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FIBONACCI NUMBERS
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,
2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025,
121393, 196418, 317811, 514229, 832040, 1346269, 2178309,
3524578, 570288, 9227465, 14930352, 2415781, 39088169 ..
F(n) =
Presented By : Soumya Gulati
{
0,
if n = 0,
1,
if n = 1,
F(n-1) + F(n-2), if n > 1
Class IX–A
Roll No. 23
About Fibonacci :
 Born in Pisa, Italy in 1175 AD
 Full name was Leonardo Pisano
 Grew up with a North African education under the
Moors
 Traveled extensively around the Mediterranean
coast
 Met with many merchants and learned their systems
of arithmetic
 Realized the advantages of the Hindu-Arabic system
Fibonacci’s Mathematical Contribution : Books and Letters
 Wrote five mathematical works ::: Four books and One preserved letter





Liber Abbaci (The Book of Calculating)
Practica geometriae (Practical Geometry)
Flos
Liber quadratorum (The Book of Squares)
A letter to Master Theodorus written around 1225
FIBONACCI’S MATHEMATICAL CONTRIBUTION
 Introduced the Hindu-Arabic number system into Europe based on ten digits
and a decimal point
 Europe previously used the Roman number system consisted of Roman
numerals
 Persuaded mathematicians to use the Hindu-Arabic number system
1 2 3 4 5 6 7 8 90
AND
I=1
V=5
X = 10
L = 50
C = 100
D = 500
M = 1000
THE FIBONACCI NUMBERS
 Were introduced in “The Book of Calculating”
 Series begins with 0 and 1
 Each subsequent number is the sum of the previous two.
 So now our sequence becomes 0,1, 1, 2.
 The next number will be 3.
 Pattern is repeated over and over
In mathematical terms, the sequence F n of Fibonacci numbers is defined by the
recurrence relation
F n = Fn-1 + Fn-2
with seed values F0 = 0 and F n = 1
N is a Fibonacci number if and only if 5 N2 + 4 or 5 N2 – 4 is a square number
THE FIBONACCI NUMBERS
The first 21 Fibonacci numbers F n for n = 0, 1, 2, ..., 20 are :
F₀ F₁ F₂ F₃ F₄ F₅ F₆ F₇ F₈ F₉ F₁₀ F₁₁ F₁₂ F₁₃ F₁₄ F₁₅ F₁₆ F₁₇
0
1
1
2
3
5
8
F₁₈
F₁₉
F₂₀
13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765
The sequence can also be extended to negative index n using
the re-arranged recurrence relation
which yields the sequence of "negafibonacci" numbers[ satisfying
Thus the complete sequence is
F−8
F−7 F−6
F−5
F−4
F−3
F−2
F−1
F0
F1
F2
F3
F4
F5
F6
F7
F8
−21
13
5
−3
2
−1
1
0
1
1
2
3
5
8
13
21
−8
FIBONACCI’S RABBITS
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.
 Therefore the number of rabbits per month =





