1. Step By Step School, Noida
FIBONACCI NUMBERS
Chaitanya khurana
Class 9th
Roll No ---08
Bhaskar House
Step By Step School, Noida

2. The Fibonacci numbers can be rather simply
Definition
The Fibonacci numbers can be rather simply
defined by the following:
1. Start with the numbers 1 and 
2. Add them together to make the next number.
3. Then form the next number in sequence by adding the previous two together.

3. More formally: [F1=1; F2=2], [F3=3], [Fn=Fn-1+Fn-2]
More formally: [F1=1; F2=2], [F3=3], [Fn=Fn-1+Fn-2]. The resulting sequence begins like this:
F1 to F19 : 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181,
The (recurrence) formula for these Fibonacci numbers is:
F(0)=0, F(1)=1, F(n)=F(n-1)+F(n-2) for n>1.

4. The First Forty Terms Of Fibonacci Numbers Are :-
Fn Number Fn Number Fn Number Fn Number F0 0 F F F F1 1 F F F F2 1 F F F3 2 F F F4 3 F F F5 5 F F F6 8 F F F7 13 F F F8 21 F F F9 34 F F F10 55 F F F11 89 F F F F F

5. Leonardo Fibonacci
Leonardo Pisano Bigollo also known as Leonardo of Pisa, Leonardo Pisano, Leonardo Bonacci, Leonardo Fibonacci, or, most commonly, simply Fibonacci, was an Italian mathematician, considered by some "the most talented western mathematician of the MiddleAges.
Fibonacci is best known to the modern world for the spreading of the Hindu–Arabic numeral system in Europe, primarily through the publication in 1202 of his Liber Abaci (Book of Calculation), and for a number sequence named the Fibonacci numbers after him, which he did not discover but used as an example in the Liber Abaci.

6. Leonardo Fibonacci was born around 1170 to Guglielmo Bonacci, a wealthy Italian merchant. Guglielmo directed a trading post (by some accounts he was the consultant for Pisa) in Bugia, a port east of Algiers in the Almohad dynasty’s sultanate in North Africa(now Bejaia, Algeria). As a young boy, Leonardo traveled with him to help; it was there he learned about the Hindu–Arabic numeral system.
Recognizing that arithmetic with Hindu–Arabic numerals is simpler and more efficient than with Roman numerals, Fibonacci traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time. Leonardo returned from his travels around In 1202, at age 32, he published what he had learned in Liber Abaci( Book of Abacus or Book of Calculation), and thereby popularized Hindu–Arabic numerals in Europe.

7. The Fibonacci numbers are Nature's numbering system
The Fibonacci numbers are Nature's numbering system. They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales of a pineapple. The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

8. Fibonacci in Plants
 Phyllotaxis is the study of the ordered position of leaves on a stem. The leaves on this plant are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Click anywhere on the slide to see larger picture

9. In the case of tapered pinecones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers. 

10. As well, many flowers have a Fibonacci number of petals
As well, many flowers have a Fibonacci number of petals. Some, like this rose, also have Fibonacci Spiral, petal arrangements.

11. Branching plants also exhibit Fibonacci numbers
 Branching plants also exhibit Fibonacci numbers. Again, this design provides the best physical accommodation for the number of branches, while maximizing sun exposure

12. Fibonacci in Animals
we have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand.

13. 2cm
Even the length of the bones of ones finger is in Fibonacci number system
3cm
5cm
8cm

14. How is all this possible ???