GoKawiilFibonacci's Composed Fractions
2026-03-26
Leonardo of Pisa (c1170-1241+), better known as Fibonacci, wrote Liber Abaci to introduce and advocate the adoption of “Indian figures” (Hindu-Arabic numerals) to Italy. The book explains how to perform arithmetic and common business problems using the digits and a positional number system. Leonardo uses a composed fractions (aka composite fractions, aka linked fractions) notation to express mixed radix quantities (i.e. positional numbers where the base varies by position). This notation lasted about 300 years until the adoption of decimal fractions. Drawing from his examples, we derive algorithms for the arithmetic operations in recipe and in formal Typescript representations.
Liber Abaci page 2v, introducing Indian numbers (Source: Museo Galileo)
Background
Little is known about Leonardo of Pisa’s life. (Also known as Leonardo Pisano, which just means Leonardo of Pisa.) He provides a short biographical note in Liber Abaci (an unusual touch) where he notes he traveled to Bugia (today’s Béjaïa, Algeria) to work with his father who was a public official in the customshouse. In Bugia, he learned about Indian figures (Hindu-Arabic numerals), Arabic mathematics, Pythagorean arcs, Euclid’s algorithm for computing the greatest common divisor, and Euclidean geometry. Later, he traveled extensively learning more mathematics. Back in Pisa, Leonardo had access to multiple high-quality mathematical works (at least some in Latin translation). He was inspired to write a work introducing the Italian people to these figures and the new methodology.
Liber Abaci (the Book of Calculation) is a thick manuscript of fifteen chapters (the Sigler translation is over 600 pages) written in Latin. The first edition was written in 1202 (no extant copies) and the second edition in 1228 (nineteen known copies). The first chapter introduces Hindu-Arabic numerals and compares them with Roman numerals (who Leonardo’s audience were used to working). The next four chapters cover multiplication, addition, subtraction, and division of integral numbers. This is followed by two chapters on fractions. Four chapters on business problems then follow, such as bartering, assigning profits to shareholders, and the alloying of metals. Chapter 12 contains many miscellaneous problems touching on topics in algebra and linear equations and includes the famous rabbits problem that leads to the Fibonacci sequence (see Aside below). Chapter 13 concerns an iterative method for solving equations. Chapter 14 concerns finding roots of equations and the final chapter discusses geometry.
The book is focused on practical concerns for merchants and other business people, with examples drawn from trade and finance. The book fulfilled a need. Pisa was an international trade hub and, even though it would lose that status soon in a war with Genoa, the various Italian city states were in the middle of a trade revolution. Hundreds of schools and textbooks were created over the next four centuries to teach calculation under the new method. This Abbacus School tradition focused on training students in mathematics useful for business and trade. The training used Hindu-Arabic numerals and arithmetic and followed the techniques found in Liber Abaci.
While historians differ on the precedence and impact of Liber Abaci to the Abbacus tradition and the Arithmetic Revolution (e.g. Høyrup 2022, Devlin 2011 Chapter 8), of the extant volumes known, Liber Abaci is the most comprehensive and mathematically sophisticated, by including proofs, more advanced problems and topics, and featuring far fewer errors.
Notation
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