E – Euler s number

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!

Related movie: A Gringo’s Guide to Dominican Damsels


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