Q. 126 - Precalculus Enhanced with Graphing Utilities

Question

Explain in your own words what the number e is. Provide at least two applications that use this number.

Step-by-Step Solution

Answer

The number e  as n takes very large values for the expression ${\left(1+\frac{1}{n}\right)}^n$.

One application is in the prime number theorem.

Another application is that it is used in standard normal distribution.

Step 1. Explain the number e.

Assume the expression,

${\left(1+\frac{1}{n}\right)}^n$

e can be defined,

$e=\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n$

This expression takes the value e as n takes very large values.

The table illustrates the value of the expression. The last number in the right column is correct to nine decimal places and is the same as the entry given on the calculator is given below,

Step 2. Application of the number e.

Consider the given question,

One application is in the prime number theorem as it states $\mathrm{\pi }\left(\mathrm{n}\right)$ which enumerates the number of prime numbers, less than or equal to n, $\frac{n}{\ln \left(n\right)}$. Here, $\ln$ function refers to the logarithm with the base e.

Another application is in standard normal distribution.

The normal distribution with zero mean and unit standard deviation is the standard normal distribution, $\Phi \left(x\right)=\frac{1}{\sqrt{2\mathrm{\pi }}}{e}^{-\frac{1}{2}{x}^2}$