By Abdiel, Chris, Javier and Juan. Fibonacci Numbers Fibonacci Numbers Introduction: The Fibonacci series is a list of numbers that starts out as: Finding The Limit The sequence itself does not approach a definitive value Finding the nth Fibonacci Number (the easy way) Using the mathematics of generating functions, Jacques-Phillipe-Marie Binet was able to derive a formula that could find any number in the Fibonacci series Fibonacci Numbers in... Right Triangles Right Triangles Starting with 5, every other Fibonacci number is the length of the hypotenuse of a right triangle with integer sides, or in other words, the largest number in a Pythagorean triple. Nature Explanation: Each number is the sum of the two preceding numbers in the series starting with 1+1 History of Fibonacci Full name - Leonardo Pisano Bigollo He wrote a book called Liber Abaci which contains a problem about an idealized (but unrealistic) rabbit population which led to the introduction of the series He learned both the mathematics of scholars and the calculating schemes popularly used at that time Fibonacci’s father, Guillermo Bonacci, was a secretary of the Republic of Pisa, a custom officer of the North African city of Bugia and a merchant, which gave Leonardo the opportunity to travel freely all over the Byzantine Empire Nature Fibonacci Rectangles and Shell Spirals The Bee Ancestry Code The Fibonacci sequences appears in biological settings, such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple, the flowering of artichoke, an uncurling fern and the arrangement of a pine cone. If one traces the ancestry of any male bee (1 bee), he has 1 parent (1 bee), 2 grandparents, 3 great-grandparents, 5 great-great-grandparents, and so on. This sequence of numbers of parents is the Fibonacci sequence. Such spirals are seen in the shape of shells of snails and sea shells Poetry Rabbit Problem You begin with one male rabbit and one female rabbit. These rabbits have just been born. Explanation F10 = F9 + F8

F8 F7

F7 F6

F6 F5

F5 F4

F4 F3

F3 F2

F2 F1 F9= F8+F7 =34 F8=21 (34+21)=

F8= F7+F6 =21 F7=13 (21+13)=

F7= F6+F5 =13 F6=8 (13+8)=

F6= F5+F4 =8 F5=5 (8+5)=

F5= F4+F3 =5 F4=3 (5+3)=

F4= F3+F2 =3 F3=2 (3+2)=

F3= F2+F1 =2 F2=1 (2+1)=

F2=1 F1=1 (1+1)= NOTES:

Fn=(Fn-1)+(Fn-2) F1=1

F2=1 However, the ratio of consecutive Fibonacci numbers does approach a certain limit 55

34

21

13

8

5

3

2 . . . What is the value of ?? This formula uses the two solutions of FIND # OF RABBITS AFTER 10 MONTHS (F10) Fibonacci Poetry

(Fibetry) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55...... A rabbit will reach sexual maturity after one month. The gestation period of a rabbit is one month. Once it has reached sexual maturity, a female rabbit will give birth every month. A female rabbit will always give birth to one male rabbit and one female rabbit. Rabbits never die. 1. 3. 2. 4. 5. 6. Shorter leg = difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. Longer side = sum of the three sides of the preceding triangle in this series of triangles The Fibonacci sequence can be applied in other areas of math such as the golden rectangle Starting with two squares, both in which length=1, another square can be added with a length equal to the sum of all the adjacent previous squares This construction has led to the golden spiral The spiral can be made by drawing a quarter of a circle in each square Fibonacci poems can embody the number sequence in two ways, either in numbers of syllables or in numbers of words Math,

Makes,

My head,

Quake with pain.

Writing a poem based

On Fibonacci does the same. 1

1

2

3

5

8 34^2= 1156

30^2=900

16^2=256

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# Fibonacci Series

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