Fibonacci (Leonardo of Pisa) — mathematician and the Fibonacci sequence

Fibonacci sequence and its properties

The Fibonacci sequence is a series of integers in which each term after the first two is the sum of the two preceding terms. Common starting conventions are 0, 1, 1, 2, 3, 5… or 1, 1, 2, 3, 5… The ratio of successive Fibonacci numbers converges to the golden ratio (phi), approximately (1 + sqrt(5))/2, a limit important in algebra and geometry. More on the sequence: Fibonacci numbers.

History and Liber Abaci

Fibonacci traveled with his merchant father and studied arithmetic in North Africa, where Hindu–Arabic numerals were already in use. In 1202 he published Liber Abaci (The Book of Calculation), which demonstrated arithmetic with the new numerals, practical problems for merchants, and examples that included the rabbit reproduction puzzle that illustrates the sequence. The book played a major role in shifting Europe away from Roman numerals. Read about Liber Abaci: Liber Abaci.

Uses, examples, and importance

The Fibonacci sequence appears across mathematics and applied fields: algorithm analysis, recursive algorithms, combinatorics, and continued fractions. It also appears in models and patterns in biology, such as phyllotaxis (leaf arrangements) and branching patterns, where the sequence and related ratios describe efficient packing. For more context on applications see: mathematical applications and historical influence.

Distinctive facts

• Fibonacci’s problem that introduced the sequence was framed as a population growth model for rabbits, not as a formal algebraic definition.
• The sequence connects to algebraic identities, matrix powers, and Binet’s formula which expresses terms via powers of the golden ratio.
• Different traditions start the list differently; both 0-first and 1-first conventions are in common use.

For further reading on his life and the transmission of mathematical knowledge in medieval Europe consult general histories and specialist notes: biographical sources and an overview resource: Leonardo of Pisa overview. Additional materials and modern expositions are available for readers interested in number theory and computational uses: Fibonacci sequence resources and Liber Abaci studies.