MATH IA

Candidate name: Cong Quoc Bao

Nguyen

Candidate number:

School: Auckland International College

School number: 001495

Session:

Teacher: Mr Alan He

MATH IA

EXPLORATION

The beauty of Euler’s

Table of Contents

Introduction... 2

1. Rationale... 2

2. Abstract... 2

3. History of number e... 3

Properties of number e... 3

1. Calculus properties of e... 3

2. Exponential-like functions... 3

3. e is irrational... 4

Applications of number e... 6

1. Compound interest... 6

2. Complex numbers... 7

3. Derangement... 9

4. Bernoulli trials... 12

5. Newton’s Law of Cooling... 12

Other representations of e... 13

1. As an infinite series... 13

2. As a symmetric limit... 14

3. As the sum of two hyperbolic functions...15

Conclusion... 15

Evaluation... 16

Introduction

1.

Rationale

One day, when I was studying about limits of sequences, my teacher asked our class to prove that

(

1+

1

_n

)

n

is an increasing sequence when n increases. While doing this problem, I also

noticed that when

n → ∞

,

(

1+

1

_n

)

n

also approaches a number approximates to 2.7182… Suddenly I recognized this number since it resembles a special number on the calculator that I had seen before. So I input every button on my calculator and I figured out that the special number is labelled e. That was my first encounter with what is called Euler’s number. Later on when I started learning calculus and logarithms, number e appeared again, this time I got to explored it in depth when I studied its calculus properties and its logarithmic meanings. What makes e a remarkable experience for me is when I had to use the differentiation definition to deduce the differentiation forms of the exponential and logarithms functions. It was not an easy task when at that time I did not know how e is related to these functions and I could not simplify the differentiation forms created. So when my teacher solved the problems by modifying the expression to get the limit of

(

1+

1

_n

)

n

when

n → ∞

, I was stunned by the solution since I could not believe that e has anything to do with what I was studying. I thought it was just a symbol for a normal limit and therefore does not have many usage in other studies. I was fascinated about e ever since because its characteristics are so unique and special. How could it possible that from a simple limit of a normal sequence possess such extraordinary qualities, such as having it derivative is itself? However this amazing number has never stopped to amuse me. During my IB mathematics course, I first came across the idea of complex number and once again number e occurred again in a totally different form with different usage. I wonder when would this number stop appearing in our lives, which maybe never.

So when I brainstormed to choose a topic for my Mathematics Internal Assessment, exploring number e is certainly one of my top choice. I really want to know further how this significant mathematical constant can be applied to our everyday lives. And on my course of gathering any information I can find about number e, I was continued to be surprised by how e relates to my life. It takes place in many activities such as savings model to scientific problems like

derangements or Newton’s Law of Cooling. Moreover, the more I learn about Euler’s number, the more I find it so graceful and disciplined. I call it disciplined because any problems related to it can be simplified to a very concise form. Even though when expressed explicitly, e seems like an “ugly”, irrelevant number (e = 2.7182…), when applying mathematics theories to investigate e, it will show distinctive and gorgeous properties that is so different from its irrational numeric form.

Another reason why I am so interested in number e is because of it official finder, Leonhard Euler. Since my first mathematics lesson in primary schools, I don’t know how many times have the name Euler appear in the books, such as geometry analysis like Euler’s theorem or Euler’s circle, and especially his famous mathematical constant, e. To me he appears as a perfect

mathematician that contribute so significantly to contemporary lives. He set the foundation of many aspects in math today and helped to solve numerous important problems. Therefore partly

I want to finish this exploration to show my admiration and respect to Euler and his accomplishments.

So from my great passion for Euler and his number, I have decided to choose number e as my topic for Mathematic IA and I hope I will get a chance to truly explore what I really love.

2. Abstract

In this exploration I will investigate number e by first considering some of it special properties, namely calculus properties, exponential-like functions, and its irrationality. Then I will consider its application into real lives in areas such as compound interest, complex numbers, derangement probability, Bernoulli trials and Newton’s Law of Cooling. After that I will explore try to prove some other ways to represent number e such as using infinite series, symmetric limit and hyperbolic functions.

