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In the Homeric epic that is mathematics, e gets a choice epithet. It is “the natural base.”
I inventoried the reasons last year (for the historic once-in-a-century e Day). And I later stumbled across an even more delicious “naturalness” argument for e.
But that’s hardly exhaustive. Here’s another thing I dig about e, an easy fact for any first-year calculus student to verify:
Also, an addendum: a fun observation from Twitter…
Addendum #2, on March 4, 2019: the inimitable Sam Shah asks how we can introduce e to algebra students, and then delivers a delightful post rounding up more lovely e facts.
Doesn’t Coya’s law hold for any logarithm, not just log base e? (except the unit law has e replaced by the base of the log.)
Good point!
Thank you. Here are two more properties that make e unique among the reals. (Does unique ==> natural? Discuss among yourselves.)
(1) e is the positive real number with the greatest “self-root.” When posed as the problem, “Determine the maximum value of f(x) = x^(1/x),” this fact is the correct answer to a standard Calculus I exercise.
(2) The second fact is closely related to your yummy “naturalness” justification. Call two positive real numbers a and b _exponential B.F.F.s_ iff a ≠ b and a^b = b^a. Then e is the unique real number > 1 without an exponential B.F.F.
x#y is an operation I discovered on my own back when I was taking a class on ring/field theory. Some other properties:
1#x = 1^ln x = 1; x#1 = x^(ln 1) = x^0 = 1; so 1 behaves like 0 does in multiplication.
x#(e^(1/ln x)) = e, meaning e^(1/ln x) is the inverse of x (though it’s undefined if x = 1).
(xy)#z = (xy)^ln z = (x^ln z)(y^ln z) = (x#z)(y#z); that is, # distributes over multiplication just like multiplication distributes over addition.
The positive real numbers with the operations × and # are a field, and that field is isomorphic to the real numbers with the usual addition and multiplication.
For any x the tangent at (x, e^x) cuts the x axis at x – 1
This law has been known for a while. A quick search provides a result 4 years old and I would guess a more thorough search would find one much, much older.
https://www.quora.com/Why-is-the-graph-of-x-ln-y-y-ln-x-the-open-first-quadrant-What-class-should-I-have-learned-that-in-and-how-could-I-prove-it