Leonardo Fibonacci or Leonardo of Pisa
/ FIB-ah-NAH-chi /
c. 1170–c. 1250
Italian Mathematician and Merchant
Mathematics Ranking 18th of 46
Fibonacci sequence on a Macau stamp from 2005.
Leonardo Fibonacci’s masterpiece, Liber abbaci, or Book of calculation, first published in 1202, was very influential in spreading the use of the Hindu-Arabic numerals in Europe. It also provided Europe’s first comprehensive introduction to Islamic mathematics including Al-Khwarizmi’s Algebra. The many surviving manuscripts testify to the wide readership the book enjoyed. Leonardo’s book also played an important role in the birth of the modern financial system and the way of doing business that depends on sophisticated banking methods.
The young Fibonacci grew up in North Africa, where his father was in charge of a customs house. As a young man, he traveled extensively around the Mediterranean, visiting Egypt, Syria, Greece, Sicily, and southern France, observing the various computational systems used by merchants, particularly in the Islamic world. He quickly recognized the enormous advantage of the Hindu-Arabic system, and introduced the new numerals with the following words:
"The nine Indian numerals are 9, 8, 7, 6, 5, 4, 3, 2, 1. With these nine, and with the sign 0, which the Arabs called zephir (cipher), any desired number can be written."
It was mainly through Fibonacci’s Liber abbaci that the Indian numerals came to be widely known in Christian Europe.(1) For a long time they were used alongside the Roman numerals. The change from the latter to the former was a slow process with a number of false starts, primarily because the abacus remained popular for carrying out calculations, and traders and others engaged in commercial activities were reluctant to adopt a new system that was difficult to comprehend.
Prior to the adoption of the Hindu-Arabic system of arithmetic, traders used one of two methods to carry out their calculations: finger arithmetic or a mechanical abacus. The former was a very sophisticated system using all the fingers and thumbs of both hands—the general English term is “digit,” of course, from the Latin digitus meaning finger or toe. (This is how finger arithmetic led to our present-day use of the word “digit” for the basic number symbols 0, 1, 2, through 9.)
Ancient finger arithmetic was capable of representing numbers up to 10,000. Unfortunately, it required considerable training to master the system and a lot of practice to become fluent. Medieval textbooks, including some surviving early copies of Liber abbaci, often had an initial page with carefully drawn diagrams showing the positions of the fingers to represent various numbers and perform different operations.
In the Muslim world and much of Europe, the abacus was a flat board with ruled lines on which small pebbles were placed and moved around. This form of early calculation gave rise to our present word “calculus” for a system of calculation procedures (and hence led to the word “calculate” itself), since the Latin word for a pebble is calculus. The more familiar form of abacus now found in toy stores, with beads on wires instead of pebbles on a ruled board, was used in China, where it was called a Xuanpan. As with finger arithmetic, calculating with an abacus required considerable practice to achieve accuracy and speed.
Traders became highly proficient with whatever method of calculation they used, so the two systems were efficient—at least for one-on-one trading. The significance of that last caveat is that neither method leaves a record of the calculation. One obvious consequence is that if either party suspected an error had been made, the only option was to repeat the calculation, and continue to do so until both traders were satisfied. But that was not the major issue. Another, less obvious drawback was far more significant. When the businessmen in twelfth-century Italy started to form trading empires, they had a clear need for an audit trail. An individual sitting in Pisa controlling a network of traders needed to be able to review the financial books on a regular basis.
Hindu-Arabic arithmetic, carried out on paper, was not only much easier to learn and master than finger arithmetic or use of an abacus board, it also left a clear audit trail that anyone could inspect and check for arithmetical accuracy. Almost certainly, then, two factors contributed to the rapid adoption of Hindu-Arabic arithmetic in thirteenth-century Italy: its relative ease of learning and use, and the automatic provision of a complete record of calculation.
At times there were dictates from above to discourage the use of the new numerals. In 1299, for example, the city of Florence passed an ordinance prohibiting the use of the new numerals since they were more easily altered (e.g., by changing 0 to 6 or 9) than Roman numerals or numbers written out in words. As late as the end of the fifteenth century, the mayor of Frankfurt ordered his officials to refrain from calculating with Hindu-Arabic numerals. Even after the decimal numeral system was well established, Charles II of Sweden (1682-1718) tried in vain to ban the decimal system and replace it with a base 64 system for which he devised sixty-four symbols! But these were temporary setbacks. Once the contest between the “abacists” (those in favor of the use of the abacus or some mechanical device for calculation) and the “algorists” (those who favored the use of the new numerals) had been won by the latter, it was only a matter of time before the final triumph of the new numerals, with bankers, traders, and merchants adopting the system for their daily calculations. On top of this, following the appearance of Liber abbaci, the teaching of arithmetic became hugely popular throughout Italy, with perhaps a thousand or more handwritten texts being produced over the following three centuries.(2)
Leonardo’s book not only provided a bridge that allowed modern arithmetic to cross the Mediterranean, but also bridged the mathematical cultures of the Arabic and European worlds, by showing the West the algebraic way of thinking that forms the basis of modern science and engineering (though not our familiar symbolic notation for algebra, which came later.) The expansion of mathematics in Europe, in particular algebra, during the thirteenth, fourteenth, and fifteenth centuries was largely driven by the business world with Leonardo’s book paving the way. Algebra was not taught as an academic discipline in European universities until the middle of the sixteenth century.
Liber abbaci also marked the beginning of the modern financial system with complex banking methods explained through numerous practical examples. The book contained not only the rules for computing with the new Hindu-Arabic numerals, but also numerous problems of various sorts in such practical topics as calculating present value, compounding interest, evaluating geometric series, dividing profits from business ventures, and pricing goods and monies involving a complex variety of weights, measures, and currencies.
