
Euler Constant eIn 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.
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 \( e \approx 2.71828182846 \) 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