3. History of number e

John Napier first referenced the constant in a table of his work on logarithms published in 1618. However this work does not contain the constant itself, it was just a list of logarithms evaluated from the constant. The table was assumed to be formulated by William Oughtred. Later on, the official discovery of the number was made by Jacob Bernoulli, who tried to find the value of the following limit (which is equal to e): _{n →∞}

lim

(

1+

1

_n

)

n

After that, the first recorded use of this constant, which was then represented as the letter b, was made by Gottfried Leibniz and Christiaan Huygens in 1690 and 1691. But it was not until

Leonhard Euler that the symbol e for this constant became well-known. Euler first used e to represent the base for natural logarithms when he wrote a letter to Christian Goldbach on 25 November 1731. From 1727 to 1728, in an unpublished document on explosive forces in guns, Euler started to use the letter e for the constant and e made its first official appearance in Euler’s Mechanica (1736). Nowadays e has become the standard for this constant and it is widely used.

Properties of number e

1. Calculus properties of e

 The derivative of a common exponential function

f ( x )=a

x is:

(

a

x

)

'

=

d

dy

a

x

=

a

x

× ln a

If a=e

→

(

e

x

)

'

=

d

dy

e

x

=

e

x

× ln e=e

x So the derivative of

e

x

is

e

x

 The derivative of a common logarithm function

f ( x )=log

a

x

is:

(

log

a

x

)

'

=

d

dy

log

a

x=

1

x × ln a

If a=e

→

(

log

e

x

)

'

=

d

dy

ln x=

1

x × ln e

=

1

x

So the derivative of ln x is

1

_x

 The integration of a common exponential function

f ( x )=a

x is:

∫

a

x

dx=

a

x

ln a

+

C

If a=e

→

∫

e

x

dx=e

x

+

C

So the antiderivative of

e

x is

e

x

+

C

From the above results, I can see the beauty of the number e in which its derivative and antiderivative is itself and it can simplify ln x into

1

_x

by the use of

differentiation.

2.

Exponential-like functions

These functions are based on exponential functions but have different shapes and properties and they can be investigated by using e :

i.

f ( x )=

√

x

x

(

¿

x

1

x

)

Find the global maximum of

f (x)

To find the global maximum of

f ( x )

I can consider a new function

g

(

x

)

=ln f

(

x

)

=ln

(

x

1

x

)

₌1

x× ln x= ln x

x Since ln x is an increasing function on

R, therefore g(x) will reach its maximum value when

f ( x )=x

1

x _is

maximum. I can find the maximum of

g (x )

by deducing

g

'(x) and solve the equation

g

'

(

x )=0

.

g

'

(

x )=

(

ln x

x

)

'

=

(

ln x )

'

× x−( x )

'

× ln x

x

2

_¿

1

x

× x−ln x

x

2

=

1−ln x

x

2 Therefore for

0<x <e , g ’(x)

_{is positive, which means}

g(x)

_{is increasing and for}

x>e , g ’(x )

_{is negative and}

g(x)

_{is decreasing. Hence, the global} maximum for

g(x)

(and also for

f (x)

) is at x=e , when g’(x) = 0. ii.

f ( x )=x

x

Find the global minimum of

f ( x )

Consider a new function

g (x )

g (x )=ln f ( x )=ln

(

x

x

₎

=

x × ln x

_{Again since}

ln x

_{is an increasing function on}

find the minimum of

g (x )

by calculating

g ’( x )

and solve the equation

g ’( x )=0

_.

g

'

(

x )=( x ×ln x )

'

=(

x )

'

_{× ln x +x × (ln x )}

'

_¿

_{ln x+x ×}

1

x

=ln x+1

Therefore for

0<x <

1

e

, g ’(x)

is positive, which means g(x) is increasing and for

x>

1

e

, g ’(x)

is negative and g(x) is decreasing. Hence at

x=

1

e

, g ’(x)=0

and g(x) reaches its global minimum and f (x) is also at its global minimum when

x=

1

_e

.

From the above examples, I can see that the natural base e can be applied to evaluate many properties of functions related to exponentials and logarithms

3.

e is irrational

The question whether e is rational or not had been an interest for mathematicians since e was first introduced by Jacob Bernoulli in 1683. Later on, Leonhard Euler, a student of Jacob’s younger brother Johann, managed to prove that e is irrational, which means is cannot be expressed as a quotient of two integers

 Proof:

First we need

¿

prove that :

∑

x=1 ∞

1

2

x

=1

LHS=

1

2

1

+

1

2

2

+

1

2

3

+

…+

1

2

n

(

n→+∞)

¿

2

n −1

+2

n−2

+

2

n−3

+

…+1

2

n

(

n→+∞)

¿

(

2−1)

(

2

n−1

+2

n−2

+2

n−3

+

…+1

)

2

n

(

n →+∞)

¿

2

n

−1

2

n

=1−

1

2

n

(

n →+∞)

using formula(i) :a

n

−

b

n

=(

a−b)(a

n −1

_{b+ a}

n−2

_b

2

+

…+a

2

_b

−2

+

a b

n−1

)

¿

¿

n →∞

lim

(

1−

1

2

n

)

=1→

∑

x=1 ∞

1

2

x

=1

Number e can also be represented as e=

∑

n=0 ∞

1

n!

