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 1175 to 1250, and traveled widely in the Mediterranean while young. He recognized the superiority of the Hindu-Arabic number system, and wrote LIber Abaci, the Book of Counting, to explain its merits (and to set down most of the arithmetic knowledge of the time).

Hindu – Arabic Numerals:

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:

x22+ x= u

x22− xv=

Fibonacci’s solution involved finding three squares that form an arithmetic sequence, say

ab22= −=d, c22b+d, 2 so d is the common difference. Then let x = b . One such solution is given by d ab22==1, 25, c2=49 so that x = 25 . 24

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 ab22+=c2, can be obtained by letting as==2 tbs22−tc=s2+t2, 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 1 2435 1 4 8 15 17 1 5102426 1 6123537 2 3 12 5 13 2 4161220 2 5202129 2 6243240 3 4 24 7 25 3 5301634 3 6362745 3 7424058 4 5 40 9 41 4 6482052 18 174 6264 29952 30600

The Fibonacci Sequence

Of course, Fibonacci is best known for his sequence, 1, 1, 2, 3, 5, 8, 13, . . . . 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 surrounded 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: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Adult Pairs: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 Young Pairs: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 Total Pairs: 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987

We should note that the earliest known appearance of this sequence was in the work of the Sanskrit grammarian Pingala, sometime between 450 and 200 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-1, or a long syllable L to a meter of length n-2. 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 un . Thus u1 = 1and u7 =13 . Some Properties of the Fibonacci Sequence

1. 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 1 as well.)

2. The gcd of any two Fibonacci numbers is also a Fibonacci number. In particular,

gcd(uunm, ) = ud, where dn= gcd( ,m) .

3. Ratios of successive terms form a convergent sequence, whose limit is the Golden Ratio φ.

1 11 22 31.5 5 1.666666667 81.6 13 1.625 21 1.615384615 34 1.619047619 55 1.617647059 89 1.618181818 144 1.617977528 233 1.618055556 377 1.618025751 610 1.618037135 987 1.618032787 1597 1.618034448 2584 1.618033813 4181 1.618034056 6765 1.618033963 10946 1.618033999 17711 1.618033985 28657 1.61803399 46368 1.618033988 75025 1.618033989

un+1 th That the limit is φ can be seen by letting Rn = be the n ratio (so un

u4 3 uunn+21un+ R3 ===1.5, for example). Then since =+, we get u3 2 uunn+11++un1 1 1 Rn+1 =+1. In the limit, both Rn+1 and Rn approach the limit L, so L =+1or, Rn L 15+ rearranging into a quadratic form, LL2 − −=10which has solutionφ = . 2 4. There are lots of interesting identities:

uu12++u3+...+unn=u+2−1

uu12++2 3u3+...+nunn=nu++2−un3+2 uu22++u2+...+u2=uu 123 nnn+1 21n− uunn=+−+11un(1−)

uu13++u5+...+u2nn−1=u2

uuu24+++6...+u2nn=u2+1−1 And so forth.

5. Because of its close connection with the golden ratio, Fibonacci numbers tend to be found 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 “found” in places where the likelihood of it being found 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 found 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 unique 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 11 times the 7th of the ten numbers.

9. The units digit of Fibonacci numbers repeat in a cycle of 60 (e.g. the ninth Fibonacci number, 34, ends in a 4. So will the 69th, 129th, etc.) The last two digits repeat in a cycle of 300, and the last three in a cycle of 1500.

10. Take any four Fibonacci numbers, e.g. 2, 3, 5, 8. The product of the outer two (16), twice the product of the inner two (30), and the sum of the squares of the inner two (34) form a Pythagorean triple: 1622+=30 256 +900 =1156 =342.