This is a frustrating article because it never explains why e is the natural logarithm base. To me, the easiest way to understand it is via continuous compound interest:
* If you invest $1 at 100% interest for 1 year, you get $2 at the end
* Compounded 2 times in a year, you get 100/2 = 50% interest every 1/2 year, which amounts to $2.25
* Compounded 4 times in a year, you get 100/4 = 25% interest every 1/4 year, which amounts to $2.44
* Compounded n times in a year, you get 100/n percent interest every 1/n year, which amounts to (1+1/n)^n dollars
* So continuous compound interest is the limit as n approaches infinity, which amounts to $2.71828 at the end of the year
(This is a great problem to give to pre-calc students to see if they can figure out the calculation for themselves.)
We did that in 11th grade pre-calculus class. We got the number 2.71828 and then our teacher said "Now go to that button on your graphing calculator that says e^x and type e^1" We did and our minds were blown. Why was this strange number that we "discovered" in our calculator already?
A few months later I talked to a super advanced math genius kid who had a signature that said e^(i*pi)+1=0 and I asked him if that was Euler's number. He was a super quiet skittish guy that rarely talked. His eyes lit up and he spent the next 2 hours teaching me about Taylor series and showed me how to prove it.
It remains the most fascinating math equation I have ever seen.
He had a lot of issues and dropped out because he couldn't pass a history class. I found that guy on Facebook 20 years later and thanked him. He didn't remember that but he was so happy that he made such a big impact on me.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it.
e arises when you ask the question: is there a function that is its own derivative? And it turns out the answer is yes. It is this infinite series:
1 + x + x^2/2 + x^3/6 + ... x^n/n! ...
which you can easily verify is its own derivative simply by differentiating it term-by-term. When you evaluate this function at x=1, the result is e. In general, when you evaluate this function at any real value of x, the result is e^x.
But the Euler equation e^iπ = -1 has nothing to do with exponentiating e, it's just a notational convention that is defined to be the series above. When you evaluate that series at x=iπ, the result is -1. In general, the value of the series for any x=iy is cos(y) + i*sin(y).
I've seen (1+1/n)^n before, but never seen an explanation of why I might ever want to use something of that form. I've used the e^ix notation extensively, but again I've never really cared, because to me it was just a compact representation of sin and cos together. Likewise, all the proofs in that article are still a bit "so at this point on this carefully chosen graph, the gradient is e" and I think "Who cares? You carefully chose the graph to prove a point, I'll never see it in the real world."
And then the example with compounding interest - immediately I can see the application. It's definitely a good way of explaining it, although maybe it'd be even more grounded if it had n=12 and n=365 as examples. When you notice that the actual values seem to be converging, then you can try plugging in ever bigger and bigger numbers. This way you can discover the value of e for yourself and that process of discovery leads to a better understanding than rote learning of an abstract thing you haven't mentally visualised yet. All the other explanations are useful later, and they allow you to see it in different situations, but having at least one "this is why it's tangibly useful" hook at the start is definitely a massive help in understanding something.
> e is the base rate of growth shared by all continually growing processes.
I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart.
I think the idea of finding a function that is its own derivative, discovering that there is a polynomial series that meets that requirement, and evaluating that series at x=1 is a much more natural explanation.
You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.
> But the Euler equation e^iπ = -1 has nothing to do with exponentiating e, it's just a notational convention that is defined to be the series above.
Can't the same thing be said about using fractions on the exponent? Exponentiation is actually just repeated multiplication (a^n=a*a*...*a, repeated n times), but you can't do that when n is a fraction or irrational anymore than you can do it when it's imaginary.
We have to define what it means for an exponent to be non-integer: for fractions we might define a^(b/c) as the root of the equation x^c=a^b, and to allow irrationals I think you need some real analysis (it's been a while, but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
There's a very intuitive way to "see" that e^ix=cos(x)+i*sin(x): all you have to do is to treat complex numbers like you would any other number, and "believe" the derivative rule for complex numbers (so (e^(ix))'=ie^(ix)). Then you can just graph f(x)=e^(ix) for real x by starting at x=0 (when clearly f(x)=1) and from there take small steps in the x axis and use the derivative to find the value of the next step with the usual formula f(x+dx)=f(x)+f'(x)*dx.
Doing that you realize the image of e^(ix) just traces a circle in the complex plane because every small step in the x direction makes e^(ix0) walk a small step perpendicular to the line going from 0 to e^(ix0), simply because multiplying by i means rotating 90 degrees.
> to allow irrationals I think you need some real analysis (it's been a while, but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
a^b, for positive a and irrational b can also be defined as lim (x -> b, x € Q) a^x - which is possible because Q is dense in R. This is a pretty natural way of extending a function to the reals.
The way we extend exponentiation to complex exponents is IMHO much less straightforward.
Ah, a vaguely wondered if it had something involving Cauchy sequences, I think that was that limit!
> The way we extend exponentiation to complex exponents is IMHO much less straightforward.
I think it depends on how much you're used to dealing with complex numbers. In college, I was always taught to prove that Euler formula by replacing ix for x in that series, and then noting that the alternating signs and presence/absence of i in the terms allowed you to separate it into two series for cosine and sine. That always felt awkward, like there was no way anyone could just come up with that naturally.
Many years later I found that construction with graphing e^(ix) by taking small steps using f(x+dx)=f(x)+f'(x)*dx, and everything clicked: how exponentials work in the real axis is pretty different from the imaginary axis, but both are completely intuitive and unavoidable once you understand that.
Even later I "discovered" the connection with group theory[1]; that one still blows my mind.
"Exponentiation is actually just repeated multiplication (a^n=a*a*...*a, repeated n times), but you can't do that when n is a fraction or irrational anymore than you can do it when it's imaginary."
That depends on your point of view. You can also view exponentiation as "really" being e's infinite series, and it so happens that that matches what you get in the case of repeated multiplication. The advantage of that is now you can start exponentiating a lot more than just numbers. Here's 3blue1brown on raising e to the power of a matrix: https://www.youtube.com/watch?v=O85OWBJ2ayo
In general there is fruit in viewing infinite series as the fundamental building block of a lot of math and non-infinite series as the special case. I won't claim which is "real" or "correct", though, just point out that there is value in viewing "repeated multiplication" as the special case rather than the "real" thing. Of course you can always view exponentiation as the generalization too.
> Can't the same thing be said about using fractions on the exponent?
Yes, it can. In fact, it is really useful to think of the "usual" definition of exponentiation as repeated multiplication as nothing more than a special case of a much more general concept, which is evaluating a function whose defining property is that it is its own derivative.