(

will be proved later

)

→

1

1

+

1

1

<

e=

1

1

+

1

1

+

1

1× 2

+

1

1× 2× 3

+

…<

1

1

+

1

1

+

1

1× 2

+

1

1× 2× 2

+

…

→2<e<2+

1

2

+

1

4

+

1

8

+

…

→2<e<2+1 →2<e<3 → e is not an integer Suppose e is a rational number

→∃ a∧b so that e=

a

b

(

b≠ 1 because e is not an integer)

Then define a number x=b !

(

e−

∑

_n=0

b

1

¿

e=

a

b

→ x=b !

(

a

b

−

∑

n=0 b

1

n!

)

=

a (b−1) !−

∑

n=0 b

b !

n !

The first term a (b−1)!is an integer since a∧b are integers∧each

element of

_∑

n=0 b

b !

n !

is also aninteger since n≤ b

→ x is aninteger

Now I will provethat 0<x <1.

¿

do that first I will use the formula

e=

_∑

n=0 ∞ 1 n !

→ x=b !

(

_∑

n=0 ∞

1

n !

−

∑

n =0 b

1

n !

)

=

b !×

n=b+1

∑

∞

1

n !

=

n=b +1

∑

∞

b !

n!

→ x >0 because each term of n=b+1

∑

∞

b !

n !is positive

Now I need

¿

prove that x <1

∀ n≥ b+1 we can obtain anupper limit

b!

n!

=

1

(

b+1) (b+2) (b+3) …

(

b+( n−b )

)

<

1

(

b+1) (b+1) (b+1) … (b+1)

b +1

¿

¿

¿

→

b !

n !

<

1

¿

Let k =n−b

→ x=

∑

n=b+1 ∞

b !

n !

<

n=b+1

∑

∞

1

(

b+1)

n−b

=

∑

k=1 ∞

1

(

b+1)

k

Consider L=

∑

k=1 ∞

1

(

b+1)

k

→ L=

1

b +1

+

1

(

b+1)

2

+

1

(

b+1)

3

+

…+

1

(

b+1)

n

(

n→ ∞)

→ L=

(

b+1)

n−1

+(

b+1)

n−2

+

(

b+1)

n−3

+

…+1

(

b+1 )

n

→ L=

[

(

b+1)−1

](

(

b+1)

n−1

₊₍

_b+1)

n−2

₊₍

_{b+1 )}

n−3

+

…+1

)

(

b+1)

n

_{× b}

→ L=

(

b+1)

n

₋₁

(

b+1)

n

_{× b}

(

using formula (i)

)

→ L=

1

b

−

1

(

b +1)

n

× b

whenn → ∞

¿

n →∞

lim

1

(

b+1)

n

× b

=0 → L=

1

b

→ x <L → x <

1

b

<

1(becase b ≠ 1)

→

{

0<x <1

x is an integer

→there is a contradiction

→The hypothesis that e isirrational is wrong∧thereforee is irrational .

Applications of number e

The number e is extremely unique in mathematics because it can be applied to many aspects of our lives. Some examples of applications of number e are explored below:

1. Compound interest

The study of compound interest was the first time that the natural exponential e had been discovered therefore I think that it is meaningful to investigate compound interest and see how it is related to e

 The formula to calculate the amount of money invested after a period of time by compound interest is

u

n

=

u

0

× (1+i)

n

with:

n isthe number of periods that theinterest is added

iis theinterest rate per period

u

₀

isthe initial deposit amount

u

_n

isthe final amount after n periods

 In the case of continuous compound interest, the interest rate is compounded per very small intervals of time and I can change the formula to take account for the small intervals of time:

Let

r be the percentage rate per year

t be the number of years

N be the number of interest added per year

→i=

r

N

∧

n=Nt

→u

n

=

u

o

×

(

1+

r

N

)

Nt

Let a=

N

r

→u

n

=

u

0

×

(

1+

1

a

)

art

→u

n

=

u

0

×

[

(

1+

1

a

)

a

]

rt

For continous compound growth, since the interest is added per very small intervals of time, the number of interest added per year (N) will get very large

N →+∞then a→+∞

(

since a=

N

r

∧

r is a constant becausei isalso a constant

)

→u

n

=

u

0

×

[

lim

a→+ ∞

(

1+

1

a

)

a

]

rt

However from the definition of e I know that

e= lim

n →+∞

(

1+

1

n

)

n

→u

n

=

u

0

e

rt

 Example: Diagram for the effect of earning 20% yearly interest on a deposit of 1000$ at different frequencies. The green curve represents the compound interest

when the interest is counted continuously and it has the shape of a

a × e

bx graph

 So from the investigation I can see that if the interest rate is paid continuously then the final amount can be calculated by the formula

u

n

=

u

0

e

rt where r is the interest

rate per year and t is the number of years.

