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I don't get it how we came up with $e$ and how can nature use this number so much! that is what I have been told and I only know that $e$ is a specific constant like $\pi$! I understand that $\pi$ just happens to be the ratio of circumference over diameter and I can accept that this is constant for any circle, but what about $e$? what is the real meaning of $e$? thanks!

leonbloy
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trig
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  • Do you know the definition of $e$? – Potato Oct 29 '14 at 15:40
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    Another definition of $e$ that I like: $e^x\ge1+x$ for all $x$, and no other number has this property. (Try graphing this.) Using this definition, and without calc, it's possible to derive $1-\frac12+\frac13-\dotsb=\log_e2$. It's a wonderful proof, but the character limit is too small to contain it. (I sound like Fermat…) – Akiva Weinberger Oct 29 '14 at 16:03

4 Answers4

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The constant $e$ appears in many different settings. The most common examples include

  • The value that $\left(1 + \frac{1}{n}\right)^n$ closes in on as $n$ gets large.
  • The value of the infinite sum $\sum_{i = 0}^\infty \frac{1}{i!}$.
  • The unique number so that $\left[e^x\right]' = e^x$.
  • The base of the natural logarithm, which again is the antiderivative of $\frac{1}{x}$.

I've seen all of them used as the actual definition in different books.

Arthur
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    Along the lines of your third point, I'd also add that it's also the value at $x = 1$ of the unique solution to the differential equation $y'(x) = y(x), y(0) = 1$. – anomaly Oct 29 '14 at 15:46
  • how can all these different settings involve the exact same constant? and what is actually meant when we say that e is involved in many natural phenomena? – trig Oct 29 '14 at 15:47
  • @trig Proving that all of these describe the same number is another question entirely. Points 2 and 3 are easily related by what is known as Taylor expansions. Point 1 can be related to point 3 by raising to the power of $x$ and differentiate. I don't remember right now how point 4 is usually handled. As for nature, I have personally never seen $e$ appear concretely in nature, but the laws of nature as described by mankind are written in calculus. As Hurkyl's answer below tells you, $e$ makes for the tidiest expressions. – Arthur Oct 29 '14 at 15:50
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$e$ is the base for exponents for which calculus is the most convenient, just like radians are the angle measure that is most convenient for calculus.

e.g.

$$ \frac{d}{dx} \left( e^x \right) = e^x \qquad \text{but} \qquad \frac{d}{dx} \left( 2^x \right) = 2^x \log_e 2$$

$$ \frac{d}{dx} \sin(x) = \cos x \qquad \text{but} \qquad \frac{d}{dx} \sin(x^\circ) = \frac{\pi}{180} \cos(x^{\circ})$$

Therefore, we tend to express exponential functions using $e$ as the base.

  • Related, $e$ is the unique base of an exponential function such that its slope at $x = 0$ is $1$. I've always liked that for some reason, even though it's very obvious from its derivative. – MT_ Oct 29 '14 at 16:05
  • @Soke That's similar to my comment on the OP. – Akiva Weinberger Oct 29 '14 at 16:14
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One place where $e$ appears is as the solution to the differential equation $$\frac{dy}{dx} = y.$$ This equation describes the growth/decay of the value $y$, and says that the growth of $y$ is proportional to how big $y$ is.

For instance, a population will produce more offspring if there are 100,000 people than if there were 2 people. Since the function $e^x$ satisfies this differential equation, you can use it as a rough model of population growth. You could make the same argument if you have money instead of people. The more money you have, the more interest you will get.

The logarithm was actually discovered by Napier before the discovery of $e$. Logarithms make doing multiplication a lot easier, when a log table is available. Later, in the 1600s, there was an investigation of the area under the curve $f(x) = 1/x$, by Fermat and others, which led to the discovery of the "natural" logarithm. Then the next question is, what is the base of that logarithm? That base is $e$.

This is one line of investigation that led to $e$. A more direct discovery was in the investigation of the compounding interest formula $(1+1/n)^n$. The limit of this is $e$. Leibniz was the first to carry out the investigation of the limit (if I recall correctly), and he actually called it $b$. It was relabeled as $e$ later on by Euler.

Joel
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As I see it, it happens to exist a constant ratio between the circumference and the diameter for any circle, and happens to exist a real number $e$ such that the slope of the tangent to the curve $e^x$ has the value $e^x$ at $x$.

The existence of those constants is not at all trivial.

Lehs
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