The Number $e$

The number $e$, sometimes called the natural number, or Euler's number, is an important mathematical constant approximately equal to 2.71828. When used as the base for a logarithm, the corresponding logarithm is called the natural logarithm, and is written as $\ln (x)$. Note that $\ln (e) =1$ and that $\ln (1)=0$.

There are a number of different definitions of the number $e$. Most of them involve calculus. One is that $e$ is the limit of the sequence whose general term is $(1+{1 \over n})^n$. Another is that $e$ is the unique number so that the area under the curve $y=1/x$ from $x=1$ to $x=e$ is $1$ square unit. 

Another definition of $e$ involves the infinite series $1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4! }+....$. It can be shown that the sum of this series is $e$.

Importance of $e$

The number $e$ is very important in mathematics, alongside $0, 1, i, \, \text{and} \, \pi.$ All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in the formulation of Euler's identity, which (amazingly) states that $e^{i\pi}+1=0.$ Like the constant $\pi$, $e$ is irrational (it cannot be written as a  ratio of integers), and it is transcendental (it is not a root of any non-zero polynomial with rational coefficients).

Compound Interest

One of the many places the number $e$ plays a role in mathematics is in the formula for compound interest. Jacob Bernoulli discovered this constant by asking questions related to the amount of money in an account after a certain number of years, if the interest is compounded $n$ times per year. He was able to come up with the formula that if the interest rate is $r$ percent and is calculated $n$ times per year, and the account originally contained $P$ dollars, then the amount in the account after $t$ years is given by $A=P(1+{r \over n})^{nt}.$ By then asking about what happens as $n$ gets arbitrarily large, he was able to come up with the formula for continuously compounded interest, which is $A=Pe^{rt}.$