Leonardo was the first to develop present-value analysis for comparing the economic value of alternative contractual cash flows. Present-value analysis is a method for comparing the relative economic value of differing payment streams, taking into account the changing value of money over time. Mathematically reducing all cash flow streams to a single point in time allows the investor to decide which is unambiguously the best. According to a 2001 survey of corporate financial officers, the present-value criterion is now used by virtually all large companies in capital budget decision making.(3)
The Rabbit Problem
Although later generations of mathematical historians would consider Liber abbaci one of the most influential books of all time, to most people its authors’ greatest fame would rest on one problem concerning rabbits.(4) It states:
“How many pairs of rabbits are created by one pair in one year?”
A certain man had one pair of rabbits together in a certain enclosed place, and one wishes to know how many are created from the pair in one year when it is the nature of them in a single month to bear another pair, and in the second month those born to bear also. Leonardo proceeded to calculate: After the first month there will be two pairs, after the second, three. In the third month, two pairs will produce, so at the end of that month there will be five pairs. In the fourth month, three pairs will produce, so there will be eight. Continuing in this fashion, he showed that there will be 377 pairs by the end of the twelfth month. Listing the sequence 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 in the margin, he noted that each number is found by adding the two previous numbers, and “thus you can do it in order for an infinite number of months.” The sequence, calculated recursively, is known today as a Fibonacci sequence. The Fibonacci numbers were given their name by the French mathematician Edouard Lucas in the 1870s, after his compatriot, the historian Guillaume Libri, gave Leonardo the nickname Fibonacci in 1838.
Fibonacci Sequence in Nature
A lot of the interest in the Fibonacci sequence is due to the surprising frequency with which they occur in nature. For example, the number of petals on flowers is a Fibonacci number more often than would be expected from pure chance: an iris has 3 petals; primroses, buttercups, wild roses, larkspur, and columbine have 5; delphiniums have 8, ragwort, corn marigold, and cineria 13, asters, black-eyed Susan, and chicory 21, daisies 13, 21, or 34; and Michaelmas daisies 55 or 89.
Sunflower heads, and the bases of pinecones, exhibit spirals going in opposite directions. The sunflower has 21, 34, 55, 89, or 144 clockwise, paired respectively with 34, 55, 89, 144, or 233 counterclockwise; a pinecone has 8 clockwise spirals and 13 counterclockwise. All are Fibonacci numbers.
Pine cone with Fibonacci sequence. Israeli stamp from 1961.
The Golden Ratio
The key mathematical fact underlying nature’s seeming preference for Fibonacci numbers is their close connection to an equally famous mathematical constant known as the Golden Ratio. Often denoted by the Greek letter Ø (phi), the Golden Ratio is, like the mathematical constant π, an irrational number—a number whose decimal expansion continues forever, without ever settling into a regular repeating pattern. The decimal expansion of π begins 3.14159; Ø starts out 1.61803.
The number Ø first appeared in Euclid’s Elements (written around 350 B.C.E.), where it is defined as the ratio into which you should divide a line so the ratio of the entire line to the longer division equals that of the longer division to the shorter. Euclid gave it the name “extreme and mean ratio.” It acquired the name “Golden Ratio” in 1835, in a book by the mathematician Martin Ohm.
Various false beliefs have attached themselves to the number: that it is the aspect ratio of the rectangle the human eye finds the most aesthetically pleasing, leading the ancient Greeks to incorporate it into much of their architecture, including the Parthenon in Athens; that various painters and musicians used it in their compositions; that if you measure the distance from the top of your head to the floor and divide by the distance from your navel to the floor you get the Golden Ratio; and several more.
However, the Golden Ratio is genuinely exhibited by the growth of plants. Nature’s inevitable preference for efficiency leads it to place petals on flowers, seeds in flowerheads, and leaves on plant stems in a fashion that depends on the Golden Ratio, which has a mathematical property that results in optimized structure.
The connection between the Fibonacci numbers and the Golden Ratio was first verified in the nineteenth century: If each Fibonacci number is divided by the one that precedes it, the answers you get grow steadily closer to the Golden Ratio—in mathematical terms, the limit of those ratios is the Golden Ratio. (The first few values work out: 2/1=2; 3/2=1.5; 5/3=1.666; 8/5=1.6; 13/8=1.625; 21/13=1.615; 34/21=1.619; 55/34=1.618.) Since Ø is an irrational number, whereas the number of petals, spirals, or stamins in any plant or flower has to be a whole number, and because of the above limit property, this will tend to be a Fibonacci number.
Lastly, keen interest with Fibonacci numbers led to the formation in 1963 of the Fibonacci Association, which continues to publish a regular mathematical journal, the Fibonacci Quarterly. The association and its journal are devoted to mathematical investigations of the Fibonacci numbers and sequences like it.
(1) George Gheverghese Joseph, The Crest of the Peacock – Non-European Roots of Mathematics (Princeton, 2011), p. 466
(2) Keith Devlin, The Man of Numbers – Fibonacci’s Arithmetic Revolution (New York, 2011), p. 6.
(3) Keith Devlin, Finding Fibonacci – The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World (Princeton, 2017), p. 194.
(4) Keith Devlin, The Man of Numbers – Fibonacci’s Arithmetic Revolution, p. 143.
Key References: The Man of Numbers – Fibonacci’s Arithmetic Revolution by Keith Devlin, 2011; Fibonacci’s Liber Abbaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation by Laurence Sigler, 2002; Fibonacci and the Financial Revolution by William N. Goetzmann, 2004.