Sure you can. You know what 2^n is and you want that 2^(1/3) 2^(1/3) 2^(1/3)= 2^1=1. That uniquely defines the exponential function on the rationals. For the real numbers you need some amount of continuity or measurability, but then it is also uniquely determined.
> but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
No, you don't just "say" that. You prove it. Big difference.
> No, you don't just "say" that. You prove it. Big difference.
Whether a^b=exp(b*log(a)) is a definition or a proof really depends on how exactly you define certain terms (e.g. exp). What's certainly a theorem that requires a proof is that the definition of a^b (for irrational b) via limits of rational exponents and the one via exp are equivalent.
The exponential function arises from multiple sources:
- the solution of the ode you just stated.
- compound interest.
- The defining property of exponential functions is f(x+y)=f(x)f(y) with some normalization.
- Moving on the unit circle is given by an exponential function because rotation is a group, i.e. a^(i(x+y)). Now choose the basis a such that you move with unit speed.
- ...
The nice thing is that all of these very different motivations lead to the same thing.
The "has nothing to do with exponentiating e" I would strongly disagree with. It has everything to do with exponentiating and is exactly the only way exponentiation can work. So afterwards you can pretend you didn't know that and define exponentiation by using e. Same for matrix exponentials, semigroups etc.
> The "has nothing to do with exponentiating e" I would strongly disagree with.
That was a bad way of phrasing it. I should have said something more along the lines of "... has nothing to do with multiplying e by itself iπ times. Multiplying something by itself n times is actually a special case of the more general concept of finding a function that is its own derivative."
> But the Euler equation e^iπ = -1 has nothing to do with exponentiating e
It has, that's the beauty of it. You can define as usual the function x -> e^x on the real line. Now, complex analysis tells us that if this function can be extended to a holomorphic function on the whole complex plane, then the extension is unique. And, in fact, this function does admit such an extension, so you can compute e^z for any complex number z, and in this way one gets e^iπ = -1.
> which you can easily verify is its own derivative simply by differentiating it term-by-term.
Yes, but it's not a priori clear that the series actually even converges for all x, and the fact that power series are differentiable term-by-term within their radius of convergence also requires proof.
Fair enough, but IIRC students are first taught about converging and non converging series in pre calc. So then you get to calc, and work on the e proof, but first you say, "if you paid attention last year, you'll remember this series converges."
Mathematics education seems to be really different depending on the country (I'm in Europe). I barely saw a proof (except maybe some geometry proofs) in any of my high school maths classes and had to learn what a mathematical proof is in university. But I think this also depends on what focus area / profile you have in high school (mine was languages).
For me these concept seem intuitively linked, simply because derivation measures the rate of change, and that compounding definition aims to find the compounding function with constant growth. The factorial definition only seems to make for an easier proof of the derivation of e^x.
That's kind of the point. The only background knowledge you need to understand my definition is how to differentiate a polynomial. Anyone capable of understanding what it even means to find a function that is its own derivative is almost certainly going to know that.
Another way to explain it, which I think is a "best" explanation in its own right, is that if you start differentiating exponential functions you can't help but discover e. The general differentiation of b^x, for b a constant and x the variable, is ln(x)*b^x. And of course to understand ln you need e.
Or, to put it another way, you will blunder into this somewhere around month two of calculus 1, unavoidably.
Of course, that doesn't show how it will show up in all sorts of other places; "the one and only function that is its own derivative" strikes me as more likely to be something we encounter everywhere.
So d(b^x) is itself multiplied by lim(∂->0):(b^∂-1)/∂. But now what? How do you evaluate that limit? How do you show that e is the magic value of b that makes that limit turn out to be 1? And in particular, how do you show that to someone whose only background knowledge is how to differentiate polynomials?
IMHO it's a lot easier to see that e is the value to which the polynomial series that is its own derivative converges at x=1.
Nope. Thanks for the pointer. And thanks for pointing out CIS. I've been coding in CL since before it was standardized and I had never run across that before. But yes, one of the things I love about CL is that I can do (expt (sqrt -1) (sqrt -1)) and get the right answer.
Yes; it is all literally exponentiation which we can approximate with concrete floating-point numbers that know nothing about the formula we are exploring.
Yes, that is true, of course. But consulting a Lisp REPL merely demonstrates that it is true. It does not explain why it is true.
You'd get further, pedagogically speaking, by pointing out that (e^i)^i = e^(i*i) = e^(-1) = 1/e, and so e^i is a number that is in some sense "half way" between e and 1/e when you are exponentiating. As an analogy, consider:
e^(1/2) * e^(1/2) = e^(1/2 + 1/2) = e^1 = e
as a demonstration that e^(1/2) is a number that is in some sense "half-way" between 1 and e when you are multiplying, a.k.a. the square root of e. But that still leaves unanswered the question of what "half-way" means when exponentiating rather than multiplying.
Yes. Multiplying any function by a constant changes its derivative by the same constant, so for any function f, if df = f then d(Cf) = Cf. So yes, you can multiply by C without losing the interesting properties of the derivative (whatever those might be), but you can do that to any function, and so the fact that you can do it to this function is true but not particularly interesting.
This is why I find e so fascinating, frustrating and puzzling.
The other constants fundamental to science, like the gravitational constant or the speed of light, can only be measured, not discovered from nothing. We aren't even sure how constant they actually are, there might be extremely tiny variations in either time or space that our instruments just can't measure yet. In theory, other universes could exist where these constants are "set" a little bit differently; whether we could live in such universes is another matter entirely.
E, on the other hand, comes from pure mathematics. As long as fractions, addition and exponentiation work the same way in another hypothetical universe, this strange E number is going to have the same strange value.
e isn't really a fundamental physical constant and doesn't show up very much in famous scientific formulas. It's a number that has very convenient properties for performing calculus operations and shows up in that context. pi isn't really an important physical constant, either, but it shows up a lot because of two things --- it's useful for calculations involving cycles, and because things being spherical or circular is a useful simplifying assumption (and also because things particularly in astronomy end up being spherical or circular (or nearly) quite often in reality).
Neither of those things are sort of empirically measured, and in any formula where they show up, you could theoretically absorb them into other constants -- and in fact the Einstein gravitational constant does exactly that -- it's defined as (8*pi*G)/c^4, absorbing pi into newton's gravitational constant (for historical reasons, when using plank units, they set G and c to 1, so it ends up just being 8pi -- _reduced_ planck units set the whole constant to 1). It's just frequently easier to separate out e and pi for the purposes of actually working out the math.
