FIBONACCI Presentation by: Angela Tersigni

1. FIBONACCIPresentation by: Angela Tersigni • Known as : Leonardo Pisanus Leonardus filius Bonacci Leonardus Bigollus • “Bigollo” – Tuscan dialect – meaning “blockhead” or “traveller” • Never referred to himself as “Fibonacci” • Medieval biographies and portraits rare: All pictures and statues of Leonardo of Pisa are only from artists’ imaginations • Exact date of birth and death not known

2. Background History • Leonardo was born 2 centuries after cultural and economic slowdown in Europe known as Dark Ages • Commercial Revolution was well underway • Both local and international trade occurred • Mediterranean Sea linked regions representing different religions, political entities and cultures • Three Italian cities dominated imports and exports: Venice, Genoa and Pisa • Pisa: population of approx. 10,000 and was a “commune” – independent republic

3. More History… • His father, Giuliemo, held a diplomatic post in Bugia (North Africa) • Leonardo travelled extensively with his father: Egypt, Syria, Constantinople, Sicily, France, Greece • Leonardo acquired much knowledge of various mathematical systems and texts during his travels • Some argue he is not a true mathematician but only an author of a very successful text (Liber Abaci) • Yet, he did also compile his own techniques, theorems and facts when he published his findings • When we think of Fibonacci, we think of his introduction on the Hindu-Arabic numerals (HAN) to the Western world and the famous Fibonacci sequence

4. Fibonacci’sContributions • His writings provided us with more than HAN and the Fibonacci sequence: • Extraction of square and cube roots • Rule of 3 and Rule of 5 for problem solving • Averages • Compound interest • Diophantine equations; especially his unique method for finding Pythagorean Triples • “res” term for unknown quantities • Chinese Remainder Theorem • Summing arithmetic and geometric series • Rule of False Position and Double False Position for problem solving • Working backwards for problem solving • Casting Out 9s to check for arithmetic accuracy

5. RomanNumerals • Not a positional system • Abacus provided place-value which was missing in their notation • No need to memorize facts • Abacus made addition and subtraction quick and easy but multiplication and division slow and tedious • Disadvantage (abacus): work vanished once done and it was impossible to explore numbers and their relationships

6. Leonardo’sContact with HAN • Arab businessmen travelled extensively (China, Africa, Russia, India) and they collected much scientific knowledge and geography • Muslims well-known for their wealth of knowledge. In the 9th century, Muslims were reading Aristotle, Euclid, Hippocrates, Galen, Ptolemy • They corrected and added their own findings to these works • In India, Arabs acquired: algebra continued from Greeks, place-value, numbers 1 to 9 and 0

7. By 7th century in India, the harmonious use of 1-9, 0 and place-value was well-established • Advantage of HAN for business: work did not vanish, easy to double-check work without calculation from the beginning • Leonardo saw a better advantage: to explore relationships between numbers • In 825 A.D., al-Khwarizmi wrote a book on Hindu numerals which was translated by Gerard of Cremona in 10th century • Translation was only known to small group of scholars • Leonardo made these “Hindu numerals” (his own words) well-known in Liber Abaci

8. LiberAbaci • Book of Abacus or Calculation • Ironically, it actually freed Western mathematics from the abacus • Written in 1202 and revised in 1228 to include mathematics for commercial use • Fortunately, the 1228 version has come down to us

9. Chapters 1-7: -Hindu-Arabic numerals -Arithmetic with HAN Chapters 8-11: Commerce: alloying, pricing, conversions, interest, wages, profit, barter, negative numbers (debt) Chapters 12-13: Recreational mathematics: puzzles, “rabbit problem” Chapter 14: Extraction of Roots: Approximation: (a^2+r)^1/2 = a + r/2a Chapter 15: Geometry: Deals with geometry arithmetically (unlike previous Greek works) Use of “res” (unknown quantity) and “census” (square of the unknown quantity

10. Lattice Multiplication 01 02 17 24 26 15 13 18 17 13 09 00 ______________ 139676498390 Answer: 23,958,233 x 5,830 = 139,676,498,390

