The Magic of Fibonacci

Group Members Name : Hassan Shahzad AheerRoll # : 15SBSCS25

Name : Syed Habib Roll # : 15SBSCS63

Name : Saadat AliRoll # : 15SBSCS55

Name : Hafsa AkramRoll # : 15SBSCS23

Name : AtiRoll # : 15SBSCS05

Topic

Way do we learn math Introduction The Fibonacci Number In term of calculation In Term of application In term of inspiration

Why do we learn mathematic ?

Essentially for three reasons : 1. Calculation 2. Application3. Inspiration

With the help of this three essential reasons we describe the Fibonacci sequence .

Fibonacci Leonardo Pisano was the greatest

European mathematician of the 12th

century. His nick name was Fibonacci. He wrote a book called Liber abaci

(The Book of Calculation) that was the first textbook in western world and use the Hindu – Arabic system of numbers.

In term of calculation From the standpoint of calculation the Fibonacci

sequence is very easy.1 1 2 3 5 8 1321 34 55 . . . .Formula :Fn + 2 = Fn + 1 + Fn

In term of Application

There are many application of Fibonacci.1. In Nature 2. In computer science 3. In animals and more . .

Fibonacci number in Nature things

Fibonacci number 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.

Fibonacci number in Computer program

• Fibonacci numbers are used in Fibonacci heaps, which are a data structure that can be used to speed up some very practical algorithms.

• Fibonacci numbers give a model for designing recursive programming algorithmswhere the time for any routine is the time within the routine itself, plus the time for the recursive calls.

Fibonacci in AnimalsA starfish has 5 arms. (5 is the 5th Fibonacci number).

In term of Inspiration

• The inspirational thing about Fibonacci number is they display beautiful number patterns.

• Suppose we like to square the Fibonacci number

1 1 2 3 5 8 13 21 34

1 1 4 9 25 64 169 441 1156

Simpe Fibonacci Numbers

Square of Fibonacci Numbers

1 1 2 3 5 8 13 21 34

1 1 4 9 25 64 169 441 11561 + 1 + 4 = 6 2 * 31 + 1 + 4 + 9 = 15 3 * 51 + 1 + 4 + 9 + 25 = 40 5 * 81 + 1 + 4 + 9 + 25 + 64 = 104 8 * 13

Fibonacci numbers and the golden rectangle

If the two smallest squares have a width and height of 1, then the box to their left has a measurement of 2 and the other boxes measure 3, 5, 8, and 13.

The golden ratio is expressed in spiraling shells.

There is a quarter of a circle in each square going from one corner to the opposite.

This is not a true mathematical spiral.

The Golden Ratio

A complex idea can be conveyed with just a single still image, namely making it possible to absorb large amounts of data quickly.

The golden ration

If you divide 13 by eight, you get 1.625. And if you divide the larger number by the smaller number, then these ratios get closer and closer to about 1.618, known to many people as the Golden Ratio

• 13 / 8 = 1.625• 21 / 13 =1.615• 34 / 21 = 1.6.19 so on • In last we get this

golden number which is known as golden ration

• 1.618033……

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Credits

Special thanks to all the people who made and released these awesome resources for free:

• Arthur Benjamin

• The magic of Fibonacci numbers

• The book of calculation