what is $e$ really? what is its meaning?

Question

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!

Answer 1

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.

Answer 2

$e$ 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.

Answer 3

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.

Answer 4

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.