That's the same with pi if you think about it, and in my eyes is what makes it a transcendental number, so to speak. The numbers themselves could be different if you used a different base instead of base 10, but they still just represent a particularly useful value for us.
One of my favorites is https://en.wikipedia.org/wiki/Feigenbaum_constants . Pops up in many situations, and as I understand it, it is not even well-understood why it pops up in as many places as it does. Intuitively one would expect that many of the places it pops up ought to have their own local constant of some sort, but instead this one keeps popping up. 7-year-old Numberphile video on it: https://www.youtube.com/watch?v=ETrYE4MdoLQ
One of my favorite parts if the excellent book Visual Complex Analysis is where he explains that e is defined in terms of the derivative of the exponential function. E is e because that’s what it has to be for the exponential function to be its own derivative.
"A base of 2 is useful because there are several small positive integers whose base-two logarithms are also integers." What? No! Base 2 is natural in exactly the same way that base e is natural, except for discrete domains instead of continuous domains. There is a unique family of functions for which the rate of change of the function is equal to the current value of the function everywhere. On discrete domains it's some scaled translation of 2^x, and on continuous domains it's some scaled translation of e^x. "Some scaled translation" here is accounting for the fact that the function is only uniquely exactly 2^x or e^x if we also add the constraint that f(0)=1.
C is a plane with multiplication. I would say, the fact that there is even a way of making this work is surprising (it doesn't work for any R^n, n > 2, at least not if you want a field). The particular way how we multiply complex numbers is also IMHO not self-evident (neither in the algebraic, nor in the geometric interpretation). It just turns out that doing it this way gives us really nice properties.
> how we multiply complex numbers is also IMHO not self-evident
Multiplication of complex number in the x + iy form treats them exactly like any other binomial factors, using the "FOIL" rule:
F O I L
(a + bi)(c + di) = ac + adi + bci + bdi^2
= ac + (ad + bc)i - bd
= ac - bd + (ad + bc)i
I don't know about self-evident; you pretty much have to do it that way. If you hand (a + bi)(c + di) to someone who knows basic algebra, but has no idea what i is, they will come up with the first FOIL expression above; after that, we have to know that i^2 is -1.
> nor in the geometric interpretation.
That isn't self-evident, but when you gemoetrically work out what the multiplication is doing, it's just "add the arguments (angles), multiply the moduli (distances from origin)".
You have to do it that way because it's the only way to get a field out of R^2, but I don't think it's at all self-evident. The "we have to know that i^2 is -1" is doing some heavy lifting in your explanation. In the context of "complex numbers are just a 2d plane" there's no a priory reason why you'd think that (0,1)*(0,1) has any particular meaning, let alone that it's equal to (-1,0).
If we interpret vectors in R^n as dislocation (or as forces, or...), then vector addition behaves exactly as one would expect. I think with enough motivation a bright school kid could come up with the formula for vector addition. And it works for every dimension.
Vector multiplication only works for 2d (ignoring the less well behaved cross product that also only works in certain dimensions) and I don't think you'd come up with it naturally.
Also, historically, the 2d plane isn't how complex numbers were discovered. They first appeared as weird quantities that made solving (real) cubic equations simpler but were regarded as just temporary artifacts. The realisation that they can be identified with R^2 came much later.
1+x is the tangent line at x=0. Since the graph of an exponential lies above its tangent line, 1+x <= e^x is another way of saying that the derivative of e^x at x=0 is 1, ie. e^x is its own derivative.
Nonsense, this equation doesn't even make any sense without a well defined notion of the exponential function, and then a well defined extension of said function into the complex numbers. You will already have e by the time you reach this equation because defining what exp(z) even means requires you already know the properties of e^x over the reals you wish to preserve. And Euler's formula comes from finding such a function and then defining it to be exp(z). Multiplication in the complex plane by a unit vector is a rotation. Exponentials "repeated multiplication" by such a vector is spinning. And it turns out spinning at a constant rate satisfies the properties of the exponential function so it makes sense to say that's what exp(ix) means.
This is perhaps the most unnatural equation (well identity) in maths. It doesn't fall out anywhere, you would never write it down and solve for e, it's a special case of a more general result you would get first, and it's symbol soup for precisely the reason that the identity itself confers no understanding.
exp/log are natural because you almost can't help but discover them as they appear in so many different seemingly unrelated places.
> And Euler's formula comes from finding such a function and then defining it to be exp(z).
That's certainly one way, but you can also define exp via its power series (which is easily proven to be convergent everywhere). Then, all the properties of exp, as well as Euler's formula, are actual theorems, not just definitions.
It doesn’t matter which definition you pick. I e.g. had it defined axiomatically when I studied, i.e. we were given a list of properties which identify exp unambiguously and then we were given a proof of its existence. The fact that those properties were part of the definition doesn’t take away from their profoundness. The function could still just not exist. Same thing with defining by formulas. The map is not the territory.
Not really. Your equation has no unknowns, but if you consider the "e" in your equation as the unknown, and solve it (here I use "a" instead as the unknown, and use "e" in its usual meaning):
We want a^(i pi) + 1 = 0. Now,
a^(i pi) = e^(ln(a) i pi) = e^(i ln(a) pi) = cos( ln(a) pi) + i sin( ln(a) pi),
so we want cos( ln(a) pi) = -1, sin( ln(a) pi) = 0,
so ln(a) = 1, 3, 5, so a = e, e^3, e^5, ...
Thus indeed e^(i pi) + 1 = 0.
But also ln(a) = -1, -3, -5 work, so for example for
a = 1/e = 0.3678794412... > 0, we have
a^(i pi) + 1 = 0;
and of course for a = e^-99 = 1.0112214926104485... × 10^-43 etc.
We measure angles in radians because it's an easy way to measure the length of the circle sector in the units of the circle's radius. It feels neat in many cases.
A more practical way to measure angles would be in rotations. 0° = 0, 360° = 1, 90° = 0.25, etc. It would remove a ton of 2π and 4π² factors from a lot of equations in physics.
For trigonometry/calculus/physics radians are by far the most practical because they are dimensionless, so no constants appear when differentiating or integrating. (By the way, these constants will involve factors of pi anyway, it's inherent.)
For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
I don't see how Taylor series specifically would be affected. Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series.
Fourier transform would have 4π² instead of 2π under the exponent, no big deal.
The Euler's formula gets a factor of 2π under the exponent though. Given its wide application, it adds plenty of noise, of course.
> Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series.
