e (Euler'southward Number)

e (eulers number)

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

The first few digits are:

2.7182818284590452353602874713527 (and more ...)

It is oftentimes chosen Euler'southward number later Leonhard Euler (pronounced "Oiler").

east is an irrational number (information technology cannot be written every bit a uncomplicated fraction).

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

due east is found in many interesting areas, so is worth learning nigh.

Calculating

There are many ways of calculating the value of east , but none of them ever give a totally exact answer, because e is irrational and its digits become on forever without repeating.

But it is known to over 1 trillion digits of accuracy!

For example, the value of (1 + ane/n)n approaches east as n gets bigger and bigger:

graph of (1+1/n)^n

n (ane + 1/north)n
i 2.00000
2 2.25000
v 2.48832
x 2.59374
100 two.70481
one,000 2.71692
10,000 two.71815
100,000 ii.71827

Try it! Put "(1 + one/100000)^100000" into the calculator:

(1 + 1/100000)100000

What exercise you get?

Some other Calculation

The value of e is also equal to one 0! + 1 one! + i ii! + i iii! + i 4! + 1 five! + 1 vi! + i 7! + ... (etc)

(Notation: "!" ways factorial)

The starting time few terms add up to: i + 1 + ane 2 + ane half dozen + 1 24 + 1 120 = two.71666...

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

You tin can endeavor it yourself at the Sigma Estimator.

Remembering

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

  • To
  • express
  • e
  • remember
  • to
  • memorize
  • a
  • sentence
  • to
  • memorize
  • this

Or y'all can remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

2.7 1828 1828

And post-obit 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 Part:

natural exponential function
Graph of f(10) = ex

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

At any point the slope of due east x equals the value of e x :

natural exponential function
when x=0, the value e 10 = 1 , and the gradient = i
when x=i, the value e x = due east , and the gradient = east
etc...

This is true anywhere for eastward x, and helps united states of america a lot in Calculus when we need to detect slopes etc.

So e is perfect for natural growth, come across exponential growth to learn more.

Surface area

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

natural exponential function

An Interesting Property

Just for fun, try "Cut Upwards Then Multiply"

Permit us say that we cutting a number into equal parts and then multiply those parts together.

Example: Cut 10 into 2 pieces and multiply them:

Each "piece" is x/two = v in size

five×5 = 25

Now, ... how could we get the answer to be as big as possible, what size should each piece be?

The reply: make the parts as close as possible to " e " in size.

Example: 10

10 cutting into 2 equal parts is 5: 5×five = v2 = 25

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

10 cut into 4 equal parts is 2.5: 2.v×two.5×2.5×ii.5 = 2.54 = 39.0625

10 cut into five equal parts is 2: 2×2×2×two×2 = 25 = 32

The winner is the number closest to " e ", in this case 2.v.

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

100 Decimal Digits

Here is due east to 100 decimal digits:

two.71828182845904523536028747135266249775724709369995957
49669676277240766303535475945713821785251664274...

Advanced: Employ of eastward in Compound Interest

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

Imagine a wonderful bank that pays 100% interest.

In 1 year you could turn $1000 into $2000.

Now imagine the depository financial institution pays twice a twelvemonth, that is l% and l%

One-half-style through the year you lot have $1500,
you reinvest for the remainder of the year and your $1500 grows to $2250

You got more money, because you reinvested one-half fashion through.

That is called compound interest.

Could we get even more if we bankrupt the year up into months?

We can use this formula:

(1+r/n)n

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

Our half yearly example is:

(1+1/2)2 = two.25

Permit'southward attempt information technology monthly:

(1+1/12)12 = 2.613...

Allow'due south try it ten,000 times a year:

(1+ane/10,000)10,000 = 2.718...

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

Why does that happen?

The answer lies in the similarity betwixt:

Compounding Formula: (1 + r/n)n
and
e (as northward approaches infinity): (1 + ane/northward)n

The Compounding Formula is very like the formula for e (as due north approaches infinity), just with an extra r (the interest rate).

When nosotros chose an interest charge per unit of 100% (= 1 every bit a decimal), the formulas became the same.

Read Continuous Compounding for more.

Euler'south Formula for Circuitous Numbers

e also appears in this nigh amazing equation:

eastward i π + 1 = 0

Read more here

Transcendental

eastward is likewise a transcendental number.

e-Mean solar day

balloons

Celebrate this astonishing number on

  • 27th January: 27/1 at 8:28 if you like writing your days kickoff, or
  • February 7th: 2/7 at 18:28 if y'all similar writing your months offset, or
  • On both days!

2011, 2012, 2013