Basic Math Fibonacci

8/7/2019 Basic Math Fibonacci

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CONTENTS

TITLE PAGE

PART A

INTRODUCTION 3-8

THREE EXAMPLES OF CREATIONS 9-15

ESSAY ABOUT FIBONACCI SEQUENCES 16-17

PART B

INTEREST 19-20

PART C

REFLECTION 22

BIBLIOGRAPHIES 23


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RESOURCES 1

In mathematics, the Fibonacci numbers are the following sequence of numbers:

By definition, the first two Fibonacci numbers are 0 and 1, and each remaining

number is the sum of the previous two. Some sources omit the initial 0, instead

beginning the sequence with two 1s.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the

recurrence relation

with seed values

The Fibonacci sequence is named after Leonardo of Pisa, who was known as

Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio".) Fibonacci's 1202 book

Liber Abaci introduced the sequence to Western European mathematics, although

the sequence had been previously described in Indian mathematics.


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Origins

The Fibonacci sequence was well known in ancient India, where it was applied to the

metrical sciences (prosody), long before it was known in Europe. Developments

have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopla

(c.1135 AD), and Hemachandra (c.1150 AD).

The Fibonacci sequence is formed by adding S to a pattern of length n 1, or L to a

pattern of length n 2; and the prosodicists showed that the number of patterns of

length n is the sum of the two previous numbers in the sequence. Donald Knuth

reviews this work in The Art of Computer Programming.

In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in

his Liber Abaci (1202). He considers the growth of an idealised (biologically

unrealistic) rabbit population, assuming that:

y In the "zeroth" month, there is one pair of rabbits (additional pairs of

rabbits = 0).

y In the first month, the first pair begets anothe r pair (additional pairs of

rabbits = 1).

y In the second month, both pairs of rabbits have another pair, and the first pair

dies (additional pairs of rabbits = 1).

y In the third month, the second pair and the new two pairs have a total of three

new pairs, and the older second pair dies (additional pairs of rabbits = 2).

The laws of this are that each pair of rabbits has 2 pairs in its lifetime, and dies.

Let the population at month n be F(n). At this time, only rabbits who were alive at

month n 2 are fertile and produce offspring, so F(n 2) pairs are added to the

current population of F(n 1). Thus the total is F(n) = F(n 1) + F(n 2).


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List of Fibonacci numbers

The first 21 Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for

n = 0, 1, 2, ... ,20 are:

F

0

F

1

F

2

F

3

F

4

F

5

F

6F7 F8 F9

F1

0

F1

1F12 F13 F14 F15 F16 F17 F18 F19 F20

0 1 1 2 3 5 81

3

2

1

3

455 89

14

4

23

3

37

7

61

0

98

7

159

7

258

4

418

1

676

5

Using the recurrence relation, the sequence can also be extended to negative index

n. The result satisfies the equation

Thus the complete sequence is


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RESOURCES 2

The Fibonacci Sequence

edieval mathematician and businessman Fibonacci ( Leonardo Pisano)

posed the following problem in his treatise Liber Abaci (pub. 1202):

How many pairs of rabbits will be produced in a year, beginning with a single

pair, if in every month each pair bears a new pair which becomes productive from

the second month on?

It is easy to see that 1 pair will be produced the first month, and 1 pair also in the

second month (since the new pair produced in the first month is not yet mature), and

in the third month 2 pairs will be produced, one by the original pair and one by thepair which was produced in the first month. In the fourth month 3 pairs will be

produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get

the following sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...

This is an example of a recursive sequence, obeying the simple rule that to calculate

the next term one simply sums the preceding two:

F(1) = 1

F(2) = 1

F(n) = F(n 1) + F(n 2)

Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on.

This simple, seemingly unremarkable recursive sequence has fascinated

mathematicians for centuries. Its properties illuminate an array of surprising topics,

from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants

(not to mention populations of rabbits!).


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RESOURCES 3

THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN

The Fibonacci sequence exhibits a certain numerical pattern which originated as the

answer to an exercise in the first ever high school algebra text. This pattern turned

out to have an interest and importance far beyon d what its creator imagined. It can

be used to model or describe an amazing variety of phenomena, in mathematics and

science, art and nature. The mathematical ideas the Fibonacci sequence leads to,

such as the golden ratio, spirals and self - similar curves, have long been appreciated

for their charm and beauty, but no one can really explain why they are echoed so

clearly in the world of art and nature.

The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a

young man in his twenties, a member of an important trading family of Pisa. In his

travels throughout the Middle East, he was captivated by the mathematical ideas that

had come west from India through the Arabic countries. When he returned to Pisa he

published these ideas in a book on mathematics called Liber Abaci, which became a

landmark in Europe. Leonardo, who has since come to be known as Fibonacci,

became the most celebrated mathematician of the Middle Ages. His book was a

discourse on mathematical methods in commerce, but is now remembered mainly

for two contributions, one obviously important at the time and one seemingly

insignificant.

The important one: he brought to the attention of Europe the Hindu system for writing

numbers. European tradesmen and scholars were still cling ing to the use of the old

Roman numerals; modern mathematics would have been impossible without thischange to the Hindu system, which we call now Arabic notation, since it came west

through Arabic lands.