Implicit in that statement is the use of the series definition of sine, which is a "meaningful" or "natural" definition insofar as it represents some function we care about. I'll address at the end what happens if we assume that definition regardless of its independent plausibility, but first consider:
The semantic meaning of sine, at least from its historical roots and how you might independently uncover it from earlier fields like geometry instead of later fields like differential equations, is that given an angle (in some units, we'll touch on that in a moment) we'd like to know the ratio two sides of a particular triangle associated with that angle inscribed in a circle. Given a choice of units for the angle, the triangle is fixed, and so the result (that ratio of side lengths) is also fixed.
Suppose you want to know how that ratio varies with respect to the angle. You can imagine a change of coordinates `y = cx` and consider the derivative of `sin(x)` vs `sin(y/c)`. The latter will have a numeric value `1/c` times less than the former. E.g., imagine a whole circle represented `1` angle instead of `2pi`. Then converting from our normal radians baseline to that new unit you have `y = (1/2pi)x`, and the derivative of the semantic ratio we're considering with respect to the new measure of angle is multiplicatively `2pi` greater than the original.
Going back to your series definition, suppose we pick that series as the definition of sine, independent of units. The problem that arises is that particular uses of sine do have units, and converting from the problem you care about to your particular from-on-high chosen definition of sine will run into the exact sort of problem our `y = cx` paragraph above touched on. The derivative with respect to the quantity of interest still has an extra `1/c` factor, and the fact that our God-blessed choice of sine is independent of units didn't actually solve anything in the composite problem.
If you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc.
I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
Without delving too far into the philosophy of math as concerns existence vs convention...
e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale.
It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
But we have an inconsistency because pi relates the diameter to the circumference, not the radius to the circumference, which leaves an annoying 2 in the 2pi radians in a circle.
Other chosen conventions are less great, like not having the circle constant be 6.283185... so that we don't need to deal with factors or divisors of 2 all the time when using radians or almost anything else involving pi
90 degrees is one fourth of a circle, it would be so much more intuitive if we'd use "1/4th of something" rather than "1/2 radians" to express this
e^z, defined as the series \sum_{n = 0}^\infty z^n/n!, can only be a function of a dimensionless number z.
sin(z) and cos(z), defined as power series, technically also work this way. And that's OK, because angles are dimensionless: a radian is just C/(2πr), where C is the circumference of a circle of radius r. But it is sometimes convenient to pick your favorite number of radians, like π/180 of them, and call that a degree, and then to say that sin(x degrees) is the same as sin(xπ/180 radians).
With this convention, where the left-hand side of e^(ix) = sin(x) + icos(x) is a function of a dimensionless variable, and the right-hand side can be viewed as a function of a dimensioned argument only in the sense written above, it really is the case that the equation written is true, but the equation e^(ix) = sin(x degrees) + icos(x degrees) is false.
(On the other hand, you could make the case that e^(ix) is really a function of an angle, where its value is the complex number that lies on the unit circle at that angle. Then you do recover a "dimensioned" version of e^(ix) = sin(x) + i*cos(x) that's valid even if you measure angles in degrees.)
Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle:
|[cos(x+δ) + i sin(x+δ)] - [cos(x) + i sin(x)]| = δ
This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians.
In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step.
I thought it was about finding conventions that are useful, surprising, or pleasing in some way. Or, given a set of conventions, finding new ways that they are useful, surprising, or pleasing.
So I'm curious to know about this other... non-conventional point.
This is a frustrating article because it never explains why e is the natural logarithm base. To me, the easiest way to understand it is via continuous compound interest:
* If you invest $1 at 100% interest for 1 year, you get $2 at the end
* Compounded 2 times in a year, you get 100/2 = 50% interest every 1/2 year, which amounts to $2.25
* Compounded 4 times in a year, you get 100/4 = 25% interest every 1/4 year, which amounts to $2.44
* Compounded n times in a year, you get 100/n percent interest every 1/n year, which amounts to (1+1/n)^n dollars
* So continuous compound interest is the limit as n approaches infinity, which amounts to $2.71828 at the end of the year
(This is a great problem to give to pre-calc students to see if they can figure out the calculation for themselves.)
We did that in 11th grade pre-calculus class. We got the number 2.71828 and then our teacher said "Now go to that button on your graphing calculator that says e^x and type e^1" We did and our minds were blown. Why was this strange number that we "discovered" in our calculator already?
A few months later I talked to a super advanced math genius kid who had a signature that said e^(i*pi)+1=0 and I asked him if that was Euler's number. He was a super quiet skittish guy that rarely talked. His eyes lit up and he spent the next 2 hours teaching me about Taylor series and showed me how to prove it.
It remains the most fascinating math equation I have ever seen.
He had a lot of issues and dropped out because he couldn't pass a history class. I found that guy on Facebook 20 years later and thanked him. He didn't remember that but he was so happy that he made such a big impact on me.
Exactly the same for me and I'm pretty sure this is how Jakob Bernoulli came to define the number as well, trying to see what the upper bound for infinitesimal compounding was.
That may be how it was arrived at historically, but it is not the best way to explain it.
e arises when you ask the question: is there a function that is its own derivative? And it turns out the answer is yes. It is this infinite series:
1 + x + x^2/2 + x^3/6 + ... x^n/n! ...
which you can easily verify is its own derivative simply by differentiating it term-by-term. When you evaluate this function at x=1, the result is e. In general, when you evaluate this function at any real value of x, the result is e^x.
But the Euler equation e^iπ = -1 has nothing to do with exponentiating e, it's just a notational convention that is defined to be the series above. When you evaluate that series at x=iπ, the result is -1. In general, the value of the series for any x=iy is cos(y) + i*sin(y).
It's that simple.
It's a pretty good way of explaining it.
I've seen (1+1/n)^n before, but never seen an explanation of why I might ever want to use something of that form. I've used the e^ix notation extensively, but again I've never really cared, because to me it was just a compact representation of sin and cos together. Likewise, all the proofs in that article are still a bit "so at this point on this carefully chosen graph, the gradient is e" and I think "Who cares? You carefully chose the graph to prove a point, I'll never see it in the real world."
And then the example with compounding interest - immediately I can see the application. It's definitely a good way of explaining it, although maybe it'd be even more grounded if it had n=12 and n=365 as examples. When you notice that the actual values seem to be converging, then you can try plugging in ever bigger and bigger numbers. This way you can discover the value of e for yourself and that process of discovery leads to a better understanding than rote learning of an abstract thing you haven't mentally visualised yet. All the other explanations are useful later, and they allow you to see it in different situations, but having at least one "this is why it's tangibly useful" hook at the start is definitely a massive help in understanding something.
