# Fibonacci Numbers

** ****Fibonacci
Number**

0,1,1,2,3,5,8,13,21,34,55…

Each number in the sequence is the sum of the two
numbers that precede it.

Here (0,1 )

0 , 1 -> 0 + 1 = 1 generates sequence ( 0, 1, 1)

Now 1 , 1 ->1+1 =2 generates sequence (0, 1, 1, 2)

Now 1 , 2 ->1+2 = 3 generate sequence (0,1,1,2,3)

2 , 3->2+3 = 5 generate sequence (0,1,1,2,3,5) and
so on…

It can be represented as:

** F _{n} = F_{n-1} +
F_{n-2}**

** ** **Fibonacci Man**

Leonardo of Pisa, or Leonardo Pisano in
Italian since he was born in Pisa,Italy (see Pisa on Google Earth), the city with the famous Leaning Tower, about 1175
AD.

well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic (like our present numbers
0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals

**Fascinating Facts**

**Population of rabbit growth**

Suppose
a newly-born pair of rabbits, one male, one 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
(one male, one female) **every month** from the second month on.
The puzzle that Fibonacci posed was...

How many pairs will there be in one year?

1. At the end of the first
month, they mate, but there is still one only 1 pair.

2. At the end of the second
month the female produces a new pair, so now there are 2 pairs of rabbits in
the field.

3. At the end of the third
month, the original female produces a second pair, making 3 pairs in all in the
field.

4. 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.

**Mysterious Patterns**

Fibonacci sequence table
and its square value

Let X represent a Fibonacci number and Y represents square of Fibonacci
number

X | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 |

Y | 0 | 1 | 1 | 4 | 9 | 25 | 64 | 169 |

Taking first X from the following table

0 + 1 + 1 = 1

Adding next square number

0 + 1 + 1 + 4 = 6

Adding next square number

0 + 1 + 1 + 4 + 9 = 15

Adding next square number

0 + 1 + 1 + 4 + 9 +25 = 40

And next

0 + 1 + 1 + 4 + 9 +25 +64 = 104
------------ Eq 1

You will see 1,6,15,40,104 …
formed as

1 = 1* 1 ( 1 is a fibonacci number occurring twice)

6 = 2 * 3 ( 2, 3 fibonacci numbers)

15 = 3 * 5 (3, 5 fibonacci numbers)

40 = 5 * 8 ( 5, 8 fibonacci numbers)

104 = 8 * 13 ( 8,13 fibonacci numbers)

= ( Product of two
Fibonacci numbers)

**Area of rectangle**

Let’s draw some squares one by one

This is a square of side length 1. Its area
is 1^2 = 1. We draw another one next to it:

Now the upper edge of the figure has length
1+1=2, so we can build a square of side length 2 on top of it:

Now the length of the rightmost edge is
1+2=3, so we can add a square of side length 3 onto the end of it.

Now the length of the bottom edge is 2+3=5:

And that makes the leftmost edge 3+5=8:

1 * 1 +1 * 1 + 2 * 2 + 3* 3 + 5 * 5 + 8 * 8 = 8 * 13

From Eq 1

0 + 1 + 1 + 4 + 9 +25+64 = 104 = 8 * 13 = Area of rectangle

Other pattern

Division of Fibonacci numbers

8 * 13 if we divide 13% 8 =
1.625…

13* 21 21%13= 1.615….

21 * 34 34%21
= 1.619…

34*55
55%34= 1.6176…

55*89
89%55= 1.618…

Here** 1.618033…. Golden ratio**

Here

a/b = (a+b ) / a **= 1.618….**

**Real World Example showing golden ratio
concept**