11. Liber Abaci Continued… • Leonardo used many examples in describing arithmetic using HAN • Tables of facts included in his writing • Memory of facts seen as a disadvantage of this numeral system • Tables for fractions reduced to unit fractions: example: 3/8 reduced to 1/8 1/4 (1/8 +1/4) Egyptian influence? • Leonardo introduced horizontal bar for fractions • He wrote mixed fractions from right to left: example: ¼ 4 instead of 4 ¼ (Arabic influence)

12. Problem Solving Methods

14. Double False Position Where should the Fountain be Placed such that 2 birds flying at The same rate And leaving from Each tower will Arrive at the Fountain at the Same time? -Assumed tall tower was 10 ft. from the fountain -Using Pythagorean theorem: calculated first bird travelled 1700 ft. and second bird 2500 ft. -Added 5 ft. to distance of the fountain from the taller tower, subtracting 5 ft. from the shorter tower -This time the squares of the hypotenuses came out to 1,825 ft.and 2,125 ft. -Original difference in squares of the hypotenuses was 800, and the alteration by 5 ft. reduced it by 500 ft., another alteration by 3 ft. in the same direction would reduce it by 300 ft., eliminating it completely. -Thus, the fountain is 18 ft. from the taller tower and 32 ft. from the shorter tower

15. Leonardo used many problem examples to illustrate his problem-solving methods • His problem-solving methods as well as the problems themselves were taken from Muslim and Hindu texts • At times, he modified the problems to make them more relevant and useful for Europeans • Note: Leonardo always drew diagrams and tables to better illustrate the problem and the solution. He introduced the lines to show which numbers were multiplied (example: Rule of 3 and 5) • Leonardo would usually provide many different strategies to solve one problem

16. Solving Backwards

17. Place-Value Game • The following game in Liber Abaci demonstrates how comfortable Leonardo was with the concept of HAN: A group of men is seated in a row –one wearing a ring on a certain joint of a certain finger of one hand. A leader counts the wearer’s position in the row, doubles it, adds 5 to the product, multiplies the sum by 5 and adds 10. To this figure, he adds a number indicating the particular finger (little finger on left hand counts as 1 and the thumb on the right hand is 10). Multiply this sum by 10. Add the number indicating the joint of the finger on which the ring is placed: using 1 for finger tip, 2 for middle and 3 for the lowest part. The “guesser”, given the result, subtracts 350. He is then able to tell the audience the ring wearer and on which hand and part of the finger he is wearing the ring on. Modern notation: 10[5(2x + 5) + 10 + y] + z where x is position in row of Ring wearer, y is finger on which he wears it and z is the joint

18. Place-Value Game • The following game in Liber Abaci demonstrates how comfortable Leonardo was with the concept of HAN: A group of men is seated in a row –one wearing a ring on a certain joint of a certain finger of one hand. A leader counts the wearer’s position in the row, doubles it, adds 5 to the product, multiplies the sum by 5 and adds 10. To this figure, he adds a number indicating the particular finger (little finger on left hand counts as 1 and the thumb on the right hand is 10). Multiply this sum by 10. Add the number indicating the joint of the finger on which the ring is placed: using 1 for finger tip, 2 for middle and 3 for the lowest part. The “guesser”, given the result, subtracts 350. He is then able to tell the audience the ring wearer and on which hand and part of the finger he is wearing the ring on. Modern notation: 10[5(2x + 5) + 10 + y] + z where x is position in row of Ring wearer, y is finger on which he wears it and z is the joint

19. Pratica Geometriae • Written in 1220 • 8 chapters of theorems mainly based on Euclid’s “Elements” and “On Divisions” • Practical measurement problems with theoretical geometry • Problems treated algebraically • Trigonometry problems aimed at surveyors

20. Prince Frederick ll • Emperor and king of the 2 Sicilies • Sicily was a meeting ground for Christian and Muslim cultures of Europe and North Africa • Frederick sent out questionnaires and scientific problems to scholars in Egypt, Syria, Iraq, Asia and Yemen in order to acquire knowledge and understanding • Indication of Leonardo’s fame at the time: Frederick travelled to Pisa with his menagerie and entourage to visit and question Leonardo • Leonardo described the 3 problems posed to him in Liber Quadratorum and Flos

21. Liber Quadratorum • “Book of Squares” written in 1225 • Not as influential but best work • Major contribution to number theory between Diophantus (4th century Alexandria) and Pierre de Fermat (17th century) • Illustrated algebraic problems with lines or geometric figures • He assigned letter labels to his lines which represented the unknowns (start of algebraic notation) • The following will show some work contained in Liber Quadratorum:

