A cute proof that makes e natural
poshenloh.com
poshenloh.com
IMO exponentials should just not be taught at all without basic notions of calculus (slopes of tangent lines suffice, as Po Shen Loh does here). The geometric intuition matters more than how to algebraically manipulate derivatives. The differential equation is by far the most natural approach, and it deserves to be taught earlier to students as is done apparently in France and Russia.
It doesn’t, but the limit as the number of splits approaches infinity is obviously an interesting (i.e. “natural”) result.
The perimeter of a polygon with an infinite number of sides is also interesting for the same reason.
> It doesn’t, but the limit as the number of splits approaches infinity is obviously an interesting (i.e. “natural”) result.
Except that the limit as the number of splits approaches infinity is just the declared rate of interest. The computation that ultimately yields e is a mistake, not a natural quantity to calculate.
Then later when you have formally introduced sequences and how to prove convergence, you can show that (1+1/n)^n is monotonically increasing and bounded above, hence convergent. This is no longer a mistake, but closer to a fun (and quite difficult) mathematical puzzle than anything practical. Naming it "e" is still premature at this point.
Then even later when you've introduced differentiation, it's time to talk about the derivative of arbitrary exponential functions, which is where that sequence reappears, and giving e a special name finally becomes appropriate.
It seems like American math curricula are typically so excited to talk about e that they try to skip over all the intermediate steps?
1. e is the exponential base for which f'(x) = f(x).
2. ln is the logarithm base e, and when f(x) = ln x, f'(x) = 1/x.
3. e is the sum of the series x^n / n! .
4. The textbook did specifically cover the fact that e is the limit of (1 + 1/n)^n as n goes to infinity, and it also specifically tied this in to the idea of computing interest by an obviously incorrect method. You could only call this a "natural mistake to make" in the same sense that it's "natural" to assume the square root of 10 must be 5, or that the geometric mean of two numbers is necessarily equal to the arithmetic mean.
5. However, the limit is important in that it illustrates that one to an infinite power is an indeterminate form.
6. As detailed in points (1) and (2), and hinted by the name "natural logarithm", we measure exponentials and logarithms by reference to e for the same reason we measure angles in radians.
It's possible that this particular definition of e is important to a proof of one of the properties of e^x or ln x, but if so I don't remember reading about it in the textbook and it wouldn't have been covered in class. In my real analysis class, we used the Maclaurin series for e; (1 + 1/n)^n was never mentioned.
(It's really easy to show that that series is monotonically increasing.)
> Then later when you have formally introduced sequences and how to prove convergence
This is not material you'd expect at all in a calculus class. If sequences are mentioned, it would only be in passing as you move to series. Several methods of testing infinite series for convergence are covered. What it means for a sequence to converge is not. Limits are not defined in terms of sequences. Infinite series would be covered after, not before, differential and integral calculus.
You have to have a name for e because otherwise it would be impossible to work with. But it is interesting and the wrong way to compute interest isn't; there's no point in trying to motivate something important with something unimportant.
The important aspect however is that the compound interest interpretation of this sequence is not only unnatural but wrong, in the sense that it starts from the wrong guess that a rate r compounded n times should be something like a rate r/n. It is unnecessarily confusing to teach such young students historical mistakes: bad students won't understand the mistake and good students will be bewildered. In both cases students come away thinking of e as a sort of "mystical" thing.
If I had to teach the compound interest, one starts by considering with a yearly interest rate r, your principal grows after m years by (1+r)^m. Now, can we find an equivalent monthly interest rate r2? To do so, we must solve (1+r2)^12 = (1+r), which requires logarithms, at which point you note immediately that r2 =/= r/12, which is perhaps unexpected. Now, it might be natural to ask what about for continuous compounding? Then, we study (1+f(n))^n as n goes to infinity, where f(n) is some function of n. We know it should decrease for large n, and the binomial expansion lets us guess (for integer n) that it should be f(n) = c/n for some constant c. Now, we are ready to compute the value of this limit.
The important part is that in studying compound interest one needs certain analytical tools, such as logarithms and asymptotic analysis, beforehand. It does not make much sense, as you say, to skip over all the intermediate steps and introduce the constant e as the solution to some unmotivated and unnatural formula relating to compound interest, but this is indeed what American schools do. In my experience, very few American high school students understand or remember (if they were ever taught it) the identity a^x = e^(x * ln a), and the concept of exponentiation is generally not well understood.
No, it doesn't. r_2 = (1 + r)^{1/12} - 1. Compound interest looks like (money) = (money_0)*r^t; you'd only need logarithms if you were trying to solve for time.
> Now, it might be natural to ask what about for continuous compounding?
I can't tell what you're getting at here. Once you've written down the equation (1 + r_2)^{12} = (1 + r), you've already provided a complete solution for continuous compounding. If the time you want to compound over is t, and Y is one year, then the solution is always given by (1 + r_2) = (1 + r)^{t/Y}. Nothing goes to infinity.
[1] This is why it's named Euler's number - because it was discovered by Bernoulli. Many of the things that Euler discovered (like Lambert's W function etc) are named after other people too in the same tradition.
General Taylor series were discovered by James Gregory (long after the first Taylor series for sine and cosine etc were written down by Madhava of Sangamagrama) who taught them to Maclaurin who taught them to Taylor.
Lambert's W function (also known as the product log function) was the function that Euler discovered that solved a problem that Lambert couldn't solve.
Gauss' law in physics was discovered by Lagrange. In turn, Lagrange's notation for derivatives was used by Lagrange, but was invented far earlier by Euler.