See the great article on e from Better Explained:
https://betterexplained.com/articles/an-intuitive-guide-to-e...
> e is the base rate of growth shared by all continually growing processes.
I don't think that's a very good explanation because all the heavy lifting is being done by the phrases "base rate" and "continually growing", nether of which are well defined. "Continually growing" could reasonably be interpreted to mean "monotonically increasing" in which case f(x)=x qualifies and the whole explanation falls apart.
I think the idea of finding a function that is its own derivative, discovering that there is a polynomial series that meets that requirement, and evaluating that series at x=1 is a much more natural explanation.
You don't need or even want well defined terms when you're trying to explain why a mathematical formula is useful or naturally applicable to real world situations.
Maybe you don't want well-defined terms in your pedagogy, but I do.
> But the Euler equation e^iπ = -1 has nothing to do with exponentiating e, it's just a notational convention that is defined to be the series above.
Can't the same thing be said about using fractions on the exponent? Exponentiation is actually just repeated multiplication (a^n=a*a*...*a, repeated n times), but you can't do that when n is a fraction or irrational anymore than you can do it when it's imaginary.
We have to define what it means for an exponent to be non-integer: for fractions we might define a^(b/c) as the root of the equation x^c=a^b, and to allow irrationals I think you need some real analysis (it's been a while, but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
There's a very intuitive way to "see" that e^ix=cos(x)+i*sin(x): all you have to do is to treat complex numbers like you would any other number, and "believe" the derivative rule for complex numbers (so (e^(ix))'=ie^(ix)). Then you can just graph f(x)=e^(ix) for real x by starting at x=0 (when clearly f(x)=1) and from there take small steps in the x axis and use the derivative to find the value of the next step with the usual formula f(x+dx)=f(x)+f'(x)*dx.
Doing that you realize the image of e^(ix) just traces a circle in the complex plane because every small step in the x direction makes e^(ix0) walk a small step perpendicular to the line going from 0 to e^(ix0), simply because multiplying by i means rotating 90 degrees.
> to allow irrationals I think you need some real analysis (it's been a while, but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
a^b, for positive a and irrational b can also be defined as lim (x -> b, x € Q) a^x - which is possible because Q is dense in R. This is a pretty natural way of extending a function to the reals.
The way we extend exponentiation to complex exponents is IMHO much less straightforward.
Ah, a vaguely wondered if it had something involving Cauchy sequences, I think that was that limit!
> The way we extend exponentiation to complex exponents is IMHO much less straightforward.
I think it depends on how much you're used to dealing with complex numbers. In college, I was always taught to prove that Euler formula by replacing ix for x in that series, and then noting that the alternating signs and presence/absence of i in the terms allowed you to separate it into two series for cosine and sine. That always felt awkward, like there was no way anyone could just come up with that naturally.
Many years later I found that construction with graphing e^(ix) by taking small steps using f(x+dx)=f(x)+f'(x)*dx, and everything clicked: how exponentials work in the real axis is pretty different from the imaginary axis, but both are completely intuitive and unavoidable once you understand that.
Even later I "discovered" the connection with group theory[1]; that one still blows my mind.
[1] This is a really nice explanation: https://www.youtube.com/watch?v=mvmuCPvRoWQ
"Exponentiation is actually just repeated multiplication (a^n=a*a*...*a, repeated n times), but you can't do that when n is a fraction or irrational anymore than you can do it when it's imaginary."
That depends on your point of view. You can also view exponentiation as "really" being e's infinite series, and it so happens that that matches what you get in the case of repeated multiplication. The advantage of that is now you can start exponentiating a lot more than just numbers. Here's 3blue1brown on raising e to the power of a matrix: https://www.youtube.com/watch?v=O85OWBJ2ayo
In general there is fruit in viewing infinite series as the fundamental building block of a lot of math and non-infinite series as the special case. I won't claim which is "real" or "correct", though, just point out that there is value in viewing "repeated multiplication" as the special case rather than the "real" thing. Of course you can always view exponentiation as the generalization too.
> Can't the same thing be said about using fractions on the exponent?
Yes, it can. In fact, it is really useful to think of the "usual" definition of exponentiation as repeated multiplication as nothing more than a special case of a much more general concept, which is evaluating a function whose defining property is that it is its own derivative.
> but you can't do that when n is a fraction
Sure you can. You know what 2^n is and you want that 2^(1/3) 2^(1/3) 2^(1/3)= 2^1=1. That uniquely defines the exponential function on the rationals. For the real numbers you need some amount of continuity or measurability, but then it is also uniquely determined.
> but I think the usual way is to first define exp and log, and then say that a^b=exp(b*log(a)), which is kind of cheating because we have to define exp first!).
No, you don't just "say" that. You prove it. Big difference.
> No, you don't just "say" that. You prove it. Big difference.
Whether a^b=exp(b*log(a)) is a definition or a proof really depends on how exactly you define certain terms (e.g. exp). What's certainly a theorem that requires a proof is that the definition of a^b (for irrational b) via limits of rational exponents and the one via exp are equivalent.
The exponential function arises from multiple sources:
- the solution of the ode you just stated.
- compound interest.
- The defining property of exponential functions is f(x+y)=f(x)f(y) with some normalization.
- Moving on the unit circle is given by an exponential function because rotation is a group, i.e. a^(i(x+y)). Now choose the basis a such that you move with unit speed.
- ...
The nice thing is that all of these very different motivations lead to the same thing.
The "has nothing to do with exponentiating e" I would strongly disagree with. It has everything to do with exponentiating and is exactly the only way exponentiation can work. So afterwards you can pretend you didn't know that and define exponentiation by using e. Same for matrix exponentials, semigroups etc.
> The "has nothing to do with exponentiating e" I would strongly disagree with.
That was a bad way of phrasing it. I should have said something more along the lines of "... has nothing to do with multiplying e by itself iπ times. Multiplying something by itself n times is actually a special case of the more general concept of finding a function that is its own derivative."
> But the Euler equation e^iπ = -1 has nothing to do with exponentiating e
It has, that's the beauty of it. You can define as usual the function x -> e^x on the real line. Now, complex analysis tells us that if this function can be extended to a holomorphic function on the whole complex plane, then the extension is unique. And, in fact, this function does admit such an extension, so you can compute e^z for any complex number z, and in this way one gets e^iπ = -1.
> which you can easily verify is its own derivative simply by differentiating it term-by-term.
Yes, but it's not a priori clear that the series actually even converges for all x, and the fact that power series are differentiable term-by-term within their radius of convergence also requires proof.
