Essay/Term paper: Fibonacci numbers

Essay, term paper, research paper:  Science

Fibonacci Numbers


The Fibonacci numbers were first discovered by a man named Leonardo
Pisano. He was known by his nickname, Fibonacci. The Fibonacci sequence is a
sequence in which each term is the sum of the 2 numbers preceding it. The first
10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers
are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in
Italy. He played an important role in reviving ancient mathematics and made
significant contributions of his own. Even though he was born in Italy he was
educated in North Africa where his father held a diplomatic post. He did a lot
of traveling with his father. He published a book called Liber abaci, in 1202,
after his return to Italy. This book was the first time the Fibonacci numbers
had been discussed. It was based on bits of Arithmetic and Algebra that
Fibonacci had accumulated during his travels with his father. Liber abaci
introduced the Hindu-Arabic place-valued decimal system and the use of Arabic
numerals into Europe. This book, though, was somewhat contraversial because it
contradicted and even proved some of the foremost Roman and Grecian
Mathematicians of the time to be false. He published many famous mathematical
books. Some of them were Practica geometriae in 1220 and Liber quadratorum in
1225.
The Fibonacci sequence is also used in the Pascal trianle. The sum of
each diagnal row is a fibonacci number. They are also in the right sequence:
1,1,2,5,8.........
Fibonacci sequence has been a big factor in many patterns of things in
nature. One has found that the fractions u/v representing the screw-like
arrangement of leaves quite often are members of the fibonacci sequence. On many
plants, the number of petals is a Fibonacci number: buttercups have 5 petals;
lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13
petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89
petals. Fibonacci nmbers are also used with animals. The first problem Fibonacci
had wehn using the Fibonacci numbers was trying to figure out was how fast
rabbits could breed in ideal circumstances. Using the sequence he was ale to
approximate the answer.
The Fibonacci numbers can also be found in many other patterns. The diagram
below is what is known as the Fibonacci spiral. We can make another picture
showing the Fibonacci numbers 1,1,2,3,5,8,13,21,.. if we start with two small
squares of size 1, one on top of the other. Now on the right of these draw a
square of size 2 (=1+1). We can now draw a square on top of these, which has
sides 3 units long, and another on the left of the picture which as side 5. We
can continue adding squares around the picture, each new square having a side
which is as long as the sum of the latest two squares drawn.

If we take the ratio of two successive numbers in Fibonacci's series, (1 1 2 3 5
8 1 3..) we find:

1/1=1; 2/1=2; 3/2=1.5; 5/3=1.666...; 8/5=1.6; 13/8=1.625;

It is easier to see what is happening if we plot the ratios on a graph:

Greeks called the golden ratio and has the value 1.61803. It has some
interesting properties, for instance, to square it, you just add 1. To take its
reciprocal, you just subtract 1. This means all its powers are just whole
multiples of itself plus another whole integer (and guess what these whole
integers are? Yes! The Fibonacci numbers again!) Fibonacci numbers are a big
factor in Math, The Golden Ratio, The Pascal Triangle, the production of many
species, plants, and much much more.