22. Leonardo’s method of finding Pythagorean triples: most original thinking about numbers at the time Notes square numbers can be constructed as sums of Odd numbers: 1 = 1^2 1+3 = 2^2 1+3+5 = 3^2 etc.. The sequence and series of square numbers always rise through regular addition of odd numbers. Algebraically: n^2 + (2n+1) = (n+1)^2 Thus, when you wish to find 2 square numbers whose addition Produces a square, take any odd square as one of the 2 square Numbers and find the other square by the addition of all odd Numbers up to but not including the odd square number. • His general solution was worked out algebraically (using words), not geometrically!

23. Some more of his original work include: • x^2 + y^2 and x^2 – y^2 could not both be squares ( Assertion: no right triangle with rational sides could equal a square with a rational side). The proof was incomplete; Fermat sketched the proof • x^4 – y^4 cannot be a square • Expressed a number as 2, 3 or 4 squared numbers or squared fractions • Congruum: A number, K, is a congruum if: K=ab(a+b)(a-b) ; if a+b is even or K=4ab(a+b)(a-b) ; if a+b is odd where a and b are integers

24. The following problem was presented to him during his meeting with Prince Frederick ll in 1225:

25. Flos • “Flower” • Written in 1225 • Contained the 2nd problem posed to him by Prince Frederick ll • Although this problem was familiar to Arab mathematicians, it was Leonardo who recognized the fact that Euclid’s method for solving such equations by square roots would not work here. The solution required cube roots which Euclid could not extract using a ruler and compass

26. Problem: x^3 +2x^2 +10x = 20 ; solve for x Leonardo’s solution: X= 1.3688081075 (modern notation) • He reasoned that 1<x<2 and narrowed down the answer by trial and error • He does not give any indication how he found it • He preserved the sexagesimal fractions (Babylonian) • This was a substitute for decimal fractions which did not come to the West until late 16th century (probably due to the fact that the monetary system did not have a decimal relationship)

27. Conclusion • At first, HAN had opposition: business could make due without it, difficult to learn tables, records could be easily altered • Pure mathematics almost at a standstill 300 years after his death • By 15th century, the HAN system was displacing the Roman numeral system • Printing press made his work better known; he paved the way for algebraic notation and aided scientific progress in the West • Many mathematicians borrowed from him (Fermat, Pascal, Descartes etc..)

28. The well-known problem…

29. Fibonacci Sequence: 1st known recursive sequence in Western world • Leonardo recognized it as such but attached no other special importance to it • He did not name this sequence after himself; Lucas coined the phrase in 1877 after rediscovering it and coming up with his own Lucas sequence • German astronomer, Johannes Kepler in 1611: as n increases, then the ratio of F(n)/F(n)+1 approaches the “golden ratio” • 17th century formula: u(n+2) = u(n+1) +u(n) • 1830: A. Braun – bracts on pinecones • 1840s: Jacques Binet: F(n)=1/(5)^1/2[[1+(5)^1/2]/2]^n - 1/(5)^1/2[[1-(5)^1/2]/2]^n • 1920: Church (botanist) – sunflower heads (spirals)

30. 1920s: Hambidge (botanist) – “dynamic symmetry” • Dynamic symmetry: represented by a logarithmic / equiangular spiral that does not change shape as it is growing • Equiangular spiral can be drawn using the Fibonacci squares and connecting their corners with a quarter circle with the square side as the radius

31. Dynamic symmetry seen in shells, animal horns, claws, beaks, ocean waves, leaves on corn plants and trees • 1963: Fibonacci Society was formed in California – mathematicians share ideas and stimulate research surrounding the Fibonacci numbers • What would Leonardo think about this peaked interest in his famous problem?

32. Summed arithmetic series: 2+4+6+8 = 4(8+2)/2 = 20 1+2+…+n = n(n+1)/2 • Sum of odd numbers: 1+3+5+7+…+n = n^2 ; where n is the rank of the series • Summed geometric series: 1+2+4+8 = 16-1 = 15 ; where 16 is the next number in the series • Solved some Diophantine equations of 2nd degree

33. Are we looking too hard???

34. If we look hard enough, will we find the pattern we are looking for all around us?