"Feynman's Trick" in calculus of parameterizing and then differentiating under the integral was also discovered by Euler. Like yeah. 250 years isn't enough to stop someone stealing the name of something you discovered. I think Euler discovered so many things people just decided at some point they couldn't name everything after Euler so started giving other people a chance.
The Gaussian distribution was discovered by de Moivre. Gaussian elimination was in textbooks in the time of Gauss so in his work he calls it "common elimination".
Arabic numerals were invented by Indian mathematicians.
Practically the only thing we know for absolute certain about Pythagoras is that he didn't discover Pythagoras' theorem (that had been known to the Babylonians centuries earlier).
Bayes never published his paper during his lifetime but it involves a very important thought experiment in probability and not the equation that everyone knows as Bayes' theorem, which was actually written by Laplace after reading Bayes paper.
Cantor didn't discover the Cantor set.
etc etc. There are hundreds or possibly thousands of examples. This is known as Stigler's law. https://en.wikipedia.org/wiki/Stigler's_law_of_eponymy
There are two more fun examples then I’ll stop. Kuiper published a paper stating that a ring of asteroids didn’t exist in the solar system. So when such a ring was discovered naturally it was named the Kuiper belt after him. Not maths, but in the same vein, in chess an early theorist called Damiano published an analysis showing that 1 e4 e5 Nf3 f5 was losing for black, so now that’s called “Damiano’s defence “
Fibonacci series goes far far back in time than Leonardo of Pisa.
>In science the credit goes to the man who convinces the world, not to the man to whom the idea first occurs.
-- in Eugenics Review April 1914 ‘Francis Galton’My assumption is that some schools think the number e should be introduced much like the number pi. Except the number pi has a much more natural definition, relating to circles, and appears in important formulas like the area of a circle. Obviously it would be possible to first introduce the special functions sin(x), cos(x), etc. for which it is a fun fact that there is some irrational number between 3 and 4 such that sin(pi) = 0, but it's clear that this would be silly.
For the number e, however, this approach is not so silly, as what is important and useful are indeed the functions exp(x) and ln(x). The constant e itself does not appear in any natural or intuitive formulas, but only in connection with the function exp(x) evaluated at x=1 (or x = i*pi, in the famous formula, which should not really be taught at the high school level IMO).
That said, it is worthwhile to go through the algebra exercise to convince yourself that, for large n, (1+x/n)^n expands out to approximately 1 + x + x^2/2 + x^3/6 + ...
Hint. The x^k terms come out to (x/n)^k (n choose k). This will turn out to be x^k/k! + O(x^k/n). As n goes to infinity, the error term drops out, and we're just left with the series that we want.
The rigorous version of this argument uses the Dominated Convergence Theorem in the special case of infinite series.
An explicit epsilon-delta style proof is not that hard to produce, it's just a little messy. What you have to do is, for a given x and 0<ε, pick N large enough that you can bound the tail after x^N/N! onwards with a geometric series adding up to at most ε/2. Now pick n large enough enough that the sum of errors in the terms up to N is also bounded by ε/2. From that n on, (1+x/n)^n is within ε of the power series for e^x.
Interestingly, a similar approach gives the shortest proof that exp(x) and ln(x) are computable functions (since integration is a computable functional, thanks to interval arithmetic), and therefore that e = exp(1) is a computable real number.
The problem with this approach is that, since we were already introduced to exponents and logarithms in algebra but via different definitions, it always left this unanswered question in my head about how we knew these two definitions were the same, since everyone quickly glossed over that fact.
I think even in complex analysis, the above definition a^b := exp(b ln(a)) makes sense, since the function ln() admits a Riemann surface as its natural domain and the usual complex numbers as its codomain.
[EDIT] Addressing your response:
> Calculus glosses over the case when a is negative
The Riemann surface approach mostly rescues this. When "a" is negative, and b is 1/3 (for instance), choose "a" = (r, theta) = (|a|, 3 pi). This gives ln(a) = ln |a| + i (3 pi). Then a^b = exp((|a| + i 3 pi) / 3) = exp(ln |a|/3 + i pi) = -|a|^(1/3), as desired.
Notice though that I chose to represent "a" using theta=3pi, instead of let's say 5pi.
Even then, general logarithms become messy. It's easy to say "just take local segments of the whole surface" in the abstract, but any calculator will have to make some choice of branch cuts. E.g., clearly (−1)^(1/3) = −1 for any sane version of exponentiation on the reals, but many calculators will spit out the equivalent of (−1)^(1/3) = −e^(4πi/3) instead.
(Just in general, analytic continuation only makes sense in the abstract realm. If you try doing it numerically to extend a series definition, you'll quickly find out how mind-bogglingly unstable it is. I think there was one paper that showed you need an exponential number of terms and exponentially many bits of accuracy w.r.t. the number of steps. Not even "it's 2025, we can crank it out to a billion bits" can save you from that.)
They just shouldn't be taught in algebra. There, one is thinking about oh how do I extend the definition of exponentiation from integers to real numbers. But thinking about continuous extensions of an integer function to the real line is really something that should be saved for calculus.
Some details here: https://en.wikipedia.org/wiki/Factorial_number_system#Fracti...
See also: https://math.stackexchange.com/questions/425963/is-there-a-s...
Ultimately, the above argument resembles Fourier's proof that e is irrational. But I find it more intuitive. A lot of other special constants can be proved irrational using similar techniques.
But the cute proof was pretty cute. I recommend calc teachers try to work it into their lecture on e.