Fair enough, but IIRC students are first taught about converging and non converging series in pre calc. So then you get to calc, and work on the e proof, but first you say, "if you paid attention last year, you'll remember this series converges."
Mathematics education seems to be really different depending on the country (I'm in Europe). I barely saw a proof (except maybe some geometry proofs) in any of my high school maths classes and had to learn what a mathematical proof is in university. But I think this also depends on what focus area / profile you have in high school (mine was languages).
For me these concept seem intuitively linked, simply because derivation measures the rate of change, and that compounding definition aims to find the compounding function with constant growth. The factorial definition only seems to make for an easier proof of the derivation of e^x.
That's kind of the point. The only background knowledge you need to understand my definition is how to differentiate a polynomial. Anyone capable of understanding what it even means to find a function that is its own derivative is almost certainly going to know that.
a power series is not a polynomial though.
That's splitting a pretty fine technical hair for pedagogy at this level.
Another way to explain it, which I think is a "best" explanation in its own right, is that if you start differentiating exponential functions you can't help but discover e. The general differentiation of b^x, for b a constant and x the variable, is ln(x)*b^x. And of course to understand ln you need e.
Or, to put it another way, you will blunder into this somewhere around month two of calculus 1, unavoidably.
Of course, that doesn't show how it will show up in all sorts of other places; "the one and only function that is its own derivative" strikes me as more likely to be something we encounter everywhere.
> The general differentiation of b^x, for b a constant and x the variable, is ln(x)*b^x.
That's true, but it's begging the question because you can't define ln without already knowing about e.
You have to go back to first principles:
d(b^x) := lim(∂->0):(b^(x+∂) - b^x)/∂ = ((b^x)(b^∂) - b^x)/∂ = (b^x)(b^∂-1)/∂
So d(b^x) is itself multiplied by lim(∂->0):(b^∂-1)/∂. But now what? How do you evaluate that limit? How do you show that e is the magic value of b that makes that limit turn out to be 1? And in particular, how do you show that to someone whose only background knowledge is how to differentiate polynomials?
IMHO it's a lot easier to see that e is the value to which the polynomial series that is its own derivative converges at x=1.
b^x = (e^ln(b))^x = e^( ln(b) * x ),
and its derivative then is (by the chain rule)
ln(b) exp( ln(b) * x ) = ln(b) * b^x.
Have you happened to have read Visual Complex Analysis?
If not I think you might like it.
And a tangent from your username: I quite like how complex numbers and functions like CIS made it into Common Lisp.
Nope. Thanks for the pointer. And thanks for pointing out CIS. I've been coding in CL since before it was standardized and I had never run across that before. But yes, one of the things I love about CL is that I can do (expt (sqrt -1) (sqrt -1)) and get the right answer.
> it's just a notational convention
I'd argue it's a bit more, it's a natural way to extend domain of exponentiation, as we do before this with naturals -> integers -> reals.
Sorry but as a layman high-school educated guy, GP's explanation was a lot easier to follow with it having a "natural" real-life example
Yes, it does have to do with exponentiating e, though.
e^(iπ) is (e^i)^π.
There is a concrete complex number e^i:
See, it's around 0.54 + 0.84i. It's on the unit circle.When you raise this number to pi, you get -1.
This means it's the pi-th root of -1; let's try it: and e must be the i-th root of this: Yes; it is all literally exponentiation which we can approximate with concrete floating-point numbers that know nothing about the formula we are exploring.> There is a concrete complex number e^i:
Yes, that is true, of course. But consulting a Lisp REPL merely demonstrates that it is true. It does not explain why it is true.
You'd get further, pedagogically speaking, by pointing out that (e^i)^i = e^(i*i) = e^(-1) = 1/e, and so e^i is a number that is in some sense "half way" between e and 1/e when you are exponentiating. As an analogy, consider:
e^(1/2) * e^(1/2) = e^(1/2 + 1/2) = e^1 = e
as a demonstration that e^(1/2) is a number that is in some sense "half-way" between 1 and e when you are multiplying, a.k.a. the square root of e. But that still leaves unanswered the question of what "half-way" means when exponentiating rather than multiplying.
Obtaining an intuition for what it means to use an imaginary exponent takes work.
Exponents give us expoential decay/growth in the real number line, but periodicity in the imaginary domain, which is strange.
In physics, this lets us analyze decaying or amplifying oscillations in a unified way. (Laplace transform and all that.)
Note that C + Cx + ... would have also worked. So why the 1? Just simplicity?
Yes. Multiplying any function by a constant changes its derivative by the same constant, so for any function f, if df = f then d(Cf) = Cf. So yes, you can multiply by C without losing the interesting properties of the derivative (whatever those might be), but you can do that to any function, and so the fact that you can do it to this function is true but not particularly interesting.
Another useful explainer -- obligatory 3b1b: https://www.youtube.com/watch?v=m2MIpDrF7Es
This is why I find e so fascinating, frustrating and puzzling.
The other constants fundamental to science, like the gravitational constant or the speed of light, can only be measured, not discovered from nothing. We aren't even sure how constant they actually are, there might be extremely tiny variations in either time or space that our instruments just can't measure yet. In theory, other universes could exist where these constants are "set" a little bit differently; whether we could live in such universes is another matter entirely.
E, on the other hand, comes from pure mathematics. As long as fractions, addition and exponentiation work the same way in another hypothetical universe, this strange E number is going to have the same strange value.
e isn't really a fundamental physical constant and doesn't show up very much in famous scientific formulas. It's a number that has very convenient properties for performing calculus operations and shows up in that context. pi isn't really an important physical constant, either, but it shows up a lot because of two things --- it's useful for calculations involving cycles, and because things being spherical or circular is a useful simplifying assumption (and also because things particularly in astronomy end up being spherical or circular (or nearly) quite often in reality).
Neither of those things are sort of empirically measured, and in any formula where they show up, you could theoretically absorb them into other constants -- and in fact the Einstein gravitational constant does exactly that -- it's defined as (8*pi*G)/c^4, absorbing pi into newton's gravitational constant (for historical reasons, when using plank units, they set G and c to 1, so it ends up just being 8pi -- _reduced_ planck units set the whole constant to 1). It's just frequently easier to separate out e and pi for the purposes of actually working out the math.
That's the same with pi if you think about it, and in my eyes is what makes it a transcendental number, so to speak. The numbers themselves could be different if you used a different base instead of base 10, but they still just represent a particularly useful value for us.
