The Fibonacci sequence and the golden ratio

Leonardo of Pisa, known as Fibonacci (“son of Bonaccio”), was the most important European mathematician of the Middle Ages. He grew up in Bugia, on the North African coast, where his father ran a trading post for Pisan merchants, and there he learned the Hindu-Arabic numeral system — the nine digits and zero — from Arab teachers. In the Liber Abaci (1202) he introduced it to the West, where merchants still calculated with Roman numerals and the abacus, showing through hundreds of problems how much more powerful it was for trade, currency exchange and bookkeeping.

That book contains the problem that made him famous: starting from a pair of rabbits that, once mature, produces a new pair every month, the number of pairs month by month is $1,\ 2,\ 3,\ 5,\ 8,\ 13,\ 21,\ 34,\ \ldots$ Each term is the sum of the two before it, that is, the sequence defined by the recurrence $F_{n+1} = F_n + F_{n-1}$. Curiously, the sequence was already known to Indian scholars (for example Pingala and Hemachandra) in connection with counting poetic rhythms, centuries before the Liber Abaci.

The link with the golden ratio was noticed only later. Already Euclid, in the Elements, had defined dividing a segment “in extreme and mean ratio”: the golden number $\varphi = \frac{1+\sqrt5}{2} \approx 1.618.$ In the seventeenth century Kepler observed that the ratio of two consecutive Fibonacci numbers approaches $\varphi$ ever more closely: $\frac{F_{n+1}}{F_n} \longrightarrow \varphi \qquad (n \to +\infty),$ a fact proved today with Binet’s formula. So a problem about rabbits, born as an exercise in merchants’ arithmetic, turned out to be one of the most studied sequences in mathematics (Boyer, Katz).

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Topics: Sequences
Concepts: Sequence · Sequence by recurrence · Fibonacci
People: Leonardo Fibonacci · Euclid · Kepler