10 days ago

> Was surprised to have that learning moment in the middle of the exam and not prior…

Better than at the end of the exam...

analog31 · 9 days ago

The foxes bring with them e^(i*x)

vmilner · 10 days ago

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...

Aardwolf · 10 days ago

> 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 · 10 days ago

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”.

filmor · 9 days ago

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 :)

Aardwolf · 10 days ago

> 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

SilasX · 10 days ago

"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.

CornCobs · 9 days ago

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)

yellowcake0 · 9 days ago

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 · 10 days ago

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.

londons_explore · 10 days ago

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 · 10 days ago

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 · 10 days ago

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 · 10 days ago

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.

ithinkso · 10 days ago

Even worse, I've seen a lot of people that where convinced the derivative of f(x)=e^7 is e^7

Rompect · 9 days ago

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?

scythe · 10 days ago

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.

mabbo · 9 days ago

I had a wonderful grade 12 calc teacher in high school who taught everything from first principles. I would leave his class feeling like I had gone to the gym from my brain. Despite his incredible teaching, I only pulled off a low 70s grade in the class.

So I retook the course the next year. Taught by a new teacher fresh from teachers college, theoretically with a specialty in math since they were teaching an upper level math course.

I don't think the new teacher even knew how to do derivatives from first principles. Just rote memorization of the different types of differentiation.

I got an A in that class the second time, having learned nothing.

tomrod · 10 days ago

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 · 10 days ago

Yes exactly! Deriving calculus from set axioms truly opened my mind to math, and more generally critical thinking.

vmilner · 8 days ago

[I should add that this was posted in 2012, and differentiation from first principles was apparently emphasised far more in the syllabus in 2017 onwards.]

nick__m · 10 days ago

And 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 ?

adunk · 10 days ago

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 · 10 days ago

Indeed. It shows that the pi formulation is actually somewhat ugly because it lacks symmetry.

Scarblac · 10 days ago

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 · 10 days ago

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 · 9 days ago

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.

f00zz · 10 days ago

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 · 9 days ago

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 · 9 days ago

Also an amazing way are Taylor polynomials, this article explains the process of thought really well:

https://betterexplained.com/articles/taylor-series/

mkl · 10 days ago

Also π and the three most important operations: addition, multiplication, exponentiation.

323 · 10 days 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

wrycoder · 10 days ago

Which only slightly obfuscates the fact that e^(iπ) = -1. Bamboozles the rubes!

323 · 10 days ago

e^(iπ) = -1 is basically the unit circle in the complex plane:

https://en.wikipedia.org/wiki/Circle_group

jhncls · 10 days ago

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

lunchladydoris · 10 days ago

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-...

_Microft · 10 days ago

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 · 10 days ago

:D

Mathematician. Thought about a new (at least to me) transcendental number ...

westcort · 9 days ago

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 · 9 days ago

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.

woopwoop · 10 days ago

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 · 9 days ago

Unfortunately, it is trivial... A joke. The constants there could amount to almost anything.

jstx1 · 10 days ago

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.

naasking · 10 days ago

> 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.

hinkley · 9 days ago

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.

Denvercoder9 · 10 days ago

The fact that e = 2.718... is a fundamental property of the exponential function, though. It's not an arbitrary choice.

ianai · 10 days ago

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.

abnry · 10 days ago

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.

jstx1 · 10 days ago

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.

neantherpi · 9 days ago

Sincerely, you completely changed the way I look at e. Interesting that ln(x) does follow such a notational style; to change the base one can either divide by ln(b) or use a separate notation (log vs ln). I also had put a lot of weight into e being transcendental but it seems like as long as x is rational the value will also be transcendental, so not that special (if someone could confirm).

agumonkey · 10 days ago

Isn't there another fix point like value for hyperoperations ?

quantum_state · 10 days ago

That's why it is physicists' best friend :-).