End of the first month = 1 pair
End of the second month = 2 pair
End of the third month = 3 pair
End of the fourth month = 5 pair
5 pairs of rabbits produced in one year.
1, 1, 2, 3, 5, 8, 13, 21, 34, …
FIBONACCI’S HONEY BEES
The number of ancestors of honey bees at each
generation follows the Fibonacci series!
FIBONACCI’S NUMBERS IN PASCAL’S TRIANGLE
 In Pascal’s Triangle, the entry is sum of the two numbers either
side of it, but in the row above “shallow” diagonal sums in
Pascal’s Triangle are the Fibonacci numbers
 Fibonacci numbers can also be found using a formula.
1
1
1
1
1
1
2
3
4
1
3
6
1
4
1
FIBONACCI’S NUMBERS AND PYTHAGORUS TRIANGLES
 Every successive series of four Fibonacci numbers can be used to generate
Pythagorean triangles
 Fibonacci numbers 1, 2, 3, 5 produce Pythagorean ∆ with sides 5, 12, 13
 First side (a) of Pythagorean triangle = 12
 Second side (b) of Pythagorean triangle = 5
 Third side (c) of Pythagorean triangle = 13
𝑎2 + 𝑏2 = 𝑐 2
METHOD :
Any four consecutive Fibonacci numbers F n, Fn+1, Fn+2 and Fn+3 can also be used to generate a
Pythagorean triple
Multiply the two middle or inner numbers (here 2 x 3 = 6); Double the result ( 6 x 2 = 12). [SIDE a]
a = 2 Fn+1 x Fn+2
Multiply together the two outer numbers ( 1 x 5 = 5). [SIDE b]
b = F n x Fn+3
The hypotenuse is found by adding together the squares of the inner two numbers (here 22=4 and
32=9 and their sum is 4+9=13). [SIDE 3]
c = (Fn+1 )² x (Fn+2 )²
FIBONACCI’S NUMBERS IN NATURE
 Fibonacci spiral found in both snail and sea shells.
A tiling with squares whose
sides are successive Fibonacci
numbers in length
A
Fibonacci
spiral
created by drawing
circular arcs connecting
the opposite corners of
squares
in
the
Fibonacci tiling; this one
uses squares of sizes 1,
1, 2, 3, 5 & 8.
FIBONACCI’S NUMBERS IN PLANTS

Sneezewort (Achillea ptarmica) shows the Fibonacci
numbers
This plant in particular
shows
the
Fibonacci
numbers in the number of
"growing points" that it
has. Suppose that when a
plant puts out a new
shoot, that shoot has to
grow two months before it
is strong enough to
support branching. If it
branches every month
after that at the growing
point, we get the picture
shown here.
FIBONACCI’S NUMBERS IN LEAVES OF PLANTS
 Plants show the Fibonacci numbers in the arrangements of
their leaves
 Three clockwise rotations, passing five leaves
 Two counter-clockwise rotations
FIBONACCI’S NUMBERS IN PINE CONE PLANT
 Pinecones clearly show the Fibonacci spiral
FIBONACCI’S NUMBERS IN TECHNOLOGY
 Fibonacci spirals can be made through the use of visual
computer programs.
 Each sequence of layers is a certain linear combination of
previous ones.
FIBONACCI’S NUMBERS IN FLOWERS
& SEEDS ON FLOWER HEADS
Lilies and Irises= 3 petals
Buttercups and Wild Roses = 5 petals
Arrangement of seeds
on flower heads
Corn marigolds
13 petals
Black-eyed Susan’s
21 petals
55 spirals
spiraling
outwards &
34 spirals
spiraling
inwards
FIBONACCI’S NUMBERS IN FRUITS & VEGETABLES
If we cut a fruit or vegetable
we will often find that the
number of sections is a
Fibonacci number:
Section of Apple
5
Section of Banana
3
Pineapple scales have
Fibonacci spirals in
sets of 8, 13, 21
Bananas have 3
or 5 flat sides
Fibonacci spiral can be found in
cauliflower
 The Fibonacci numbers can be found in the human hand and
fingers
 Every human has 2 hands, which contain 5 fingers
 Each finger has 3 parts separated by 2 knuckles
All of these numbers fit into the sequence. However we need to keep in mind,
that this could simply be coincidence
FIBONACCIS’S NUMBERS IN GRAPHS
‡ The Fibonacci numbers arise from the golden
section
‡ The graph shows a line whose gradient is Phi
‡
‡
‡
‡
First point close to the line is (0, 1)
Second point close to the line is (1, 2)
Third point close to the line is (2, 3)
Fourth point close to the line is (3, 5)
‡ The coordinates are successive
Fibonacci numbers
MAKING MODEL ON FIBONACCI NUMBERS
BIBLIOGRAPHY
http://www.mathsisfun.org
http://www.wikipedia.com
http://www.mcs.surrey.ac.uk
http://www.evolutionoftruth.com
http://www.pass.maths.org.uk
http://www.braungardt.com
http://www.sigmaxi.org
http://www.violin.odessa.ua