There's a good number of constants, and some of them are quite fun: https://en.wikipedia.org/wiki/List_of_mathematical_constants
One of my favorites is https://en.wikipedia.org/wiki/Feigenbaum_constants . Pops up in many situations, and as I understand it, it is not even well-understood why it pops up in as many places as it does. Intuitively one would expect that many of the places it pops up ought to have their own local constant of some sort, but instead this one keeps popping up. 7-year-old Numberphile video on it: https://www.youtube.com/watch?v=ETrYE4MdoLQ
One of my favorite parts if the excellent book Visual Complex Analysis is where he explains that e is defined in terms of the derivative of the exponential function. E is e because that’s what it has to be for the exponential function to be its own derivative.
"A base of 2 is useful because there are several small positive integers whose base-two logarithms are also integers." What? No! Base 2 is natural in exactly the same way that base e is natural, except for discrete domains instead of continuous domains. There is a unique family of functions for which the rate of change of the function is equal to the current value of the function everywhere. On discrete domains it's some scaled translation of 2^x, and on continuous domains it's some scaled translation of e^x. "Some scaled translation" here is accounting for the fact that the function is only uniquely exactly 2^x or e^x if we also add the constraint that f(0)=1.
Things like
Relationship to pi; Euler's formula:
derivative is itself:And e^x is the inverse of \int dx/x, which pops up a lot.
f'(x) = f(x), f(0) = 1 _uniquely defines_ f(x) = e^x, the linked article basically says this.
is it fair to describe e as the neutral element of a kind of derivative monoid ?
Sorry, you cannot use "imaginary" numbers to define something as "natural"!
I guess imagination doesn't come naturally to you, then?
Ah, that explains why you didn't see this obvious one coming.
TIL a two dimensional plane isn't natural.
C is a plane with multiplication. I would say, the fact that there is even a way of making this work is surprising (it doesn't work for any R^n, n > 2, at least not if you want a field). The particular way how we multiply complex numbers is also IMHO not self-evident (neither in the algebraic, nor in the geometric interpretation). It just turns out that doing it this way gives us really nice properties.
> how we multiply complex numbers is also IMHO not self-evident
Multiplication of complex number in the x + iy form treats them exactly like any other binomial factors, using the "FOIL" rule:
I don't know about self-evident; you pretty much have to do it that way. If you hand (a + bi)(c + di) to someone who knows basic algebra, but has no idea what i is, they will come up with the first FOIL expression above; after that, we have to know that i^2 is -1.> nor in the geometric interpretation.
That isn't self-evident, but when you gemoetrically work out what the multiplication is doing, it's just "add the arguments (angles), multiply the moduli (distances from origin)".
Very simple!
> you pretty much have to do it that way
You have to do it that way because it's the only way to get a field out of R^2, but I don't think it's at all self-evident. The "we have to know that i^2 is -1" is doing some heavy lifting in your explanation. In the context of "complex numbers are just a 2d plane" there's no a priory reason why you'd think that (0,1)*(0,1) has any particular meaning, let alone that it's equal to (-1,0).
If we interpret vectors in R^n as dislocation (or as forces, or...), then vector addition behaves exactly as one would expect. I think with enough motivation a bright school kid could come up with the formula for vector addition. And it works for every dimension.
Vector multiplication only works for 2d (ignoring the less well behaved cross product that also only works in certain dimensions) and I don't think you'd come up with it naturally.
Also, historically, the 2d plane isn't how complex numbers were discovered. They first appeared as weird quantities that made solving (real) cubic equations simpler but were regarded as just temporary artifacts. The realisation that they can be identified with R^2 came much later.
Transcendental numbers like e are a lot more "unnatural" than imaginary numbers.
e is the unique real number satisfying 1 + x <= e^x for all x.
1 - x <= e^{-x} so e^x <= 1/(1 - x) for x < 1
(1 + x/n)^n <= e^x <= (1 - x/n)^{-n} for x < 1
Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.
1+x is the tangent line at x=0. Since the graph of an exponential lies above its tangent line, 1+x <= e^x is another way of saying that the derivative of e^x at x=0 is 1, ie. e^x is its own derivative.
> e is the unique real number satisfying 1 + x <= e^x for all x.
I think if you replace e by 2e, this still holds. Therefore your definition of e is not unique.
1 + (-0.1) = 0.9
(2e)^(-0.1) = 0.844243...
The former is not <= the latter.
e is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0.
In high school we were taught that it is because The slope of the function e^x is equal to e^x. Growth being proportional to magnitude is natural.
Nonsense, this equation doesn't even make any sense without a well defined notion of the exponential function, and then a well defined extension of said function into the complex numbers. You will already have e by the time you reach this equation because defining what exp(z) even means requires you already know the properties of e^x over the reals you wish to preserve. And Euler's formula comes from finding such a function and then defining it to be exp(z). Multiplication in the complex plane by a unit vector is a rotation. Exponentials "repeated multiplication" by such a vector is spinning. And it turns out spinning at a constant rate satisfies the properties of the exponential function so it makes sense to say that's what exp(ix) means.
This is perhaps the most unnatural equation (well identity) in maths. It doesn't fall out anywhere, you would never write it down and solve for e, it's a special case of a more general result you would get first, and it's symbol soup for precisely the reason that the identity itself confers no understanding.
exp/log are natural because you almost can't help but discover them as they appear in so many different seemingly unrelated places.
> And Euler's formula comes from finding such a function and then defining it to be exp(z).
That's certainly one way, but you can also define exp via its power series (which is easily proven to be convergent everywhere). Then, all the properties of exp, as well as Euler's formula, are actual theorems, not just definitions.
It doesn’t matter which definition you pick. I e.g. had it defined axiomatically when I studied, i.e. we were given a list of properties which identify exp unambiguously and then we were given a proof of its existence. The fact that those properties were part of the definition doesn’t take away from their profoundness. The function could still just not exist. Same thing with defining by formulas. The map is not the territory.
Not really. Your equation has no unknowns, but if you consider the "e" in your equation as the unknown, and solve it (here I use "a" instead as the unknown, and use "e" in its usual meaning):
We want a^(i pi) + 1 = 0. Now,
a^(i pi) = e^(ln(a) i pi) = e^(i ln(a) pi) = cos( ln(a) pi) + i sin( ln(a) pi),
so we want cos( ln(a) pi) = -1, sin( ln(a) pi) = 0,
so ln(a) = 1, 3, 5, so a = e, e^3, e^5, ...
Thus indeed e^(i pi) + 1 = 0.
But also ln(a) = -1, -3, -5 work, so for example for
a = 1/e = 0.3678794412... > 0, we have
a^(i pi) + 1 = 0;
and of course for a = e^-99 = 1.0112214926104485... × 10^-43 etc.
