Calculating

There are many means of calculating the worth of e, but none of them ever provide a completely specific answer, because e is irrational and its digits go on forever before without repeating.

You are watching: What is e to the -1

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

For instance, the worth of (1 + 1/n)n ideologies e as n gets bigger and also bigger:

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n(1 + 1/n)n
12.00000
22.25000
52.48832
102.59374
1002.70481
1,0002.71692
10,0002.71815
100,0002.71827


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

(1 + 1/100000)100000

What do you get?


Anvarious other Calculation

The worth of e is additionally equal to 10! + 11! + 12! + 13! + 14! + 15! + 16! + 17! + ... (etc)

(Note: "!" indicates factorial)

The initially few terms add up to: 1 + 1 + 12 + 16 + 124 + 1120 = 2.71666...

In reality Euler himself used this approach to calculate e to 18 decimal places.

You deserve to try it yourself at the Sigma Calculator.

Remembering

To remember the worth of e (to 10 places) just remember this saying (count the letters!):

Toexpresseremembertomemorizeasentencetomemorizethis

Or you have the right to remember the curious pattern that after the "2.7" the number "1828" appears TWICE:

2.7 1828 1828

And following THAT are the digits of the angles 45°, 90°, 45° in a Right-Angled Isosceles Triangle (no genuine reason, simply exactly how it is):

2.7 1828 1828 45 90 45

(An prompt means to seem really smart!)

Growth

e is offered in the "Natural" Exponential Function:

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Graph of f(x) = ex

It has this wonderful property: "its slope is its value"


At any type of allude the slope of ex equals the worth of ex :

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as soon as x=0, the value ex = 1, and the slope = 1once x=1, the worth ex = e, and the slope = eand so on..


This is true everywhere for ex, and helps us a lot in Calculus once we must discover slopes and so on.

So e is perfect for herbal growth, check out exponential growth to learn more.

Area

The location up to any x-value is additionally equal to ex :

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An Interelaxing Property

Just for fun, attempt "Cut Up Then Multiply"

Let us say that we reduced a number right into equal parts and then multiply those components together.


Example: Cut 10 right into 2 pieces and multiply them:

Each "piece" is 10/2 = 5 in size

5×5 = 25


Now, ... how can we get the answer to be as significant as possible, what dimension must each piece be?

The answer: make the components as cshed as feasible to "e" in dimension.


Example: 10


10 cut into 2 equal parts is 5:5×5 = 52 = 25
10 cut into 3 equal parts is 313:(313)×(313)×(313) = (313)3 = 37.0...
10 reduced right into 4 equal parts is 2.5:2.5×2.5×2.5×2.5 = 2.54 = 39.0625
10 cut into 5 equal components is 2:2×2×2×2×2 = 25 = 32

The winner is the number closest to "e", in this situation 2.5.

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

100 Decimal Digits

Here is e to 100 decimal digits:

Advanced: Use of e in Compound Interest

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


Imagine a wonderful bank that pays 100% interemainder.

In one year you might rotate $1000 into $2000.

Now imagine the financial institution pays twice a year, that is 50% and also 50%

Half-method via the year you have actually $1500, you reinvest for the remainder of the year and also your $1500 grows to $2250

You got more money, because you reinvested half way with.

That is called compound interemainder.


Could we gain even even more if we broke the year up right into months?

We have the right to use this formula:

(1+r/n)n

r = yearly interemainder price (as a decimal, so 1 not 100%)n = variety of periods within the year

Our half ybeforehand example is:

(1+1/2)2 = 2.25

Let"s attempt it monthly:

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

Let"s try it 10,000 times a year:

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

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

Why does that happen?

The answer lies in the similarity between:

Compounding Formula:(1 + r/n)n
and also
e (as n ideologies infinity):(1 + 1/n)n

The Compounding Formula is very like the formula for e (as n approaches infinity), simply via an added r (the interemainder rate).

When we decided an interest rate of 100% (= 1 as a decimal), the formulas came to be the exact same.

Read Continuous Compounding for even more.

See more: So Why Are Doctor Who Dvds So Expensive, Doctor Who: Series 1

Euler"s Formula for Complex Numbers

e likewise appears in this the majority of impressive equation:

eiπ + 1 = 0

Read more here

Transcendental

e is additionally a transcendental number.


Euler"s Formula for Complex Numbers Irrational Number Numbers Index