The other: hidden away in a list of brain-teasers , Fibonacci posed the following

question:


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2.EXAMPLE IN NATURE 1

Probably most of us have never taken the time to examine very carefully the number

or arrangement of petals on a flower. If we were to do so, several things would

become apparent. First, we would find that the number of petals on a flower is often

one of the Fibonacci numbers. One-petalled ...

white calla lily

and two-petalled flowers are not common.

euphorbia

Three petals are more common.

trillium


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There are hundreds of species, both wild and cultivated, with five petals.

columbine

Eight-petalled flowers are not so common as five-petalled, but there are quite a

number of well-known species with eight.

bloodroot

Thirteen, ...

black-eyed susan


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twenty-one and thirty-four petals are also quite common. The outer ring of ray floret s

in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13,

21, 34, 55 or 89 petals are quite common.

shasta daisy with 21 petals

Ordinary field daisies have 34 petals ... a fact to be

taken in consideration when playing "she loves me, she loves me not". In saying that

daisies have 34 petals, one is generalizing about the species - but any individual

member of the species may deviate from this general pattern. There is more

likelihood of a possible under development than over -development, so that 33 is

more common than 35.


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EXAMPLE IN NATURE 2

With the scale patterns of pinecones, the seed patterns of sunflowers and even the

bumps on pineapples we have something rather different.

The seed-bearing scales of a pinecone are really modified leaves, crowded together

and in contact with a short stem. Here we do not find phyllotaxis as it occurs with

true leaves and suchlike. However, we can detect two prominent arrangements of

ascending spirals growing outward from the point where it is attached to the branch.


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In the pinecone pictured, eight spirals can be seen to be ascending up the cone in a

clockwise direction ...

while thirteen spirals ascend more steeply in a counterclockwise direction.


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EXAMPLE IN NATURE 3

Pineapple scales are also patterned into spirals and, ...

because they are roughly hexagonal in shape, three distinct sets of spirals may be

observed.

One set of 5 spirals ascends at a shallow angle to the right, ...

a second set of 8 spirals ascends more steeply to the left, ...


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and the third set of 13 spirals ascends very steeply to the right.


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ESSAY

Fibonacci sequence is the sequence of numbers, 1, 1, 2, 3, 5, 8, 13, . . . , in which

each successive number is equal to the sum of the two preceding numbers.The

Fibonacci sequences are the sequence of numbers defined by the linear

recurrence equation

with . As a result of the definition,it is conventional to define .

The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ...

Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials

with . Fibonacci numbers are implemented in Mathematica as

Fibonacci[n].

The Fibonacci sequence makes its appearance in other ways within mathematics as

well. For example, it appears as sums of oblique diagonals in Pascals triangle:


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1.SIMPLE INTEREST

Simple Interest Made Simple!

y The formula for simple interest is I = Prt

y In order to determine any of the variables (I, P, r, t) you only need the other 3

y When you know three of the four values, here's how you calculate the

unknown.

o To find principal (P) P = I/rt

o To find the interest rate (r) r = I/Pto To find the period of time ( t) t = I/Pr

The easy way to remember the above is to look at it as Simple Interest -

Triangular Forumlas

IIIII

Prt

Think of I at the top of the triangle and P rtat the bottom. The formulas above

are easy to remember when you keep the triangle in your mind as your visual.

A word about time. When the time is 6 months. It will be entered a .6 If you

have specific days such as 215 then you will divide the days by 365 which

gives you .58 for time. Don't forget to count the days of the month properly -

30 days in September, April, June, November and except for Feb. all the rest

have 31!


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2.COMPOUND INTEREST

Today, calculators will do the computational work for you, however, here's a

breakdown of how to calculate compound interest:

Compound interest is interest that is paid on both the principal and also on any

interest from past years. Its often used when someone reinvests any interest they

gained back into the original investment. For example, if I got 15% interest on my

$1000 investment, the first year and I reinvested the money back into the original

investment, then in the second year, I would get 15% interest on $1000 and the $150

I reinvested. Over time, compound interest will make much more money than simple

interest. The formula used to calculate compound interest is:

M = P( 1 + i )n

M is the final amount including the principal.

P is the principal amount.

i is the rate of interest per year.

n is the number of years invested.

Applying the Formula

Let's say that I have $1000.00 to invest for 3 years at rate of 5% compound interest.

M = 1000 (1 + 0.05) 3 = $1157.62.

You can see that my $1000.00 is worth $1157.62.


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REFLECTION

First I would like to say thank you to Allah S.W.T because I can finish my

basic mathematic task before the final date to send the task. I also would like to

thank the lecturer and also my tons of friends who give me great help and support.

This basic mathematics task make me know deeper and know more about

mathematics and the way to solve the mathematics problems such as which involve

the Fibonaccis and simple also compound interest.

I also get know about mathematics are around me in the nature of Gods

creation. By doing this tasks, I have learnt about the most distinguished

mathematicians of the middle ages who is Leonardo of Pisa(1170-1250). . I also

found that the structure of pascals triangle is very interesting and useful .

I learnt how to use the formula to calculate compound interest, total instalment

price, finance charge. It will help me in the future when it is about problems in solving

money and all related to it in my daily life.

I also can cooperate better with my friends when it is about teamworking

condition to solve problems, my critical thinking skills are growing and expanding and

it sure give me and my friends help so then we can keep going better and better.

Positif thinking is generated from our hard working and teamwork. We now believe

that we can do other task better because of this ba sic mathematics experience task.


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BIBLIOGRAPHIES

y http://www.math.temple.edu/~reich/Fib/fibo.html

y http://en.wikipedia.org/wiki/Fibonacci_sequence

y http://www.mathacademy.com/pr/prime/articles/fibonac/

y http://www.about.com

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Basic Math Fibonacci