Fibonacci Numbers

WHAT IS
FIBONACCI NUMBERS
"Fibonacci Sequence" redirects here. For the chamber ensemble,
see Fibonacci Sequence (ensemble)”.
 A tiling with squares whose side lengths are successive
Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13 and 21.
In mathematics, the Fibonacci numbers, commonly denoted Fn,
form a sequence, called the Fibonacci sequence, such that each
number is the sum of the two preceding ones, starting from 0 and
1.
The sequence starts:[2]

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
HOW THE ARE SPECIAL
Fibonacci numbers are special kinds of
numbers that form a special sequence.
This sequence is one of the famous
formulas in mathematics. You can find
Fibonacci numbers in plant and animal
structures. These numbers are also called
nature's universal rule, and nature's secret
code
GOLDEN TEASURE
Fibonacci numbers are strongly related to the golden ratio:
Binet's formula expresses the nth Fibonacci number in terms
of n and the golden ratio, and implies that the ratio of two
consecutive Fibonacci numbers tends to the golden ratio
as n increases.
Fibonacci numbers are named after the Italian mathematician
Leonardo of Pisa, later known as Fibonacci. In his 1202
book Liber Abaci, Fibonacci introduced the sequence to
Western European mathematics, although the sequence had
been described earlier in Indian mathematics, as early as 200
BC in work by Pingala on enumerating possible patterns of
Sanskrit poetry formed from syllables of two lengths.
PASCEL TRIANGLE
O Fibonacci numbers in the
Pascal triangle
O The Fibonacci Series is found in Pascal’s Triangle.

Pascal’s Triangle, developed by the French Mathematician
Blaise Pascal , is formed by starting with an apex of 1.
Every number below in the triangle is the sum of the two
numbers diagonally above it to the left and the right, with
positions outside the triangle counting as zero.
EXAMPLES OF APPLICATON OF
FIBONACCI NUMBERS
One of the main applications of Fibonacci numbers
outside of the realm of mathematics is in the area of stock
market analysis. Many investors use what is called the
Fibonacci Retracement Technique to estimate the action
that the price of a
particular stock will take,
based on certain
ratios
found within the
Fibonacci numbers
GALAXY IN THE FORM OF
FIBONACCI SPIRAL
Spirals arise from a property of growth called self-
similarity or scaling - the tendency to grow in size but to
maintain the same shape. Not all organisms grow in this self-
similar manner. We have seen that adult people, for example,
are not just scaled up babies: babies have larger heads,
shorter legs, and a longer torso relative to their size. But if
we look for example at the shell of the chambered nautilus
we see a differnet growth pattern. As the nautilus outgrows
each chamber, it builds new chambers for itself, always the
same shape - if you imagine a very long-lived nautilus, its
shell would spiral around and around, growing ever larger
but always looking exactly the same at every scale.
Here is where Fibonacci comes in - we can build a
squarish sort of nautilus by starting with a square of size 1
and successively building on new rooms whose sizes
correspond to the Fibonacci sequence:
Running through the centers of the squares in order

with a smooth curve we obtain the nautilus spiral = the
sunflower spiral.
HOW FIBONACCI NUMBERS ARE
RELATED TO GOLDEN RATIO
There is a special relationship between the
Golden Ratio and Fibonacci Numbers (0, 1, 1, 2,
3, 5, 8, 13, 21,... etc, each number is the sum of
the two numbers before it). When we take any
two successive
(one after the other) Fibonacci
Numbers,
their ratio is very close
to the Golden Ratio:
HOW IT THEY BEGIN
The Fibonacci numbers were first discovered by a man
named Leonardo Pisano . He was known by his nickname,
Fibonacci. The Fibonacci sequence is a sequence in which
each term is the sum of the 2 numbers preceding it.
The Fibonacci Numbers
are defined by the
recursive
relation
defined by the
equations (F)
USES OF FIBONACCI
NUMBERS
O The Fibonacci sequence is widely used in
engineering applications including computer data
structures and sorting algorithms, financial
engineering, audio compression, and architectural
engineering. The Fibonacci sequence can be seen
in nature in the spirals of a sunflower's seeds and
the shape of a snail's shell.
O Fibonacci retracement levels are widely used in
technical analysis for financial market trading.
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WHAT IS

The sequence starts:[2]

O The Fibonacci Series is found in Pascal’s Triangle.

Running through the centers of the squares in order

SUBMITTED TO: Seema Garg

SUBMITTED TO: Seema Garg

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