FIBONACCI by ewghwehws

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```									Biography (1170-1250)

Fibonacci is a short for the Latin "filius Bonacci" which means
"the son of Bonacci" but his full name was Leonardo of Pisa, or
Leonardo Pisano in Italian since he was born in Pisa (Italy).
He was educated in North Africa where his father worked as a
merchant.
Fibonacci travelled widely with his father around the
Mediterranean coast .
In 1200 he returned to Pisa and used the knowledge he had gained
on his travels to write his books.
Books

 Liber Abaci (1202), The Book of Calculation
 Practica Geometriae (1220), The Practice of
Geometry
 Flos (1225), The Flower
 Liber Quadratorum (1225),The Book of Square
Numbers
The Flower

the approximate solution of the following cubic equation:

x³+2x²+10x=20

in sexagesimal notation is 1.22.7.42.33.4.40 , equivalent to
The Book of Square Numbers

Method to find Pythogorean triples:
When you wish to find two square numbers whose addition
produces a square number, you take any odd square
number as one of the two square numbers and you find the
other square number by the addition of all the odd numbers
from unity up to but excluding the odd square number. For
example, you take 9 as one of the two squares mentioned;
the remaining square will be obtained by the addition of all
the odd numbers below 9, namely 1, 3, 5, 7, whose sum is
16, a square number, which when added to 9 gives 25, a
square number.
Liber Abaci

The book introduced the Hindu-Arabic number system into
Europe , the system we use today, based on ten digits with
its decimal point and a symbol for zero:

1234567890

The book describes (in Latin) the rules for adding numbers,
subtracting, multiplying and dividing.
Rabbits
Suppose a newly-born pair of rabbits ( male + female) are put in a
field. Rabbits are able to mate at the age of one month so that at
the end of its second month a female can produce another pair
of rabbits. Suppose that our rabbits never die and that the female
always produces one new pair ( male + female) every month
from the second month on.

How many pairs will
there be in one year?!

At the end of the first month, they mate, but there is still only 1 pair
At the end of the second month the female produces a new pair, so
now there are 2 pairs of rabbits in the field.
At the end of the third month, the original female produces a second
pair, making 3 pairs in all in the field.
At the end of the fourth month, the original female has produced yet
another new pair, the female born two months ago produces her first
pair also, making 5 pairs…..
We get the following sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34 ...

This sequence, in which each number is a sum of two
previous is called Fibonacci sequence so there is the
simple rule: add the last two to get the next!

The Fibonacci numbers are the sequence of numbers defined
by the linear recurrence equation

F(n)=F(n-1)+F(n-2)
Fibonacci Rectangles

We start with two small squares of size 1 next to each other. On top
of both of these we draw a square of size 2 (=1+1).
We can now draw a new square - touching both a unit square
and the latest square of side 2 - so having sides 3 units long; and
then another touching both the 2-square and the 3-square (which
has sides of 5 units). 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 square's sides. This set of rectangles
whose sides are two successive Fibonacci numbers in length and
which are composed of squares with sides which are Fibonacci
numbers, we call the Fibonacci Rectangles.
Rectangles
Fibonacci spirals

A spiral drawn in the squares, a quarter of a circle in each square.
PASCAL‘S TRIANGLE
Nature
One of the most fascinating things about the Fibonacci
numbers is their connection to nature.

the number of petals, leaves and branches
spiral patterns in shells
pineapple scales
Flowers
Nautilus
Sun

Flower
Pineapple
Conclusion

The greatest European mathematician of the
middle age, most famous for the Fibonacci
sequence, in which each number is the sum of
the previous two and for his role
in the introduction to Europe
of the modern
Arabic decimal system.
HAPPY EASTER!!!
THE END

BYE…

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