Is E An Irrational Number

e (Euler's Number)

e (eulers number)

The number e is one of the almost of import numbers in mathematics.

The outset few digits are:

two.7182818284590452353602874713527 (and more ...)

It is frequently called Euler'due south number later on Leonhard Euler (pronounced "Oiler").

e is an irrational number (it cannot be written equally a elementary fraction).

e is the base of operations of the Natural Logarithms (invented by John Napier).

e is establish in many interesting areas, so is worth learning about.

Computing

There are many ways of calculating the value of east , but none of them e'er give a totally verbal reply, because e is irrational and its digits go on forever without repeating.

But it is known to over one trillion digits of accuracy!

For instance, the value of (1 + 1/n)ⁿ approaches e as n gets bigger and bigger:

graph of (1+1/n)^n

n	(one + 1/due north)^northward
1	2.00000
2	2.25000
5	2.48832
10	2.59374
100	2.70481
1,000	ii.71692
10,000	ii.71815
100,000	2.71827

Endeavor information technology! Put "(ane + i/100000)^100000" into the reckoner:

(1 + one/100000)¹⁰⁰⁰⁰⁰

What do you get?

Another Calculation

The value of e is also equal to i 0! + one 1! + i 2! + 1 3! + 1 4! + 1 5! + 1 6! + 1 7! + ... (etc)

(Annotation: "!" means factorial)

The get-go few terms add together up to: 1 + 1 + 1 2 + ane vi + 1 24 + i 120 = two.71666...

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

You can try it yourself at the Sigma Computer.

Remembering

To remember the value of e (to ten places) just remember this saying (count the letters!):

To
express
eastward
retrieve
to
memorize
a
judgement
to
memorize
this

Or yous tin remember the curious pattern that after the "two.7" the number "1828" appears TWICE:

two.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 information technology is):

2.7 1828 1828 45 xc 45

(An instant way to seem actually smart!)

Growth

due east is used in the "Natural" Exponential Role:

natural exponential function
Graph of f(x) = e^x

Information technology has this wonderful property: "its slope is its value"

At any indicate the gradient of eastward ^ten equals the value of east ^x :

natural exponential function
when x=0, the value e ¹⁰ = ane , and the slope = i
when x=ane, the value e ^x = eastward , and the slope = east
etc...

This is true anywhere for e ^x, and helps u.s. a lot in Calculus when we demand to find slopes etc.

So e is perfect for natural growth, see exponential growth to learn more than.

Area

The area upwardly to whatever x-value is too equal to e ^ten :

natural exponential function

An Interesting Holding

Just for fun, attempt "Cut Up Then Multiply"

Let us say that we cut a number into equal parts and and so multiply those parts together.

Example: Cut ten into 2 pieces and multiply them:

Each "slice" is 10/ii = v in size

5×5 = 25

Now, ... how could we become the reply to be as big as possible, what size should each piece exist?

The answer: brand the parts equally close as possible to " e " in size.

Instance: ten

x cut into 2 equal parts is five: 5×5 = 5ⁱⁱ = 25

x cutting into 3 equal parts is 3 1 3 : (3 ane 3 )×(3 1 3 )×(3 ane three ) = (3 1 three )³ = 37.0...

10 cutting into 4 equal parts is two.v: 2.v×2.5×2.5×ii.v = two.v^four = 39.0625

10 cut into five equal parts is 2: 2×2×2×2×2 = 2^v = 32

The winner is the number closest to " eastward ", in this example 2.5.

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

100 Decimal Digits

Hither is e to 100 decimal digits:

2.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274...

Advanced: Utilise of e in Compound Interest

Often the number e appears in unexpected places. Such as in finance.

Imagine a wonderful depository financial institution that pays 100% interest.

In one year you could plough $thou into $2000.

Now imagine the banking concern pays twice a year, that is l% and 50%

Half-mode through the twelvemonth you lot take $1500,
you lot reinvest for the remainder of the year and your $1500 grows to $2250

You got more coin, considering you reinvested half way through.

That is called chemical compound interest.

Could we become even more if we broke the yr up into months?

Nosotros can use this formula:

(i+r/n)ⁿ

r = annual involvement rate (as a decimal, so i non 100%)
n = number of periods within the year

Our half yearly example is:

(i+1/2)² = 2.25

Allow'south try it monthly:

(1+1/12)¹² = 2.613...

Let'southward try information technology 10,000 times a yr:

(1+i/ten,000)^10,000 = 2.718...

Yes, it is heading towards e (and is how Jacob Bernoulli first discovered it).

Why does that happen?

The respond lies in the similarity betwixt:

Compounding Formula:		(1 + r/n)ⁿ
and
e (every bit northward approaches infinity):		(1 + 1/n)ⁿ

The Compounding Formula is very like the formula for due east (as n approaches infinity), just with an extra r (the involvement charge per unit).

When nosotros chose an interest rate of 100% (= 1 as a decimal), the formulas became the same.

Read Continuous Compounding for more.

Euler'south Formula for Complex Numbers

due east also appears in this near amazing equation:

eastward ^{i π} + one = 0

Transcendental

e is also a transcendental number.

e-Day

balloons

Celebrate this amazing number on

27th January: 27/one at 8:28 if you lot similar writing your days first, or
Feb 7th: ii/7 at 18:28 if you like writing your months showtime, or
On both days!

2011, 2012, 2013