**e** (Euler's Number)

**e**

The number * e* is one of the most important numbers in mathematics.

The first few digits are:

**2.7182818284590452353602874713527** (and more ...)

*It is often called Euler's number after Leonhard Euler (pronounced "Oiler").*

** e** is an irrational number (it cannot be written as a simple fraction).

** e** is the base of the Natural Logarithms (invented by John Napier).

** e** is found in many interesting areas, so is worth learning about.

## Calculating

There are many ways of calculating the value of ** e**, but none of them ever give a totally exact answer, because

**is irrational and its digits go on forever without repeating.**

*e*But it **is** known to over 1 trillion digits of accuracy!

For example, the value of (1 + 1/n)^{n} approaches * e* as n gets bigger and bigger:

n | (1 + 1/n)^{n} |

1 | 2.00000 |

2 | 2.25000 |

5 | 2.48832 |

10 | 2.59374 |

100 | 2.70481 |

1,000 | 2.71692 |

10,000 | 2.71815 |

100,000 | 2.71827 |

## Another Calculation

The value of * e* is also equal to \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + \frac{1}{6!} + \frac{1}{7!} + ... (etc)

*(Note: "!" means factorial)*

The first few terms add up to: 1 + 1 + \frac{1}{2} + \frac{1}{6} + \frac{1}{24} + \frac{1}{120} = 2.71666...

In fact Euler himself used this method to calculate * e* to 18 decimal places.

You can try it yourself at the Sigma Calculator.

## Remembering

To remember the value of * e* (to 10 places) just remember this saying (count the letters!):

- To
- express
*e*- remember
- to
- memorize
- a
- sentence
- to
- memorize
- this

Or you can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

**2.7 1828 1828**

And following THAT are the digits of the angles 45°, 90°, 45° in a Right-Angled Isosceles Triangle (no real reason, just how it is):

**2.7 1828 1828 45 90 45**

*(An instant way to seem really smart!)*

## Growth

** e** is used in the

**"Natural**" Exponential Function:

Graph of **f(x) = e ^{x}**

It has this wonderful property: "its slope is its value"

At any point the slope of **e**^{x} equals the value of **e**^{x} :

when x=0, the value **e**^{x} = * 1*, and the slope =

**1**when x=1, the value

**e**^{x}=

*, and the slope =*

**e**

**e**etc...

This is true anywhere for **e**^{x}, and makes some things in Calculus (where we need to find slopes) a whole lot easier.

## Area

The area **up to** any x-value is also equal to *e*^{x} :

## An Interesting Property

### Just for fun, try "Cut Up Then Multiply"

Let us say that we cut a number into equal parts and then multiply those parts together.

### Example: Cut 10 into 2 pieces and multiply them:

Each "piece" is 10/2 = **5** in size

5×5 = **25**

Now, ... how could we get the answer to be **as big as possible**, what size should each piece be?

The answer: make the parts as close as possible to "* e*" in size.

### Example: **10**

^{2}= 25

^{3}= 37.0...

^{4}=

**39.0625**

^{5}= 32

The winner is the number closest to "* e*", in this case 2.5.

Try it with another number yourself, say 100, ... what do you get?

## 100 Decimal Digits

Here is * e* to 100 decimal digits:

**2.7182818284590452353602874713526624977572470936999595749669676277240766303535475945713821785251664274...**

## Advanced: Use of **e** in Compound Interest

**e**

Often the number * e* appears in unexpected places. Such as in

**finance**.

Imagine a wonderful bank that pays 100% interest.

In one year you could turn $1000 into $2000.

Now imagine the bank pays twice a year, that is 50% and 50%

Half-way through the year you have $1500,

you reinvest for the rest of the year and your $1500 grows to $2250

You got **more money**, because you reinvested half way through.

That is called compound interest.

Could we get even *more* if we broke the year up into months?

We can use this formula:

(1+r/n)^{n}

**r** = annual interest rate (as a decimal, so **1** not 100%)**n** = number of periods within the year

Our half yearly example is:

(1+1/2)^{2} = 2.25

Let's try it monthly:

(1+1/12)^{12} = 2.613...

Let's try it 10,000 times a year:

(1+1/10,000)^{10,000} = 2.718...

Yes, it is heading towards * e* (and is how Jacob Bernoulli first discovered it).

### Why does that happen?

The answer lies in the similarity between:

Compounding Formula: | (1 + r/n)^{n} | |

and | ||

(as n approaches infinity):e | (1 + 1/n)^{n} |

The Compounding Formula is **very like** the formula for **e*** (as n approaches infinity)*, just with an extra **r** (the interest rate).

When we chose an interest rate of 100% (= 1 as a decimal), the formulas became the same.

Read Continuous Compounding for more.

## Euler's Formula for Complex Numbers

** e** also appears in this most amazing equation:

*e*^{iπ} + 1 = 0

## Transcendental

** e** is also a transcendental number.