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The arxiv preprint linked in this is really good. I'm American so I got my education on e from the compound interest limit which isn't natural at all, as Loh points out. Why should it matter how many times I "split up" my compounding?

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.

▲ seanhunter 5 days ago | parent | next [-]

I think the reason that the compound interest limit is used may well be the history - that was how Jacob Bernoulli derived e initially[1] - and that around the time in your mathematics education when you might be learning the exponential and natural log functions is probably about the right time to teach series and it's a lovely example.

[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.

▲ dawnofdusk 4 days ago | parent | next [-]

Actually historically John Napier discovered the logarithm before Jacob Bernoulli computed e by his compound interest formula. IMO it would be more fruitful to teach the special functions exp(x) and ln(x) just as functions, which is more historical, with the important and unique property that they turn multiplication into addition and vice versa. The fact that exp(1) is some irrational number in between 2 and 3 is just a fun fact at this point, and the connection to compound interest is an even more fun fact.

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).

▲ pests 5 days ago | parent | prev [-]

Your footnote, wut?

▲ seanhunter 4 days ago | parent [-]

It's a thing in maths that stuff gets named after whoever people decide at the time deserves it, not necessarily the person who discovered it.

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 “

	▲	srean 4 days ago \| parent \| next [-]
		Funniest but is that Stigler was not the first to discover Stigler's law. Fibonacci series goes far far back in time than Leonardo of Pisa.
	▲	gsf_emergency 3 days ago \| parent \| prev \| next [-]
		Most of these fall under the math version of Darwin's Law? >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’`
	▲	sebastiennight 4 days ago \| parent \| prev [-]
		That was great reading, thank you! TIL

▲ munchler 5 days ago | parent | prev | next [-]

> Why should it matter how many times I "split up" my compounding

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.

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ogogmad 5 days ago | parent | next [-]

Looking at it in terms of compound interest seems random. That said, I think the expression lim((1+x/n)^n) is better motivated within Lie theory, since every Lie group admits a faithful linear representation in which the expression lim((1+x/n)^n) makes sense (even if infinite-dimensional Hilbert spaces might be needed, as in the metaplectic group). Then the subexpression 1+x/n approximates a tangent vector to 1.

	▲	dawnofdusk 4 days ago \| parent [-]
		A nice approach (which is also in Po Shen Loh's preprint) is that this limit can be seen as the Euler scheme for solving the differential equation f' = f. This is related to your viewpoint about Lie theory... maybe slightly more digestible for high school students.

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thaumasiotes 5 days ago | parent | prev [-]

>> Why should it matter how many times I "split up" my compounding

> 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.

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yorwba 4 days ago | parent [-]

It is a natural mistake to make, so spending some time showing that it gives the wrong result is probably appropriate. Of course giving that wrong result a special name (especially something short like "e") or even calculating its value to a high degree of precision are pointless at this stage.

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?

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dawnofdusk 4 days ago | parent | next [-]

The sequence (1+1/n)^n can be seen as a natural thing to study (indeed, tautologically, because we know that e is a natural thing to study) by various viewpoints.

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.

	▲	thaumasiotes 3 days ago \| parent [-]
		> To do so, we must solve (1+r2)^12 = (1+r), which requires logarithms 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.

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thaumasiotes 4 days ago | parent | prev [-]

Well, I don't have my calculus textbook to hand, but I can tell you what I took from the class.

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.

▲ rossant 3 days ago | parent | prev [-]

More than 15 years ago, I was tutoring a few high school students, including one who was particularly strong in math. I put together a special multi-page assignment for him that introduced the exponential function through the concept of compound interest, aiming to convey the intuition behind it and derive some key properties. He absolutely nailed it. I’ve often wondered what became of him.