Europe Starts to Wake up: Leonardo of Pisa

Transcription

1 Europe Starts to Wake up: Leonardo of Pisa Leonardo of Pisa, also known as Fibbonaci (from filius Bonaccia, son of Bonnaccio ) was the greatest mathematician of the middle ages. He lived from 75 to 50, and traveled widely in the Mediterranean while yog. He recognized the superiority of the Hindu-Arabic number system, and wrote LIber Abaci, the Book of Coting, to explain its merits (and to set down most of the arithmetic knowledge of the time). Hindu Arabic Numerals:

2 Fibbonacci also wrote the Liber Quadratorum or Book of Squares, devoted entirely to second-degree Diophantine equations. Example: Find rational numbers x, u, v satisfying: x + x= u x x= v Fibonacci s solution involved finding three squares that form an arithmetic sequence, say a b d, c b d = = +, so d is the common difference. Then let x = b. One such solution is given by d a =, b = 5, c = 49 so that x = 5. 4 In Liber Quadratorum Fibonacci also gave examples of cubic equations whose solutions could not be rational numbers. Finally, Fibonacci proved that all Pythagorean triples (i.e. positive integers a, b, c satisfying a + b = c, can be obtained by letting a = st b= s t c= s + t, where s and t are any positive integers. It was known in Euclid s time that this method would always general Pythagorean triples; Fibonacci showed that it in fact produced all Pythagorean triples. Here are a few Pythagorean triples to amaze your friends and confuse your enemies: s t a b c

3 The Fibonacci Sequence Of course, Fibonacci is best known for his sequence,,,, 3, 5, 8, 3,.... Eduard Lucas, a nineteenth century French number theorist, attached Fibonacci s name to the sequence which arose from a trivial problem in the Liber Abaci. The problem was this: A certain man put a pair of rabbits in a place surroded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive? End of Month Number: Adult Pairs: Yog Pairs: Total Pairs: We should note that the earliest known appearance of this sequence was in the work of the Sanskrit grammarian Pingala, sometime between 450 and 00 BC. He was studying the number of meters of a given overall length could be made with the Long (L) and short (S) vowels used in syllables (with the long vowel twice as long as the short). To get a meter of length n, you could add a short syllable S to a meter of length n-, or a long syllable L to a meter of length n-. Adding these two gave you the total meters of length n and since it started out with: S SS, L SSS, LS, SL It is easy to see that the recurrence relations produced what we know as the Fibonacci sequence. The Fibonacci sequence has been extensively studied and has a number of remarkable properties. In what follows we denote the nth Fibonacci number by u. Thus u = and u 7 = 3. n

4 Some Properties of the Fibonacci Sequence. Any two successive elements of the sequence are relatively prime. (Suppose otherwise; if d divided two successive elements it would divide their difference, which is the element just before them both. Working our way back in the sequence, eventually d would have to divide as well.). The gcd of any two Fibonacci numbers is also a Fibonacci number. In particular, gcd( u, u ) = u, where d = gcd( n, m). n m d 3. Ratios of successive terms form a convergent sequence, whose limit is the Golden Ratio φ That the limit is φ can be seen by letting Rn = be the n th ratio (so u4 3 u R3.5 n+ + = = =, for example). Then since = +, we get u

5 R = n+ R +. In the limit, both R and n+ Rn approach the limit L, so n L = + or, L + 5 rearranging into a quadratic form, L L = 0which has solutionφ =. 4. There are lots of interesting identities: And so forth. u + u + u u = u 3 n n+ u + u + 3 u nu = nu u + 3 n n+ n+ 3 u + u + u u = u u 3 n n n+ u = u u + ( ) n n n n+ u + u + u u = u 3 5 n n u + u + u u = u 4 6 n n+ 5. Because of its close connection with the golden ratio, Fibonacci numbers tend to be fod in lots of places in nature spiral shells, the number of petals on flowers, the growth patterns of pine cones and seed heads, and so on. It is helpful to note that the golden ratio has often been fod in places where the likelihood of it being fod by chance alone is pretty great. Thus, we will take with a grain of salt at least some of the more spectacular claims made about the golden ratio and the Fibonacci sequence being fod in nature, art, and so on. 6. Starting with 5, every Fibonacci number is the largest member of some Pythagorean triple. 7. Every positive integer can be written in a ique way as the sum of one or more distinct Fibonacci numbers in such a way that the sum does not include any two consecutive Fibonacci numbers. This is known as Zeckendorf's theorem, 8. The sum of any ten consecutive Fibonacci numbers is times the 7 th of the ten numbers. 9. The its digit of Fibonacci numbers repeat in a cycle of 60 (e.g. the ninth Fibonacci number, 34, ends in a 4. So will the 69 th, 9 th, etc.) The last two digits repeat in a cycle of 300, and the last three in a cycle of Take any four Fibonacci numbers, e.g., 3, 5, 8. The product of the outer two (6), twice the product of the inner two (30), and the sum of the squares of the inner two (34) form a Pythagorean triple: = = 56 = 34.