Fibonacci Numbers


1. Introduction 
 1. About Fibonacci 
 1. Time period (Hog 1) 
 2. Fibonacci's Life (Hog 1), (Vaj 9) 
 3. The Liber Abaci. 
 2. The Fibonacci Sequence 
 1. When discovered (Vaj 7) 
 2. Where it appears (Vaj 9) 
 3. The Golden Ratio 
 1. When discovered 
 2. Where it appears 
 4. Thesis: The Fibonacci Sequence and The Golden Ratio
are very much a part of one another, and these two
phenomena have many applications in math and nature. 
 2. The Simple explanations 
 1. How we arrive at the Fibonacci Sequence 
 1. Explained in English 
 2. Formula 
 2. The Simplest Properties of the Fibonacci Numbers 
 1. u1+u2+...+un=un+2-1 (Vor 6) 
 2. The sum of Fodd: u1+u3+u5+...+u2n-1=u2n (Vor 7) 
 3. Sum of Feven: u2+u4+...+u2n=u2n+1-1 (Vor 7) 
 4. u12+u22+...+un2=unun+1(Vor 8) 
 3. How we arrive at the Golden Ratio 
 1. Explanation of it. 
 1. Explanation of Golden Rectangle 
 2. Formula to get it. - Solve for X with a formula
and Diagram 
 3. How to get from Fibonacci series to Golden Ratio. 
 4. Properties of the Golden Ratio 
 1. a 2=a+1 (Hog 9) 
 2. 1/a =a -1 
 3. b 2=b +1 
 3. Fibonacci Numbers 
 1. In Math 
 2. In Nature 
 4. The Golden Ratio 
 1. In Math 
 2. In Nature 
 5. Conclusion 
One would be hard pressed to name any significant
mathematicians from the middle ages, the thirteenth
century. Indeed, they were few and far between. The
solitary light during this dark age of mathematical and
scientific void was Leonardo Fibonacci, also known as
Leonardo of Pisa (Hog 1). During his lifetime, c 1170-1250,
Fibonacci produced one great work, the Liber Abaci (Vaj 9).
The title of this work, which was written in 1202,
literally means "a book about the abacus" (Vor 1). It is an
excessively comprehensive work, encompassing nearly all the
arithmetic and algebraic knowledge of the time. This book
played an important role in the evolution of Western
European mathematics, but in particular, it familiarized
the Europeans with the Hindu, or Arabic numerals (1).
Although his book spoke of many different topics, Fibonacci
is most known for the sequence of numbers
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...,
to which his name has been given. Even today, this sequence
continues to be researched (Hog 1). This report should give
a basic understanding of some aspects of the Fibonacci
The Fibonacci Sequence was discovered less than 800 years
ago. The "deceptively simple definition implies a large
variety of relationships," and appears as the solution to
many intricate mathematical problems (Vaj 7). Known of
since antiquity, the Golden Ratio far predates the
Fibonacci numbers. The Golden Ratio is special. A rectangle
with sides of this ratio will retain this ratio when a
square of its width is removed from it. I will explain more
on this later. This ratio appears abundantly in nature and
math, as it is a very nature-oriented number.
The Fibonacci Sequence and The Golden Ratio are very much a
part of one another, and these to phenomena have many
applications in math and nature, which I will discuss in
this paper.
The Fibonacci Numbers are always being studied because of
their complex nature. They seem simple, but are really far
from it. We arrive at the Fibonacci Sequence through a
method of adding the previous two terms. Starting with 1,
and adding the two previous terms to get each successive
term, we end up with a sequence that looks like this:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
Therefore, if Fn is the nth term in the Fibonacci Sequence,
Fn+2=Fn+1+Fn or Fn=Fn-1+Fn-2
(Vaj 9). Following this formula will result in the
Fibonacci sequence. Fibonacci explained his sequence with a
very famous problem concerning rabbits breeding. The
following is a translation of pages 123-124 of the
manuscript from 1228: 
Someone placed a pair of rabbits in a certain place,
enclosed on all sides by a wall, to find out how many pairs
of rabbits will be born there in the course of one year, it
being assumed that every month a pair of rabbits produces
another pair, and that rabbits begin to bear young two
months after their own birth.
As the first pair produces isue in the first month, in this
month there will be 2 pairs. Of these, one pair, namely the
first one, gives birth in the following month so that in
the second month there will be 3 pairs. Of these, 2 pairs
will produce issue in the following month, so that in the
third month 2 more pairs of rabbits will be born, and the
number of pairs of rabbits in that month will reach 6; of
which 3 pairs will produce issue in the fourth month, so
that the number of pairs of rabbits will then reach 8...
(Vor 2). From this, we find the following:
First (Month) 
1 (Pairs) 
These numbers (obviously) comprise the Fibonacci Sequence
Although there are hundreds of formulas to be derived from
this sequence, only few simple ones are within this paper's
scope. First, we will calculate the sum of the first n
Fibonacci numbers. We will show that
(Vor 6). In English, adding all terms until Fn will equal
Fn+2, or the term two ahead, minus 1. Next, we have sum
formulas for Feven and Fodd. They are:
The sum of Fodd: F1+F3+F5+...+F2n-1=F2n.
Sum of Feven: F2+F4+...+F2n=F2n+1-1
(7). Adding terms one by one will prove the equation. And
finally, we square each term to come to the following
(8). Obviously, deriving these is beyond the extent of this
paper, but replacing the variables with valid terms will
prove it works.