 This finding is significant because we can know the limit of an amount of money saved in a bank and get interest paid per different lengths of time. If the interest is computed more frequently during a year, we can get more money from the same amount of deposit money and the maximum limit we can get is when the interest is paid continuously and the final amount is calculated by

u

n

=

u

0

e

rt

2. Complex numbers

Another important application of e is that it can be used in expressing complex numbers in a simpler form

 In the study of complex number I have learned that a complex number can be written in Euler’s form by using Euler’s formula:

Euler

'

s formula: e

ix

=

cosx+isinx

 Proving Euler’s formula

Consider

e

ix as a complex number. Since any complex number z can be express in polar form as

z=

|

z

|

×(cosα +isinα )

we can rewrite Euler’s formula as

e

ix

=

r (cosθ +isinθ)

(

r ( x )∧θ ( x )are functionsof x

)

_{Differentiating both sides gives}

→

(

e

ix

₎

'

=(

r )

'

(

cosθ+isinθ )+(cosθ+isinθ )

'

_×r

→ie

ix

=

dr

dx

(

cosθ+isinθ)+

(

−

sinθ×

dθ

dx

+

icosθ ×

dθ

dx

)

× r

¿

since e

ix

=

r (cosθ+isinθ )

→i ×r (cosθ+isinθ)=

dr

dx

× cosθ−sinθ ×

dθ

dx

× r +isinθ ×

dr

dx

+

icosθ×

dθ

dx

×r

→ircosθ−rsinθ=

dr

dx

×cosθ−rsinθ ×

dθ

dx

+

i

(

sinθ×

dr

dx

+

rcosθ ×

dθ

dx

)

Equatingreal∧imaginary parts

→

{

−

rsinθ=

dr

dx

cosθ−rsinθ

dθ

dx

(1)

rcosθ=sinθ

dr

dx

+

rcosθ

dθ

dx

(2)

→

{

−

rsinθ

(

1−

dθ

dx

)

=

dr

dx

cosθ (3)

rcosθ

(

1−

dθ

dx

)

=

dr

dx

sinθ(4)

If

(

1−

dθ

dx

)

≠

0∧dr

dx

≠ 0 we can divide (3 )by (4 )

(

r ≠ 0 because r isthe modulus of e

ix

)

→−

_cosθ

sinθ

=

cosθ

sinθ

→

cos

2

θ+sin

2

θ

cosθ × sinθ

=0

→

1

cosθ × sinθ

=0, which is impossible!

→

(

1−

dθ

dx

)

=

0∧dr

dx

=0

→

dθ

dx

=

1∧dr

dx

=0

And we also have

e

i 0

=

e

0

=1

→r (cosθ +isinθ)=1, when x=0

→

{

rcosθ=1

rsinθ=0

→ sinθ=0(r ≠ 0 becauseif r =0 thenrcosθ wilbe 0)

→θ (0 )=0∨θ (0)=π → cosθ=1∨cosθ=−1, when x=0

→r

(

0

)

=1 whenθ

(

0

)

=0∧r (0)=−1 whenθ (0)=π

dθ

dx

=1→ θ ( x )=x+c wherec is a constant

dr

dx

=0→ r ( x )=d

∀ x where d is a constant

If r (0)=1∧θ(0)=0 :

→r ( x )=r (0)=1∧θ ( x )=x (becauseθ ( 0)=0 → c=0)

→ e

ix

=

r (cosθ+isinθ )=cosx+isinx

If r(0)=−1∧θ (0)=π

→r

(

x

)

=r

(

0

)

=−1∧θ

(

x

)

=x+π (becauseθ

(

0

)

=π → c=π )

→ e

ix

=

r (cosθ+isinθ )=−cos ( x +π )−isin(x +π )

→ e

ix

=

cosx+isinx

So in conclusion, both cases lead to the confirmation of Euler’s formula

 Euler’s formula is extremely useful in representing complex numbers in Euler’s form:

z=x +iy=

|

z

|

(

cosθ +isinθ)=

|

z

|

e

iθ

where

x=ℜ{z

}is the real part

(

x coordinate

)

of z

y=ℑ

{

z

}

is theimaginary ( y coordinate) of z

|

z

|

=

√

x

2

+

y

2

_{is the modulus of z}

vector z measured counterclockwise∧

¿

radians

Type equation here .

