# The Natural Base e

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The number $e,$ commonly called the natural base, is a mathematical constant named by the mathematician Leonhard Euler. It is an irrational number, meaning it can't be expressed as a fraction of two integers.

$e = 2.7182818284\ldots$

## Deriving $e$

The value of the number $e$ can be found in different ways. Here, compound interest will be used to find its value. The formula for compound interest is the exponential growth function $y=P\left(1+\frac{r}{n}\right)^{nt},$ where the constant $r$ is the interest rate in decimal form. If the interest rate was $100\,\%,$ which is extremely profitable, the value of $r$ equals $1.$ $y=P\left(1+\frac{1}{n}\right)^{nt}.$ The number $n$ is the amount of times the interest is compounded each year. That is, how often the accrued interest is added to the balance. The more often the interest is compounded, the higher the profit will be each year. What happens if the interest is compounded very often? That is, when $n$ is large. To examine this, the function will be rewritten using the the power of a power property. $y=P\left(1+\frac{1}{n}\right)^{nt}=P\left(\left(1+\frac{1}{n}\right)^n\right)^t$ For the rewritten function, the expression inside the outer parentheses is a constant depending only on $n.$ To analyze what happens when $n$ increases, larger and larger values will be substituted into the expression.

$n$ | $\left(1+\dfrac{1}{n}\right)^n$ | Expression value |
---|---|---|

$10$ | $\left(1+\dfrac{1}{10}\right)^{10}$ | $2.59374\ldots$ |

$100$ | $\left(1+\dfrac{1}{100}\right)^{100}$ | $2.70481\ldots$ |

$1000$ | $\left(1+\dfrac{1}{1000}\right)^{1000}$ | $2.71692\ldots$ |

$10\,000$ | $\left(1+\dfrac{1}{10\,000}\right)^{10\,000}$ | $2.71814\ldots$ |

$100\,000$ | $\left(1+\dfrac{1}{100\,000}\right)^{100\,000}$ | $2.71826\ldots$ |

The table shows that for high values of $n,$ the expression seems to approach $\sim2.718.$ In fact, as $n\rightarrow \infty,$ the value of the expression approaches the natural base $e$: $e=2.71828\ldots$ Substituting $e$ for the expression gives the function $y=Pe^{t},$

which applies when the interest rate is $100\,\%,$ compounded infinitely often. When interest is compounded infinitely often, it is said to be*continuously compounded*. Using this exponential function, instead of the original formula, makes this growth easier to work with.

## Natural Base Exponential Function

Exponential functions are commonly expressed using the natural base $e$, as they then exhibit useful characteristics. These functions are called natural base exponential functions and are written in the form $f(x) = ae^{rx}.$ If $a$ and $r$ are both positive, the function is an exponential growth function. If $a$ is positive and $r$ is negative, the resulting function is instead an exponential decay function.

## Continuously Compounded Interest

When an interest rate of $100\,\%$ is compounded continuously, meaning it is compounded infinitely often, the resulting function is $y = Pe^t.$ When the interest rate is something other than $100,$ the arbitrary rate $r$ is used. $y = P \left( 1 + \dfrac r n \right)^{nt}.$ Suppose that the interest rate doubles, $r = 2.$ Because interest is compounded continuously, this effectively leads to the same growth in half the time. Similarly, three times the interest leads to the same growth in a third of the time. This corresponds to a horizontal stretch or shrink, leading to the following function.

$y = Pe^{rt}$

## Exercises

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