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Biography ( ) Fibonacci is a short for the Latin "filius Bonacci" which means "the son of Bonacci" but his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy). He was educated in North Africa where his father worked as a merchant. Fibonacci travelled widely with his father around the Mediterranean coast . In 1200 he returned to Pisa and used the knowledge he had gained on his travels to write his books.

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**Books Liber Abaci (1202), The Book of Calculation**

Practica Geometriae (1220), The Practice of Geometry Flos (1225), The Flower Liber Quadratorum (1225),The Book of Square Numbers

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The Flower the approximate solution of the following cubic equation: x³+2x²+10x=20 in sexagesimal notation is , equivalent to

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**The Book of Square Numbers**

Method to find Pythogorean triples: When you wish to find two square numbers whose addition produces a square number, you take any odd square number as one of the two square numbers and you find the other square number by the addition of all the odd numbers from unity up to but excluding the odd square number. For example, you take 9 as one of the two squares mentioned; the remaining square will be obtained by the addition of all the odd numbers below 9, namely 1, 3, 5, 7, whose sum is 16, a square number, which when added to 9 gives 25, a square number.

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Liber Abaci The book introduced the Hindu-Arabic number system into Europe , the system we use today, based on ten digits with its decimal point and a symbol for zero: The book describes (in Latin) the rules for adding numbers, subtracting, multiplying and dividing.

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**Rabbits How many pairs will there be in one year?!**

Suppose a newly-born pair of rabbits ( male + 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. Suppose that our rabbits never die and that the female always produces one new pair ( male + female) every month from the second month on. How many pairs will there be in one year?!

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Answer… At the end of the first month, they mate, but there is still only 1 pair At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs…..

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**We get the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34 ...**

This sequence, in which each number is a sum of two previous is called Fibonacci sequence so there is the simple rule: add the last two to get the next! The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation F(n)=F(n-1)+F(n-2)

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Fibonacci Rectangles We start with two small squares of size 1 next to each other. On top of both of these we draw a square of size 2 (=1+1). We can now draw a new square - touching both a unit square and the latest square of side 2 - so having sides 3 units long; and then another touching both the 2-square and the 3-square (which has sides of 5 units). We can continue adding squares around the picture, each new square having a side which is as long as the sum of the latest two square's sides. This set of rectangles whose sides are two successive Fibonacci numbers in length and which are composed of squares with sides which are Fibonacci numbers, we call the Fibonacci Rectangles.

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Rectangles

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**A spiral drawn in the squares, a quarter of a circle in each square.**

Fibonacci spirals A spiral drawn in the squares, a quarter of a circle in each square.

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PASCAL‘S TRIANGLE

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Nature One of the most fascinating things about the Fibonacci numbers is their connection to nature. the number of petals, leaves and branches spiral patterns in shells spirals of the sunflower head pineapple scales

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Flowers

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Nautilus

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Sun Flower

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Pineapple

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**in the introduction to Europe**

Conclusion The greatest European mathematician of the middle age, most famous for the Fibonacci sequence, in which each number is the sum of the previous two and for his role in the introduction to Europe of the modern Arabic decimal system.

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HAPPY EASTER!!!

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THE END BYE…

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Lecture 4,5 Mathematical Induction and Fibonacci Sequences.

Lecture 4,5 Mathematical Induction and Fibonacci Sequences.

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