**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.*

*And Euler is spoken like "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 an 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 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + 1/7! + ... (etc)

*(Note: "!" means factorial)*

The first few terms add up to: 1 + 1 + 1/2 + 1/6 + 1/24 + 1/120 = 2.718055556

And you can try that yourself at 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
- simplify
- 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 20 into 4 pieces and multiply them:

Each "piece" is 20/4 = **5** in size

5×5×5×5 = 5^{4} = **625**

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

### Example continued: try 5 pieces

Each "piece" is 20/5 = **4** in size

4×4×4×4×4 = 4^{5} = **1024**

Yes, the answer is bigger! But is there a **best** size?

The answer: make the parts "* e*" (or as close to

*as possible) in size.*

**e**### Example: **10**

10 cut into 3 parts is 3.3... | 3.3...×3.3...×3.3... (3.3...)^{3} = 37.037... |

10 cut into 4 equal parts is 2.5 | 2.5×2.5×2.5×2.5 = 2.5^{4} = 39.0625 |

10 cut into 5 equal parts is 2 | 2×2×2×2×2 = 2^{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?

## 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

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 |

By substituting **x = n/r **:

**r/n**becomes**1/x**and**n**becomes**xr**

And so:

(1+r/n)^{n} |
becomes | (1+(1/x))^{xr} |

Which is **just like** the formula for **e*** (as n approaches infinity)*, with an extra **r** as an exponent.

So, as **x** goes to **infinity**, then (1+(1/x))^{xr} goes to e^{r}

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

## Transcendental

e is also a transcendental number.