July 2nd, 2024

What makes e natural? (2004)

The number e in mathematics, introduced by Euler, has unique properties like transcendence and relation to trigonometric functions. Historical figures like Bürgi, Napier, and Briggs contributed to logarithmic advancements. E's significance lies in its role in exponential and logarithmic functions, crucial in calculus.

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The article discusses the concept of the number e in mathematics, its properties, and historical significance. While bases 2 and 10 are commonly used, the natural logarithm e is introduced by Euler and has unique properties. Defined as the limit of $\displaystyle \left(1+\frac{1}{n}\right)^{n}$, e is transcendental and closely related to trigonometric functions like $\displaystyle \pi$. The article delves into the history of mathematics, including the work of Joost Bürgi and John Napier on logarithms. Henry Briggs further developed Napier's work, leading to the decimal logarithm. The article also mentions the irrationality of e and efforts to calculate more digits. The key property of e is its relationship to exponential and logarithmic functions, where the slope of $\displaystyle e^{x}$ is $\displaystyle e^{x}$. The article concludes by emphasizing e's importance in various mathematical contexts and its role in calculus.

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11 comments

By @munchler - 5 months

This is a frustrating article because it never explains why e is the natural logarithm base. To me, the easiest way to understand it is via continuous compound interest:

* If you invest $1 at 100% interest for 1 year, you get $2 at the end

* Compounded 2 times in a year, you get 100/2 = 50% interest every 1/2 year, which amounts to $2.25

* Compounded 4 times in a year, you get 100/4 = 25% interest every 1/4 year, which amounts to $2.44

* Compounded n times in a year, you get 100/n percent interest every 1/n year, which amounts to (1+1/n)^n dollars

* So continuous compound interest is the limit as n approaches infinity, which amounts to $2.71828 at the end of the year

(This is a great problem to give to pre-calc students to see if they can figure out the calculation for themselves.)

By @User23 - 5 months

One of my favorite parts if the excellent book Visual Complex Analysis is where he explains that e is defined in terms of the derivative of the exponential function. E is e because that’s what it has to be for the exponential function to be its own derivative.

By @9question1 - 5 months

"A base of 2 is useful because there are several small positive integers whose base-two logarithms are also integers." What? No! Base 2 is natural in exactly the same way that base e is natural, except for discrete domains instead of continuous domains. There is a unique family of functions for which the rate of change of the function is equal to the current value of the function everywhere. On discrete domains it's some scaled translation of 2^x, and on continuous domains it's some scaled translation of e^x. "Some scaled translation" here is accounting for the fact that the function is only uniquely exactly 2^x or e^x if we also add the constraint that f(0)=1.

By @kazinator - 5 months

Things like

Relationship to pi; Euler's formula:

   ix
  e   = cos x + isin x

derivative is itself:

  d    x     x
  --  e   = e 
  dx

By @cperciva - 5 months

e is natural because it's the smallest positive root of the equation e^(i pi) + 1 = 0.

By @keithalewis - 5 months

e is the unique real number satisfying 1 + x <= e^x for all x.

1 - x <= e^{-x} so e^x <= 1/(1 - x) for x < 1

(1 + x/n)^n <= e^x <= (1 - x/n)^{-n} for x < 1

Letting n go to infinity gives e^x = \sum_{n=0}^infy x^n/n! using Newton's binomial formula.

By @reliablereason - 5 months

The simplest way that I could put it using words is:

The number is what it is cause it's "increment of increment" is the same as it's "increment" when you are using exponentiation.

By @bjornsing - 5 months

As I remember it e^x is the only function which is its own derivative, which I guess makes e “natural” in some sense.

By @RicoElectrico - 5 months

Much in math is a matter of convention. But, to paraphrase - all conventions are possible, but some are useful.

Why do we measure angles in radians? Because then d/dx (sin x) = 1 at x = 0, and sin x ≈ x for small x.

In my opinion drilling down too much on conventions misses the point of math.

By @smitty1e - 5 months

e, 3, and pi be

Something of a trinity

(Wink from Diety)

What makes e natural? (2004)

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