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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