# 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 e is irrational (not the ratio of two integers).

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 e as possible) in size.

### 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

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)) xr goes to e r

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