Fibonacci 1170-1250

1. Fibonacci1170-1250 By: ShirlFarkas and Kristina Calhoun

2. Who was Fibonacci? • The "greatest European mathematician of the middle ages“ • His full name was Leonardo of Pisa or Leonardo Pisano in Italian since he was born in Pisa, Italy. • He was the son of GuglielmoBonacci, a customs officer.

3. What’s in a Name? • Leonardo of Pisa is now known as Fibonacci short for filiusBonacci. • FiliusBonacci was used in the title of his book Libar Abaci which means "the son of Bonaccio". • Leonardo himself wrote both "Bonacci" and "Bonaccii" however he did not use the word Fibonacci. • Others believe Bonacci was a nick-name meaning "lucky son".

4. Fibonacci’s Childhood • Pisa was an important commercial town and had links with many Mediterranean ports. • Leonardo's father, GuglielmoBonacci, was a customs officer in the present-day Algerian town of Béjaïa, where wax candles were exported to France. • Leonardo grew up with a North African education under the Moors and later traveled extensively around the Mediterranean coast. • He met with many merchants, learned of their systems of doing arithmetic, and soon discovered the many advantages of the "Hindu-Arabic" system.

5. Hindu-Arabic System • Fibonacci was one of the first people to introduce the Hindu-Arabic number system into Europe. • This positional system is still used today - based on ten digits with its decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 0 • His book on how to do arithmetic in the decimal system, called Liberabbaci persuaded many European mathematicians of his day to use this new system.

6. LiberAbbaci • Book of the Abacus or Book of Calculating • Completed in 1202 and revised in 1228 • The book describes the rules for adding, subtracting, multiplying and dividing numbers. • In Chapter 12, he introduced the following problem: How Many Pairs of Rabbits Are Created by One Pair in One Year A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also.

7. Fibonacci’s Solution Because the above written pair in the first month bore, you will double it; there will be two pairs in one month. One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs;of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month; there will be 144 pairs in this [the tenth] month; to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month. To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above written pair in the mentioned place at the end of the one year.

8. Fibonacci’sExplanation You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above written sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

9. Sequence Chart

10. Did Fibonacci Invent this Sequence? • Fibonacci stated in his book Liber Abaci that he had studied the "nine Indian figures" and their arithmetic as used in various countries around the Mediterranean and wrote about them to make their use more commonly understood. • He included the "rabbit problem" in his book from one of his contacts. • He did not invent either the problem or the series of numbers. • The French mathematician Edouard Lucas gave the name Fibonacci Numbers to this series.

11. What is the Fibonacci Sequence? • In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation. Fn = Fn-1 + Fn-2

12. A tiling with squares whose side are successive Fibonacci numbers in length. A Fibonacci spiral created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling.

13. The Golden Ratio • A value, closely related to the Fibonacci series, is called the golden ratio. This value is obtained by taking the ratio of successive terms in the Fibonacci series:

14. Golden Ratio • If you plot a graph of these values you'll see that they seem to be approaching a limit. This limit is the positive root of a quadratic equation and is called the golden ratio.

15. Phi (φ) • The golden ratio is often denoted by the Greek letter phi. • It is an irrational mathematical constant, approximately 1.6180339887 • If a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ For example: 987/610 ≈ 1.6180327868852

16. The Golden Ratio is not just a number. It seems to be Nature's perfect number. For some reason, it appeals to our natural instincts. For centuries, designers of art and architecture have recognized the significance of the Golden Ratio in their work.

17. The Golden Ratio appears in everyday objects: • Index Cards • Photographs • Textbooks • Picture Frames • Computer Screens • Televisions

18. Fibonacci in Math • The Fibonacci numbers are studied as part of number theory and have applications in the counting of mathematical objects such as sets, permutations and sequences and to computer science. • He is also recognized for his important work in root findings.

19. Root Finding • Fibonacci was also capable of remarkable calculating feats. He was able to find the positive solution of the following cubic equation: • What is even more remarkable is that he carried out all his working using the Babylonian system of mathematics which uses base 60. He gave the result as 1;22,7,42,33,4,40 which is equivalent to: • It is not known how he obtained this, but it was 300 years before anybody else could find such accurate results.

20. The End!