The Golden Ratio, or Golden Section, is closely linked to
the Fibonacci Numbers. Suppose we have a line segment AB.
We must find point C on it, such that the ratio of the
total length to the larger segment is the same as that of
the larger segment to the smaller segment, as in the
following figure:



where none of the line segments measure 0 in length (Hog
9). Let us say that AC=1 for reasons of simplicity in
finding the ratio. So, keeping in mind that AC=1 in this
x = AB/AC = (AC+CB)/AC = 1+CB/AC = 1+1/(AC/CB) =
1+1/(AB/AC) = 1+1/x.
So... x = 1+1/x
(9). By bringing everything to one side and setting the
equation equal to 0, we get
x2 - x - 1 = 0.
Using the quadratic formula, we find that the roots of this
equation are, as you can verify,

a = (1+Ö 5)/2 and b = (1-Ö 5)/2
(9). By computation, you can find that a » 1.618 and b »
-.618. Thus we take the positive root a as the desired
ratio. Now that we know that the sections AC and AB must be
in the ratio 1:1.618, we may move on to the Golden
This shape contains sides of the same ratio, creating a
figure somewhat like this:
In this figure, AC/CF=CF/FG=FG/GE. In short, when a square
is removed from the end of a Golden Rectangle, the
remaining rectangle has the same proportions as the
original, that is, 1:1.618 (Hog 12).
The Golden Rectangle's proportions appear frequently
throughout classical Greek art and architecture (12). The
German psychologists Gustav Theodor Fechner (1801-1887) and
Wilhelm Max Wundt (1832-1920) have shown in a series or
psychological experiments that most people do unconsciously
favor "golden dimensions" when selecting pictures, cards,
mirrors, wrapped parcels, and other rectangular objects.
Neither artists nor psychologists fully know why, but the
Golden Rectangle holds great aesthetic appeal (12).
There is also another way to find the Golden Mean: It is
the ratio of any number in the Fibonacci Sequence to the
previous number. The following equation will return the
Golden Section number, 1.618, or a :
a » Fn+1/Fn
(Hog 28). Precisely speaking, the limit of this function is
the Golden Mean; as n approaches infinity, the value of the
function approaches the real value of a . The function's
values do rise and fall, above and below the real a , but
also get closer and closer to a .
Math and Nature exhibit the Fibonacci Numbers and the
Golden Ratio very often. First I will discuss the
appearances of Fibonacci Numbers. In nature, such
appearances are abundant concerning the Fibonacci Numbers.
On pinecones and pineapples, spirals spin out of the center
when looked at from the bottom, and the number of spirals
spinning one way is one Fibonacci number, and the number
spinning the other way is a different Fibonacci number (Hog
81-82). Also, many plants have leaves that extrude from the
stem at various, random-seeming points, so that each leaf
in the pattern has no other leaf directly above it until
the pattern repeats. This is useful to the plant in that
the leaves get as much sunlight and water as possible. A
diagram follows:
, we will arrange Pascal's Triangle as follows:
1 1
1 2 1
1 3 3 1
Works Cited - Bibliography
Hoggatt, Verner E., Jr. Fibonacci and Lucas Numbers. The
Fibonacci Association, University of Santa Clara.
California, 1969.
Knott, Ron. Fibonacci Numbers and the Golden Section.
Vajda, S. Fibonacci & Lucas Numbers, and the Golden
Section. John Wiley & Sons Limited. New York, 1989.
Vorob'ev, N. N. Fibonacci Numbers. Blaisdell Publishing
Company. New York, 1961.


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