E (Euler's Number)

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

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:

 2
 5
 10
 100
 1.000
 10.000
 100.000

**Try it!** Put "(1 + 1/100000)^100000" into thecalculator:

(1 + 1/100000)100000

What do you get?

## 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

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

You can try it yourself at the Sigma Calculator.

== Remembering

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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) = ex**

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 = ***e***, and the slope = ***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**

10 cut into 2 equal parts is 5:5×5 = 52 = 25

10 cut into 3 equal parts is 3*1***3**:(3*1***3**)×(3*1***3**)×(3*1***3**) = (3*1***3**)3 = 37,0...

10 cut into 4 equal parts is 2,5:2,5×2,5×2,5×2,5 = 2,54 = **39,0625**

10 cut into 5 equal parts is 2:2×2×2×2×2 = 25 = 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

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:

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**i***π + 1 = 0