 Euler’s form can simplify the multiplication of complex numbers

zw=

|

z

|

e

iθ

_×

_|

_w

_|

_e

i∅

=

|

z

||

w

|

e

i(θ+ ∅)

_→

_|

_zw

_|

₌

_|

_z

_||

_w

_|

_∧

_{arg ( z+w)=θ+∅=argz+argw}

 It can also be used to prove De Moivre’s Theorem:

A complex number can be written in polar form: z=

|z|cisθ where

cisθ=cosθ+isinθ=e

iθ

De Moivr e

'

s Theorem:

(

|

z

|

cisθ

)

n

=

|

z

|

n

cisnθ for all rational n

Pr oof:

¿

z∨

¿

n

e

i(nθ)

=

|

z

|

n

cis (nθ)

(

|

z

|

cisθ)

n

=(

z)

n

=(

|

z

|

e

iθ

)

n

=

¿

 Another use of number e in complex number is to define the logarithm of a complex number

From the definition and characteristics of logarithms, I know that

a=e

ln a

∧

e

a

e

b

=

e

a+b

→ z=

|

z

|

e

iθ

=

e

ln|z|

_{× e}

iθ

=

e

ln|z|+iθ

_{→ ln z=ln}

_|

_z

_|

₊

_iθ

→this can be used as a definition for the complex number logarithm∧I can see that it has multiple values since θ ismulti−valued .



Also using Eule r

'

sformula when x=π we will get a famous equation∈

¿

mathematics , call Euler

'

_{sidentity :}

e

iπ

=

cosπ +isinπ

∴ e

iπ

+1=0

Thisequation is so beautiful∧famous becauseit links3 important

number of mathematics (e ,i∧π )∈such an elegant∧simple way .

 From the above investigation I have found out that number e is a significant part in the study of complex numbers and once again it has shown its beauty when it can be applied to so many aspects of math.

3. Derangement

Number e also occur in combinatorial mathematics and probability where it is related to counting the derangement of a set with n elements when n approaches +∞

Derangement is a rearrangements of a set such that no elements appear in its original position.

The total number of derangements of a n size sets is usually denoted as Dn, dn or !n and is

named “derangement number” or “de Montmort number”  Counting derangement:

o For a set of n elements, the formula to calculate Dn is:

D

_n

=

n !

_∑

i=0 n

₍₋₁₎

i

i!

o Proof:

 To prove the above formula first I will prove the recurrence relationship of derangement:

D

n

=(

n−1)

(

D

n−1

+

D

n−2

)



Consider a set S=

{

1,2,3, … , n

}

with a cardinality n



D

0

=1 since if n=0 → S=∅∧thereis only one permutation

¿this is alsoa derangement becauseno elements are∈¿ original position



D

1

=0 sinceif n=1 → S=

{

s

}

→there isonly one

permutation for S∧it is not a derangement



If n=2 → S=

{

s;t

}

→ thereare two permuations for S which

are

{

s ;t

}

∧

{

t ;s

}

∧

onlythe latter is a derangement

→ D

₂

=1

 Now let f : S → S be a derangement for set S ,

which means f ( s) ≠ s∧consider a random s∈ S suchthat

s ≠1∧f (s )=1.

By the summation rule of counting , thetotal number of f will the number of f when

be :

(

the number of f when f

(

1

)

≠ s

)

+¿

f (1)=s

¿



Situation 1: f (1)≠ s

¿

Let G

1

=

S }=

{

1, 2,3, … , s−1, s+1,… ,n

}

∧

take a

derangement g

1

:G

1

→ G

1

, which means g

1

(

t)=f (t )

∀ t ∈G

1

→ g

₁

is a derangement of a set of (n−1) elements

→The number of derangements for G

1

is D

n−1

→ by the product rule of counting , there are(n−1) D

_n−1

number of f when f (s )≠ 1



Situation 2: f (1)=s

Similarly but this timelet

¿

G

2

=

S s}=

{

2,3, … , s−1, s +1, … , n

}

Take a derangement g

₂

:G

₂

→G

₂

by taking g

₂

(

t )=f (t )

