He was the greatest European mathematician of the Middle Ages. He lived in Pisa, at a time—about 1170 to about 1250—when Europeans were beginning to use Arabic numerals. His book Liber Abaci was a widely used introduction to this exciting new technology.
What to call him can be a confusing matter. His first name is easy: Leonardo, or Leonard, or Leonardus. But Italian usage of last names was very flexible, then and for a long time afterward. Here are some versions that were official enough to appear on manuscripts of his books.
The Fibonacci sequence (Fn, n= 0,1, ... ) satisfies F0 = 0, F1 = 1, and
Fn+2 = Fn+1 + Fn .
Here is a brief table of values of Fn .
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
| Fn | 0 | 1 | 1 | 2 | 3 | 5 | 8 | 13 | 21 | 34 | 55 | 89 |
The Fibonacci numbers are related to the golden ratio, which will be denoted here by τ.
The recurrence relation An+2 = An+1 + An is satisfied if we let An = Fn; it is also satisfied if we let An = τn. This is because τ is a solution of
x2 = x + 1 .
Now that is a quadratic equation, so it has a second solution, which turns out to be -ρ = -τ-1 . Therefore the recurrence relation is also satisfied if we let An = (-τ)-n.
That recurrence relation is a linear equation, so two solutions to it can be combined, with arbitrary constant factors. If we put together the two solutions we just found in this way:
un = (τn - (-τ)-n)/ √5 ,
then we find that u0 = 0 and u1 = 1; from this it follows that Fn = un.
Because τ > 1, positive powers of τ dominate negative powers, and so that expression for Fn implies that for large n, Fn is close to (τn)/√5. Therefore Fn+1/Fn is approximately τ.
Although τ is an irrational number, computation with it is not difficult, and expressions involving τ can often be simplified. For instance, consider the powers of τ. We already have used the fact that τn+2 = τn+1 + τn ; we can apply it to reduce powers of τ thus:
| τ0 | = | 1 | ||
| τ1 | = | τ | ||
| τ2 | = | τ | + | 1 |
| τ3 | = | 2τ | + | 1 |
| τ4 | = | 3τ | + | 2 |
| τ5 | = | 5τ | + | 3 |
| τ6 | = | 8τ | + | 5 |
| τ7 | = | 13τ | + | 8 |
and so on; in general,
τn = Fnτ + Fn-1 .
The negative powers of τ are found similarly, using the recurrence relation in the form τ−n = −τ−(n−1) + τ−(n−2). We find:
| τ0 | = | 1 | ||
| τ−1 | = | τ | − | 1 |
| τ−2 | = | −τ | + | 2 |
| τ−3 | = | 2τ | − | 3 |
| τ−4 | = | −3τ | + | 5 |
| τ−5 | = | 5τ | − | 8 |
| τ−6 | = | −8τ | + | 13 |
| τ−7 | = | 13τ | − | 21 |
and so on; in general,
τ−n = (−1)nFnτ + (−1)n+1Fn+1 .
From these expressions for powers of τ, we can get a better idea of how good the approximate values of τ are. For example, we see that 13τ − 21 = τ−7. This implies
τ = 21/13 + 1/(13τ7).
How big is the error term? Well, we have an expression for τ7, namely 13τ+8; and we know that 13τ is close to 21. Therefore the error term is close to
1/(13(21+8)) = 1/377.
Hmm, 377 = F14. Coincidence? I think not...
Conway, John Horton and Guy,Richard K., The Book of Numbers, New York, Copernicus Books, 1997; pp. 113-126.
Gardner, Martin, The 2nd Scientific American Book of Mathematical Puzzles and Diversions, New York, Simon & Schuster, 1961; chapter 8.
Gies, Joseph and Gies, Frances, Leonard of Pisa, Gainesville, Georgia, Elliott Wave International, Inc., 1983.
Graham, Ronald L., Knuth, Donald E., and Patashnik, Oren , Concrete Mathematics, second ed., Reading, Massachusetts, Addison-Wesley Longmans, 1997; section 1.2.8.
Knuth, Donald E., The Art of Computer Programming, vol. 1, Reading, Massachusetts, Addison-Wesley Longmans, 1997; section 1.2.8.
Sigler, Laurence, Fibonacci's Liber Abaci: A Translation into Modern English of Leonardo Pisano's Book of Calculation, New York, Springer Verlag, 2004.
Stewart, Ian, Letters to a Young Mathematician, New York, Basic Books, 2006.