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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 po

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.

munchler 2y ago
> 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.
- ogogmad 2y ago
  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 1y ago
    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.
- thaumasiotes 2y ago
  >> 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.
  - yorwba 2y ago
    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?
    - thaumasiotes 2y ago
      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.
seanhunter 2y ago
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.
- pests 2y ago
  Your footnote, wut?
  - seanhunter 2y ago
    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 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 “
rossant 1y ago
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.

munchler 2y ago

> 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.

ogogmad 2y ago
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 1y ago
  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.
thaumasiotes 2y ago
>> 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.
- yorwba 2y ago
  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?
  - thaumasiotes 2y ago
    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.

seanhunter 2y ago

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.

pests 2y ago
Your footnote, wut?
- seanhunter 2y ago
  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 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 “

rossant 1y ago

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.