∀ t ∈ G

₂

→ g

₂

is a derangement of a set of (n−2) elements

→The number of derangements for G

₂

is D

_n−2

Also s can be chosen∈(n−1) ways

→ by the product rule of counting , there are(n−1) D

_n−2

number of f when f (s )=1

 So ∈total by summing upthe two results , the totalnumber of

derangemen for a set of n elementsis

D

_n

=(

n−1)

₍

D

_n−1

+

D

_n−2

₎

 Now from the recurrence relationship

D

n

=(

n−1)

(

D

n−1

+

D

n−2

)

, I will prove

the formula by first subtracting

n D

n−1 from both sides

→ D

_n

−

n D

_n−1

=(

n−1) D

_{n −1}

+(

n−1) D

_n−2

−

n D

_n−1

¿

D

_n−1

(

n−1−n )+(n−1) D

_n−2

¿

−

₍

D

_n−1

−(

n−1) D

_{n −2}

₎

Let A

_n

=

D

_n

−

n D

_n−1

→ A

_n

=−

A

_n−1

A

0

=

D

0

−0 × D

−1

=1∧we have a sequence A

n

=−

A

n−1

, A

0

=1

→ A

_n

=−

A

_n−1

=

A

_n−2

=

…

→ A

_n

=(−1)

1

_A

n−1

=

(−1)

2

_A

n−2

=

…=(−1)

n

_A

0

→ A

_n

=(−1)

n

_{→ D}

n

−

n D

n−1

=(−1)

n

_{Divdingboth sides by n !}

→

D

n

n !

−

n D

_n−1

n !

=

(−1)

n

n!

→

D

_n

n !

−

D

_n−1

(

n−1) !

=

(−

1 )

n

n!

Let B

n

=

D

n

n !

→ B

_n

−

B

_n−1

=

(−1)

n

n !

 Probability to have a derangement:

o

The totalnumber of ways

¿

rearrange a set of n elements is n !

→The probability¿have a derangement is:

P

_n

=

D

n

n !

=

∑

i=0 n

(−1)

i

o

Whenn → ∞, the probability becomes :

P

n

=

∑

i=0

∞

₍₋₁₎

i

i !

=

1

e

(

which will be proved later )

 O nce againthe miraculous number e appeared∈another subject of mathematics ,

which is combinatorial math∧ probability . Moreover¿this investigation I have raised some more questions on whether number e can be related¿some more problems about probability∧I will explore further on this subject by considering Bernoulli trials below .

4. Bernoulli trials

Bernoulli trial is a random experiment with only two possible outcomes, “success” and “failure”, and the probability for “success” is the same for each experiment. A typical example of Bernoulli trials is to imagine a gambler plays a slot machine with a chance of winning of 1/n and plays it for definite times or indefinitely. I will investigate Bernoulli trials when the chance of “success” is 1/n and the experiment is conducted n times (n → ∞) and see how it is related to number e.



First I will formulate the probability

¿

win k ×is Bernoullitrials .

o

Suppose the probability

¿

get a “ success ” for each experiment is

1

_p

o The experiment is conducted n

o

Then

¿

n experiments conducted , there are C

nk

ways

¿

choose k successful

experiments

¿

n of them

o

The probability

¿

get k successful experiments is

(

1

_p

)

k

o

The probability

¿

get (n−k ) failed experiments is

(

1−

1

_p

)

n−k

→the probability of winningk ×out of n×is :

P

_k

=

C

_nk

_×

(

1

p

)

k

×

(

1−

1

p

)

n−k



If n= p

→ probability for each successis

1

_n

∧

the experiment is conducted n×

¿

→the formulabecomes : P

_k

=

C

_nk

×

(

1

n

)

k

×

(

1−

1

n

)

n−k



If k=0

→the formulawill indicate the probability

¿

success 0×out of n×

¿

P

0

=

(

1−

1

n

)

n 

Whenn → ∞

→ P

0

=

lim

n →∞

(

1−

1

n

)

n

=

1



Therefore ,∈the special case of Bernoullitrials when the experiments with

the probability

1

n

is conducted n×.When n→ ∞ , the probability

¿

get zero

success is

1

e

 So¿this explorationabout Bernoulli trials , I have found out another application

of number e , even thoughthe relationshipis not so clear as∈the study about

compound interest ∨complex numbers

5. Newton’s Law of Cooling

Besides interesting relationship to statistics and probability mathematics, number e can also be applied to other fields of math-related science by making use of it to solve differential equation. A typical example is Newton’s Law of Cooling.

 Newton’s Law of Cooling: the rate of change of the temperature of an object is directly proportional to the difference between its temperature and the temperature

of the surroundings.

