Euler Constant e


Euler Constant e

In compounding of interest it was shown that if, for example, an amount of money P (principal) is invested at an annual percentage rate r, the total amount of money A after t years is given by
\( A = P(1 + r)^t \)
It was also shown that if the interest is compounded n times during each year, the amount of money after t years is given by
\( A = P(1 + r/n)^{nt} \)
Let \( N = n / r \) , then \( r / n = 1 / N \) and \( n = r N \) , hence the formula for A becomes
\( A = P(1 + 1 / N)^{N r t} \)
Which can be written as
\( A = P ( (1 + 1 / N)^N )^{r t} \)
The question that one may ask is that what if we increase n indefinitely?
As the number of compounding n increases, \( N \) also increases, the term \( (1 + 1 / N)^N \) approaches a constant value called \( e \) (after the swiss mathematician
Leonhard Euler) and is approximately equal 2.718282....
The table of values below shows the values of \( (1 + 1 / N)^N \) as \( N \) increases.
\( N \) \( (1 + 1 / N)^N \)
1 2
2 2.25
3 2.37037
10 2.59374
20 2.65329
40 2.68506
100 2.70481
200 2.71149
400 2.71488

Below is shown the graph of \( y = (1 + \dfrac{1}{N})^N \) as a function of \( N \) and we can see that as \( N \) increases, \( y = (1 + \dfrac{1}{N})^N \) approaches a constant \( e \approx 2.71828182846 \)


Euler constant e

More rigorously, e is defined as the limit of \( (1 + 1/N)^N \) as \( N \) approaches infinity which is written as

\[ e = \lim_{N \to \infty} (1 + \dfrac{1}{N})^N \] Let \( m = \dfrac{1}{N} \) and rewrite another definition of the Euler constant \(e \) as follows
\[ e = \lim_{m \to 0} (1 + m)^{\dfrac{1}{m}} \]
The continuous compounding is defined for N very large and in this case the amount of money after t years is given by
\( A = P e^{r t} \)

Exponential and Logarithmic Functions to the Base e

The Euler constant \( e \) defined above plays an important role in applied mathematics. Many mathematical models used in physics, engineering, chemistry, economics,..., are described by exponential functions to the base \( e \) defined by
\( f(x) = e^x \)
and it inverse, the logarithm to the base \( e \), defined by
\( g(x) = ln(x) \)
Fundtion f is called the natural exponential function and the function g is called the natural logarithmic function. Both are graphed below.

graphs of natural exponential and logarithm

More References and Links

Compound Interests and Continuous Compounding of Interest
Leonhard Euler