MATH IA

MATH IA EXPLORATION The beauty of Euler’s number e Candidate name: Cong Quoc Bao Nguyen Candidate number: School: Auckland International College School number: 001495 Session: Teacher: Mr Alan He

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 Bibliography.................................................................................................................................. 17

1

Introduction 1. Rationale One day, when I was studying about limits of sequences, my teacher asked our class to prove that

1 n

n

( ) 1+

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

noticed that when

n→∞ ,

1 n

n

( ) 1+

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 2

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 →∞

1 n

n

( )

lim 1+

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

( a x ) = d a x =a x × ln a '

dy

is 

e

'

→( ex ) =

If a=e

f ( x )=a

x

is:

d x x e =e × ln e=e x So the derivative of dy

x

The derivative of a common logarithm function '

d

1

( loga x ) = dy log a x= x × ln a So the derivative of

ln x

If a=e is

1 x 3

→ ( log e x )' =

f ( x )=log a x

is:

d 1 1 ln x= = dy x × ln e x

ex



The integration of a common exponential function x

∫ a x dx= lna a + C

x

is:

x

→∫ e dx=e +C

If a=e

So the antiderivative of

f ( x )=a x

e

x

is

x

e +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.

x

1

( )

f ( x )=√ x ¿ x x

f (x)

Find the global maximum of

f (x)

To find the global maximum of

I can consider a new function

1

( )

1 ln x g ( x ) =ln f ( x )=ln x x = × ln x= x x Since R, therefore

g( x)

( )

0< x e , g ’( x ) is negative and maximum for

g (x )

f ( x )=x

1 x

g ( x )=0 .

ln x ' ( ln x ) × x−( x ) × ln x g ( x )= = x x2 '

is an increasing function on

will reach its maximum value when

maximum. I can find the maximum of the equation

ln x

g( x)

g( x)

is increasing and for

g( x) is decreasing. Hence, the global

(and also for

f (x) ) is at

x=e , when g’(x) = 0.

f ( x )=x x Find the global minimum of Consider a new function

f (x)

g (x )

g ( x ) =ln f ( x )=ln ( x x ) =x × ln x Again since R,

g (x )

will reach its minimum when 4

ln x f (x)

is an increasing function on is at its minimum too. I can

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 )' 1 0 , g ’ ( x) e

is positive, which means

is negative and

1 x= , g ’ (x)=0 e

1 ¿ ln x+ x × =ln x+1 Therefore for x

g( x)

g( x)

is increasing and for

is decreasing. Hence at

f ( x)

and g(x) reaches its global minimum and

its global minimum when

x=

is also at

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: ∞

1 =1 x x=1 2

First we need ¿ prove that : ∑ n −1

¿

2

n−2

+2

n−3

+2 2n

+…+1 (

n→+ ∞ )

LHS=

¿

1 1 1 1 + 2 + 3 +…+ n ( n→+ ∞) 1 2 2 2 2

( 2−1 ) ( 2 n−1 +2n−2 +2n−3 +…+1 ) ( n →+ ∞ ) n 2

using formula ( i ) :an−bn=(a−b)(an −1 b+ an−2 b 2+ …+a2 b−2 +a b n−1) ¿ ∞

Number e can also be represented as e=∑

n=0

2 −1 1 =1− n ( n →+ ∞ ) n 2 2 ∞

1 1 ¿ lim 1− n =1→ ∑ x =1 n →∞ 2 x=1 2

( )

1 ( will be proved later ) n!

1 1 1 1 1 1 1 1 1 1 → + < e= + + + + …< + + + +… 1 1 1 1 1× 2 1× 2× 3 1 1 1× 2 1× 2× 2 →2< e