**e** (Euler's Number)

**e**

The number * e* is a famous irrational number, and 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 the base of the Natural Logarithms (invented by John Napier).

** e** is found in many interesting areas, so it 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 (not the ratio of two integers).**

*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} = 2.718055556

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 angles 45°, 90°, 45° in a Right-Angled Isosceles (two equal angles) Triangle:

**2.7 1828 1828 45 90 45**

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

## 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**

^{5}= 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:

**2.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274...**

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

**e**

Often the number * e* appears in unexpected places.

For example,* e* is used in Continuous Compounding (for loans and investments):

Formula for *Continuous* Compounding

### Why does that happen?

Well, the formula for Periodic Compounding is:

FV = PV (1+r/n)^{n}

where **FV** = Future Value

**PV** = Present Value

**r** = annual interest rate (as a decimal)

**n** = number of periods within the year

But what happens when the number of periods heads to infinity?

The answer lies in the similarity between:

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

Compounding Formula | (as n approaches infinity)e |

The Compounding Formula is **very like** the formula for **e*** (as n approaches infinity)*, just with an extra **r** (the interest rate). When the number of periods** n** approaches infinity we can substitiute * e* into the fomula.

And that is why* e* makes an appearance in interest calculations!

## Transcendental

e is also a transcendental number.