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A cute proof that makes e natural

by georgecmu on 4/17/25, 4:19 PM with 43 comments

• by dawnofdusk on 4/17/25, 7:04 PM

  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.

• by btilly on 4/17/25, 6:51 PM

  While this may convince students, you haven't actually prove that any exponential function has a slope. The usual presentation doesn't even demonstrate that such functions are defined at irrational numbers.

  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.

• by LegionMammal978 on 4/17/25, 7:56 PM

  How my high-school calculus textbook did it was to first define ln(x) so that ln(1) = 0 and d/dx ln(x) = 1/x, then take exp(x) as the inverse function of ln(x), and finally set e = exp(1). It's definitely a bit different from the exp-first formulation, but it does do a good job connecting the natural logarithm to a natural definition. (It's an interesting exercise to show, using only limit identities and algebraic manipulation, that this is equivalent to the usual compound-interest version of e.)

• by ogogmad on 4/17/25, 8:44 PM

  Tangential fact (har har): The Taylor series for e^x, combined with the uniqueness of representing a real number in base-factorial, immediately shows that e is irrational.

• by pwdisswordfishz on 4/18/25, 6:24 AM

  And here I thought it was irrational.

• by nathan_douglas on 4/17/25, 10:26 PM

  That was lovely; I really enjoyed it. Thank you.

• by analog31 on 4/17/25, 6:00 PM

  It also makes f flat.