Let

T (t )=temperature of theobject at timet ( s )

T (0)=T

₀

=

initial temperature at T =0

T

_s

=

temperature of the surrondings

_{→ By Newto n}

'

s Law of Cooling

dT

_dt

(

T−T

_s

)

where

dT

dt

is the differentiation of T withrespect

¿

t∧it representsthe

rate of change of the temperature

Note that if T −T

_s

>

0 thenthe object is cooling →

dT

dt

will be negative

→

dT

dt

=−

k (T −T

s

)

 Now I will try to deduce a formula to calculate T with respect to t by solving a differential equation that relates to number e.

Let

y (t )=T (t)−T

_s

=

the temperaturedifference at time t

y

₀

=

T (0)−T

_s

=

initial temperaturedifference at time t=0

Takethe derivative of y (t)

→

dy

dt

=

d

dt

(

T (t )−T

s

)

=

dT

dt

−

d T

_s

dt

=

dT

dt

=−

k

(

T −T

s

)

=−

ky

(

d T

_s

dt

=0 because T

s

is a constant

)

→

dy

dt

=−

ky

Thisis a differential equation∧canbe solved

→

dy

y

=−

kdt

→

∫

1

y

dy =

∫

−

k dt

→ ln

|

y

|

=−

kt +C (C is a constant )

→

|

y

|

=

e

−kt +C

Taking the positive value of y¿get a situation of a cooling object

→ y (t )=e

C

× e

−kt

At t=0→ y

0

=

e

C

→ y (t )= y

0

e

−kt

→T (t)−T

s

=

(

T

0

−

T

s

)

e

−kt

→T (t)=T

s

+

(

T

0

−

T

s

)

e

−kt

if we have the values of T

_s

(

thetemperature of the surroundings) ,

(

T

₀

−

T

_s

)

a constant related

¿

(

theinitial temperature difference )∧k

¿

process

¿

Other representations of e

Number e is also special because it can be represented in many ways and some of which are explored below.

1. As an infinite series

e=

_∑

n=0 ∞

1

n !

 From the definition of e I know that:

e=lim

n → ∞

(

1+

1

n

)

n

 And according to binomial expansion:

(

1+

1

n

)

n

=

∑

i =0 n

C

_ni

×

(

1

n

)

i  Then consider

C

n i

_×

(

1

n

)

i

¿

n !

(

n−i)!× i!

×

1

n

i

¿

n(n−1) (n−2) … (n−i+1)

i!× n

i

¿

n

n

×

n−1

n

×

n−2

n

× … ×

n−i+1

n

i!

¿

1

(

1−

1

n

)(

1−

2

n

)

…

(

1−

i−1

n

)

i!

because lim n→ ∞

(

1− a n

)

=1∀ a → lim n →∞Cn i ×

(

1 n

)

i = 1 i!¿ 

e=lim

_{n → ∞}

(

1+

1

_n

)

n

→ e=lim

n→ ∞

∑

_{i =0} n

C

_ni

_×

(

1

n

)

i

→ e=

∑

i=0 ∞

[

lim

n → ∞

C

n i

×

(

1

n

)

i

]

→ e=

∑

i=o ∞ 1 i !

 Therefore e can be represented as

e=

_∑

n=0 ∞ 1 n !  Also we have:

e

x

=

[

lim

n→ ∞

(

1+

1

n

)

n

]

x

=

lim

n → ∞

(

1+

1

n

)

nx

Let

1

n

=

x

t

→ nx=t →t → ∞ when n→ ∞

→ e

x

=lim

(

1+

x

t

)

t

→ e

−1

=

1

e

=lim

t → ∞

(

1−

1

t

)

t

(

whichwas used ∈Bernoulli trialinvestigation)

¿

now similar

¿

our method

¿

deduce e=

∑

n=0 ∞

1

n !

→

1

e

=

∑

n=0 ∞

₍₋₁₎

n

n !

(

which wasused∈the derangement probability formula)

2.

As a symmetric limit

e=lim n → ∞

[

(

n+1

)

n+1 nn − nn

(

n−1

)

n−1

]

 We have:

(

n+1)

n+1

n

n

−

n

n

(

n−1)

n−1

¿

(

n+1

n

)

n

×( n+1)−

(

n

n−1

)

n−1

×n

¿

(

1+

1

n

)

n

×( n+1)−

(

1+

1

n−1

)

n−1

× n

→ lim n→ ∞

[

(

n+1

)

n +1 nn − nn

(

n−1

)

n−1

]

¿

lim

n →∞

(

1+

1

n

)

n

×(n+1)−lim

n →∞

(

1+

1

n−1

)

n−1

× n

¿

e × (n+1)−e ×n

¿

e

→ e=lim

n→ ∞

[

(

n+1)

n+1

n

n

−

n

n

(

n−1)

n−1

]

3.