Oops, I meant the smallest root greater than 1, not the smallest root greater than 0.
(Off-by-one errors, they're not just for programmers!)
The simplest way that I could put it using words is:
The number is what it is cause it's "increment of increment" is the same as it's "increment" when you are using exponentiation.
As I remember it e^x is the only function which is its own derivative, which I guess makes e “natural” in some sense.
Trivial, but y=0 is also its own derivative.
Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful.
Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x.
In my opinion drilling down too much on conventions misses the point of math.
We measure angles in radians because it's an easy way to measure the length of the circle sector in the units of the circle's radius. It feels neat in many cases.
A more practical way to measure angles would be in rotations. 0° = 0, 360° = 1, 90° = 0.25, etc. It would remove a ton of 2π and 4π² factors from a lot of equations in physics.
For trigonometry/calculus/physics radians are by far the most practical because they are dimensionless, so no constants appear when differentiating or integrating. (By the way, these constants will involve factors of pi anyway, it's inherent.)
For example, try to work out the Taylor series for sin(x) using degrees (or rotations). It's awful.
I don't see how Taylor series specifically would be affected. Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series.
Fourier transform would have 4π² instead of 2π under the exponent, no big deal.
The Euler's formula gets a factor of 2π under the exponent though. Given its wide application, it adds plenty of noise, of course.
> Differentiation of sin x and cos x is the same independent of the unit of x, and nothing else is used in the series.
Implicit in that statement is the use of the series definition of sine, which is a "meaningful" or "natural" definition insofar as it represents some function we care about. I'll address at the end what happens if we assume that definition regardless of its independent plausibility, but first consider:
The semantic meaning of sine, at least from its historical roots and how you might independently uncover it from earlier fields like geometry instead of later fields like differential equations, is that given an angle (in some units, we'll touch on that in a moment) we'd like to know the ratio two sides of a particular triangle associated with that angle inscribed in a circle. Given a choice of units for the angle, the triangle is fixed, and so the result (that ratio of side lengths) is also fixed.
Suppose you want to know how that ratio varies with respect to the angle. You can imagine a change of coordinates `y = cx` and consider the derivative of `sin(x)` vs `sin(y/c)`. The latter will have a numeric value `1/c` times less than the former. E.g., imagine a whole circle represented `1` angle instead of `2pi`. Then converting from our normal radians baseline to that new unit you have `y = (1/2pi)x`, and the derivative of the semantic ratio we're considering with respect to the new measure of angle is multiplicatively `2pi` greater than the original.
Going back to your series definition, suppose we pick that series as the definition of sine, independent of units. The problem that arises is that particular uses of sine do have units, and converting from the problem you care about to your particular from-on-high chosen definition of sine will run into the exact sort of problem our `y = cx` paragraph above touched on. The derivative with respect to the quantity of interest still has an extra `1/c` factor, and the fact that our God-blessed choice of sine is independent of units didn't actually solve anything in the composite problem.
If you define variants sint(x) = sin(2πx) and cost(x) = cos(2πx) that takes x in units of turns instead of radians, then d/dx sint(x) = 2π cost(x) etc.
I agree with you that this is completely fine though. I also find it more natural to think of “how many percent of a turn” an angle is than how many “degrees” or “radians” something is, since we use base-10 everywhere else. My workaround is to mostly write everything in terms of sin(2πτ), cos(2πτ), and exp(2πiτ) when I can, where τ measures turns.
> Differentiation of sin x and cos x is the same independent of the unit of x,
That’s not true. If the unit is degrees, d/dx sin(x) = pi/180 * cos(x).
Without delving too far into the philosophy of math as concerns existence vs convention...
e pops up quite often when taking limits on a surprising number of varied phenomena. It is much more than a mere convention, unless you subscribe to the nihilistic, anti-epistemological notion that all of mathematics is merely convention. It seems to be the center of the conceptual space particularly around questions of relative and absolute scale.
It's true you can use any base for computing things but some are more natural than others in that specific parametrizations have natural interpretations especially when it comes to physics (timescales, information-theoretic optimality, etc.).
No! We measure angles in radians because it's the simplest way to link the length of an arc to the radius of the circle.
But we have an inconsistency because pi relates the diameter to the circumference, not the radius to the circumference, which leaves an annoying 2 in the 2pi radians in a circle.
Other chosen conventions are less great, like not having the circle constant be 6.283185... so that we don't need to deal with factors or divisors of 2 all the time when using radians or almost anything else involving pi
90 degrees is one fourth of a circle, it would be so much more intuitive if we'd use "1/4th of something" rather than "1/2 radians" to express this
No, we measure angles in radians so that d^4/dx^4 (sin x) = sin x.
No, we measure angles in radians so that e^(ix) = cos x + i sin x.
But this does not depend in the unit.
> But this does not depend in the unit.
It does!
e^z, defined as the series \sum_{n = 0}^\infty z^n/n!, can only be a function of a dimensionless number z.
sin(z) and cos(z), defined as power series, technically also work this way. And that's OK, because angles are dimensionless: a radian is just C/(2πr), where C is the circumference of a circle of radius r. But it is sometimes convenient to pick your favorite number of radians, like π/180 of them, and call that a degree, and then to say that sin(x degrees) is the same as sin(xπ/180 radians).
With this convention, where the left-hand side of e^(ix) = sin(x) + icos(x) is a function of a dimensionless variable, and the right-hand side can be viewed as a function of a dimensioned argument only in the sense written above, it really is the case that the equation written is true, but the equation e^(ix) = sin(x degrees) + icos(x degrees) is false.
(On the other hand, you could make the case that e^(ix) is really a function of an angle, where its value is the complex number that lies on the unit circle at that angle. Then you do recover a "dimensioned" version of e^(ix) = sin(x) + i*cos(x) that's valid even if you measure angles in degrees.)
If we use a degrees version of sin and cos (call them sind and cosd), then we cannot have e on the left side without a conversion factor.
Probem is, f doesn't have nice properties like: There is something uniquely special about the unit circle, and about using the unscaled distance around the unit circle as the measure of the angle.Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle:
This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians.In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step.
That's related to how e^x is its own derivative.
> drilling down too much on conventions misses the point of math.
What is "the point of math?"
I thought it was about finding conventions that are useful, surprising, or pleasing in some way. Or, given a set of conventions, finding new ways that they are useful, surprising, or pleasing.
So I'm curious to know about this other... non-conventional point.
Well, the point, in math, is a zero dimensional object.
e, 3, and pi be
Something of a trinity
(Wink from Diety)