JANUARY 2013 VOL 1 NO 1

Or actually, there were mathematical object that enabled us to count, label, measure and compare things of varying size, dimensions and makeup. By assigning a value to an object, we were able to compare the “value” of one object against another either directly or indirectly for the purpose of exchange.

As time progressed, man noticed that certain patters were emerging from such comparisons and contemporary

mathematicians tried to formulate statements about these abstract concepts and such formulation were subject to verification by proof. Hence, for centuries, Mathematics was considered to be the science of numbers, magnitude and form.

Even today, people are trying to analyze the financial markets so as to be able to “predict” and preempt the variations, so as to make some financial gain on their

daily trading activities. These “proofs” were usually predicated on some form of pattern recognition and relationships were assumed based on the generic mathematical models that were known to the individuals at that time.

As these newsletters are an attempt to make the modeling process accessible to students, we will begin by first modeling a very well known number sequence.

In the beginning there were numbers ...

THE BELOVED APPLE

ON A LIGHTER NOTE ...

The Final Word ...

EXAMPLES OF THIS PATTERN

Ab Initio

To begin with, we will first need to

define what is the fibonacci sequence

1 1 2 3 5 8 13 21 34 55 89 ...

We can see that the first 2 terms of the

sequence is 1 and 1 and each

subsequent term is the sum of the

previous 2 terms. This can be

expresses mathematically as a LINEAR

SECOND ORDER DIFFERENCE

EQUATION

fn = fn!1 + fn!2 (1)

WHAT MIGHT SOME PROBLEMS BE

IN EXPRESSING THE RELATIONSHIP

IN THIS MANNER?

The main problem with this format is

that in order to find a specific term in

the sequence, we will need to generate

all of the previous terms. This can be

time consuming and there has to be a

better way.

HOW DO WE EVEN START

GENERATING THE ALGEBRAIC

RELATIONSHIP?

Initially, we will need to compare the

RATIO of each term against its

previous and subsequent term. We

can use NUMBERS to make this job

easier. What we notice is that as we

generate more of the sequence, the

ratio of the different terms of the

sequence will CONVERGE towards a

certain value.

fnfn!1

=1.61803398... (2)

fn!1fn

= 0.61803398... (3)

NOTE – (2) is the GOLDEN RATIO

If you plotted the points in Numbers or

GeoGebra, you will see that the graph

looks EXPONENTIAL.

Hence, we are able to HYPOTHESIZE

that the Fibonacci number series is an

exponential relationship.

THE INITIAL THOUGHT

It’s hard to work with an order 2 linear

difference equation, so initially, can we

simplify the relationship to LOOK LIKE

and simpler FIRST ORDER

DIFFERENCE EQUATION?

fn = afn!1 ; n "1 (4)

Because

f1 = af0f2 = af1 = a

2 f0f3 = af2 = a

2 f1 = a3 f0

!

fn = an f0

! fn " kn where k #$+

Hence, if

fn = fn!1 + fn!2

this means that

kn = kn!1 + kn!2

k 2 = k +1k 2 ! k !1= 0

k =1± !1( )

2! 4 1( ) !1( )2 1( )

k = 1± 52

Hence, the 2 roots are written as

k1 =

1+ 52

= !

& k2 =

1! 52

= !

Hence, the general equation needed to

generate a specific term in the

Fibonacci Sequence would take the

form

fn = fn!1 + fn!2

and this is a linear combination of the 2

solutions already found

fn = A!n + B" n (5)

where A and B are real numbers.

THE FINAL LAP

To find the value of A and of B, we will

need an INITIAL condition for f0 .

For instance,

f0 = 0 and f1 =1

This means that

A+ B = 0!!!!! (6)!A+"B =1!!! (7)

Solving (6) and (7) simultaneously,

A= !15

and B = 15

Therefore

fn =!15! n +

15! n

=15! n !! n( )

This is still very cumbersome, and

hence, there needs to be a simpler

approximation.

Now, because !1< ! < 0,

! n! 0 as n!"

hence,

fn !15! n( ) .

and the numbers were related ...

Ab Initio

EXAMPLES OF THIS PATTERN

If you searched the internet for examples and application of the Fibonacci spiral and sequence, you would find a plethora of examples from a variety of different sources.

You will begin to appreciate that the Fibonacci sequence is applicable to topics as distinct as plant ecology and the nature of the movement in the financial and stock markets.

You will begin to understand that in the apparent chaos of the world, there is some order to the “madness” and that order can be structured and summarised according to the Fibonacci number sequence.

As a closing remark and activity, there is a worksheet attached to this article

that enables you to explore the different applications that utilises the Fibonacci number sequence.

The Final Word ...

FUTURE EDITIONS RIPPLES IN THE POND

If you threw 2 stones into the school pond (and I’m not asking you to do this), it will cause ripples to emerge. These ripples will cascade outwards towards the edges, intersecting at certain point(s).

WHERE WILL THE RIPPLES INTERSECT?

HOW FAR FROM THE IMPACT POINT WILL THE INTERSECTION(s) TAKE PLACE?

WHAT FACTORS WILL AFFECT THE OUTCOME OF YOUR EXPERIMENT?

In conclusion –

As this newsletter is to highlight the applicability of Mathematics in your daily lives, YOU are welcomed to write articles, thoughts and reflections for the future editions of the newsletter. Find something that interests you and we’ll find the Mathematics that goes with it. Email ideas and submissions to jason_ingham@sst.edu.sg