# E – Euler s number

The number * e* is a famous irrational number, and is one of the most significant numbers in mathematics.

The very first few digits are:

**Two.7182818284590452353602874713527** (and more . )

*It is often called Euler’s number after Leonhard Euler.*

** 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 response, 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 fatter and thicker:

## Another Calculation

The value of * e* is also equal to 1/0! + 1/1! + 1/Two! + 1/Three! + 1/Four! + 1/Five! + 1/6! + 1/7! + . (etc)

The very first few terms add up to: 1 + 1 + 1/Two + 1/6 + 1/24 + 1/120 = Two.718055556

And you can attempt that yourself at Sigma Calculator.

## Remembering

To recall the value of * e* (to Ten places) just recall this telling (count the letters!):

Or you can recall the nosey pattern that after the ",Two.7", the number ",1828", shows up TWICE:

**Two.7 1828 1828 45 90 45**

*(An instant way to seem indeed brainy!)*

## An Interesting Property

### Just for joy, attempt ",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 Four chunks and multiply them:

Each ",chunk", is 20/Four = **Five** in size

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

### Example continued: attempt Five chunks

Each ",chunk", is 20/Five = **Four** in size

Yes, the reaction is thicker! But is there a **best** size?

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

*as possible) in size.*

**e**### Example: **Ten**

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

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

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

**e**

Often the number * e* shows up 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:

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 goes to infinity?

The response lies in the similarity inbetween:

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

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)) x**r** goes to e **r**

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

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