As the sum of two hyperbolic functions

e

x

=sinh (x)+cosh (x )

 Deriving the hyperbolic functions: From Euler’s formula we have

e

ix

=

cosx+isinx

e

−ix

=

cosx−isinx

→ cosx=

e

ix

+

e

−ix

2

∧

sinx=

e

ix

−

e

−ix

2 i

→ cos (ix)=

e

x

+

e

−x

2

=cosh (x )

¿

sin (ix)=

e

−x

−

e

x

2i

=

−

e

x

−

e

−x

2i

=

isinh(x )

 Formal definitions of hyperbolic functions:

sinh ( x )=

e

x

−

e

−x

2

(

x )=

¿

e

x

+

e

−x

2

cosh

¿

Conclusion

In conclusion, this exploration has helped me to learn deeper about the mathematics of compound interest, complex numbers, derangement, Bernoulli trials and Newton’s Law of Cooling by making use of the properties of the amazing mathematical constant, e. I have also learned how e can be otherwise represented, even in some extraordinary expressions.

Moreover, this investigation has given me a memorable experience when I explored as far as I could about e and it also taught me many valuable lessons. I had improved my mathematical skills and knowledge as I tried to individually solve many problems involving e. Also I have learned that e is a pivotal element of solutions to many problems in our lives today and through this meaningful investigation, I have seen the beauty of e and mathematics and therefore I appreciate how lucky I am to have received such education in mathematics and other subjects as well.

My above study about number e has shown its significance in real life when I had demonstrated some of its application, which mostly is to calculate realistic models in an ideal condition, such as calculating the compound interest when the interest is computed continuously or formulating Newton’s Law of Cooling on a small range of temperature difference.

Evaluation



   Strengths:

o My exploration has some strengths in that I have:

 Try to implement e on a wide range of topics

 Provide concrete and explicit proof for each problem

 Make use of materials learned during IB Diploma Program

 Weaknesses:

o However there are some flaws in my work as well because I haven’t:

 Test all theory on realistic data

 The applications are mostly based on ideal situations such as:

 e only appears in compound interest formula when the interest is

compounded continuously

 For derangement and Bernoulli trials formulae, e only appears when

n → ∞

 In these cases, e can only be used to model the problem in the above cases, not when the interest is counted yearly or n in derangement or Bernoulli trials is too small (for example n=1,2)

 Study more about relative mathematicians, such as Leonhard Euler or Jacob Bernoulli.

o I acknowledge my problems and I have try to minimize them and I will try to do it better for my next assignment.

Bibliography

 Websites and Webpages

o J J O'Connor and E F Robertson September 2001 http://www-history.mcs.st-and.ac.uk/HistTopics/e.html (accessed 10-4-2015) o WolframMathWorld April 17 2015 http://mathworld.wolfram.com/SteinersProblem.html (accessed 20-4-2015) o ProofWiki 19 September 2013 https://proofwiki.org/wiki/Recurrence_Relation_for_the_Number_of_Derangements_o n_a_Finite_Set (accessed 15-4-2015) o MathematicsStackExchange August 28 2012 http://math.stackexchange.com/questions/83380/i-have-a-problem-understanding-the-proof-of-rencontres-numbers-derangements (accessed 15-4-2015)

o UBC Calculus Online Course N/A

http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html

(accessed 17-4-2015)

o WolframMathWorld April 17 2015 http://mathworld.wolfram.com/e.html (accessed 19-4-2015)

o WolframMathWorld April 17 2015

http://mathworld.wolfram.com/HyperbolicFunctions.html

(accessed 20-4-2015)

o MathForum N/A http://mathforum.org/isaac/problems/eproof.html (accessed 15-4-2015)

 Books and Encyclopedia

o Remnert, Reinhold 1991 Springer Science + Business Media “Holomorphy of power series” Theory of Complex Functions Berlin, Heidelberg p136

o Martin, D. Haese, R. Haese, S. Haese, M. Humphries, 2012 M. HAESE MATHEMATICS “Exponentials” IB Mathematics High Level Australia pp116-118

o Gilbert, S. 1991 Wellesley-Cambridge Press “The Taylor Series for e^x, sin x, and cos x” Calculus United States p389

o Papoulis, A. 1984 McGraw-Hill “Bernoulli Trials” Probability, Random Variables, and Stochastic Processes New York pp57-63