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:

                                                                               Fn = Fn-1 + Fn-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

 

           Fig: Number of petals in flowers following Fibonacci sequence

 

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.

 

 


 Fig: Rabbits growth per month wise depicts Fibonacci numbers 


 

 

 

Mysterious Patterns

         Fibonacci sequence table and its square value

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

X011235813
Y011492564169


 

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


 

 


Online Instructor

Tanisha Medewala

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