Fibonacci
Fast Fact
Leonardo of Pisa, known as Fibonacci (Lived c. 1170 – c. 1245.) was the greatest Western mathematician of the Middle Ages. In the absence of his contributions, the scientific revolution started by Nicolaus Copernicus in 1543 would not have been possible. Fibonacci introduced the Hindu-Arabic numeral system to Europe through his book "Liber Abaci" (The Book of Calculation) in 1202, which ultimately allowed science and mathematics to flourish. This work included the famous sequence now named after him, but its primary purpose was to promote the use of Arabic numerals (modern number system) over Roman numerals.
Leonardo Pisano Fibonacci
- Known For: Noted Italian mathematician and number theorist; developed Fibonacci Numbers and the Fibonacci Sequence
- Also Known As: Leonard of Pisa
- Born: 1170 in Pisa, Italy
- Father: Guglielmo
- Died: Between 1240 and 1250, most likely in Pisa
- Education: Educated in North Africa; studied mathematics in Bugia, Algeria
- Published Works: Liber Abaci (The Book of Calculation), 1202 and 1228; Practica Geometriae (The Practice of Geometry), 1220; Liber Quadratorum (The Book of Square Numbers), 1225
- Awards and Honors: The Republic of Pisa honored Fibonacci in 1240 for advising the city and its citizens on accounting issues.
- Notable Quote: “If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circumspect in all matters.”
Beginnings
Fibonacci lived in the Middle Ages. One consequence of this is his biographical details are rather sketchy. Born in the Italian city of Pisa sometime in the years 1170 to 1175, we know his given name was Leonardo Bonacci. Later he became Leonardo of Pisa and then Fibonacci. He was not known as Fibonacci in his lifetime.
His father’s name was Guglielmo Bonacci, a public official concerned with taxation of trade between Pisa and North Africa. Fibonacci’s father spent a lot of time in the Arab port city of Bugia (now in Algeria). His work in taxation of trade led him to believe the future would be bright for people who understood numbers thoroughly.
He had his son schooled in mathematics for a short time in Bugia.
Discovering a New Way
The young Fibonacci was fascinated when he learned that Arab mathematicians did not use the Roman system of numbers: I, II, III, IV, V, etc, used in Europe for over a thousand years.
Western mathematics had in fact declined into deep hibernation after the fall of Ancient Greece. And although Ancient Greek mathematics had been stunningly brilliant – particularly in geometry – it was far from fully developed. It was severely hampered by the Greek number system, where numbers were represented by letters of the alphabet. To see the awkwardness of this system, think about calculating 17 × 19; it’s easy to do using modern numbers. Imagine, though, trying to multiply Q × S (the 17th and 19th letters of the alphabet). Suddenly what’s easy becomes awkward.
In the Roman system 17 × 19 would be XVII × XIX. The clumsy notation and the lack of the place value concept of ones, tens, hundreds, thousands, etc, made life as difficult for Roman mathematicians as it had been for Greeks.
In addition to their unwieldy numbers, Ancient Greeks and Romans also lacked the number zero; this made arithmetic and mathematics awkward, and would have made the development of modern mathematics impossible.
Fibonacci immersed himself in the new number system he learned in Bugia, realizing it was a huge improvement on Roman numerals. In addition to what he learned in Bugia, Fibonacci later traveled around the Mediterranean to Egypt, Greece, Sicily, Southern France, and Syria, learning more mathematics.
It Began in India
The number system Fibonacci fell in love with was devised in India, where the Hindi symbols for 0 to 9 are:
Most recognizable to western eyes are the Hindi symbols for zero, two and three. The properties of the number zero were defined in Indian mathematics by Brahmagupta.
Numbers on the Move
From India the new numbers traveled west to Persia, then to the Middle East and North Africa, and then, as we’ll see, to Europe. As the numbers moved westward, their shapes changed somewhat.
In Europe people called the new numbers Arabic numbers. Today the system is most often called the Hindu-Arabic number system.
The new number system spread westward.
Fibonacci’s Book of Calculation
Fibonacci believed the Indian number system had huge advantages over the Roman system and believed the people of Europe should adopt it. In 1202 he published Liber Abaci – the Book of Calculation – which began the spread of the modern number system in the West. Fibonacci updated the book and published a new edition in 1228.
Near the beginning of the Book of Calculation he wrote:
I received an excellent education in the methods of the nine Indian numbers; the knowledge of these methods pleased me more than anything else… Therefore strictly embracing the Indian method, and adding some of my own ideas, and more still from Euclid’s geometry, I assembled them in this book as understandably as I could.
Leonardo of Pisa, Book of Calculation, 1228
His Book of Calculation showed how calculations in commerce, finance, and pure mathematics could be carried out with the new number system.
How Important was Fibonacci’s Book?
Fibonacci’s book was vital in planting a seed in European minds. Popularizing the new numbers was a long process; widespread adoption began only after the twin events of:
- Gutenberg’s invention of the printing press in 1440 (only hand-written copies of Fibonacci’s works had been available before then)
- the fall of Constantinople in 1453
The fall of Constantinople resulted in its refugees arriving in Italy. Some of the refugees brought with them Ancient Greek texts that had been locked away for many centuries in Constantinople. These Greek texts helped trigger the Renaissance in Italy.
An illustration from Gregor Reisch’s 1503 work Margarita Philosophica. The man on the left working with the new number system is happy, while the other (Pythagoras), using a counting board, is sad. In the center the female Arithmetica wears a dress emblazoned with the new numbers.
Fibonacci’s Book of Calculation was also important for European commerce and finance. In Arab lands the new number system had been used only by mathematicians and scientists. Fibonacci saw the superiority of the new system for businesses and devoted several chapters of his book to show calculations of profit, interest, and currency conversions. In fact, the book’s immediate impact on the commercial world was much greater than on the scientific world.
Some of the topics Fibonacci considered in his book were: the new numbers; multiplication and addition; subtraction; division; fractions; rules for money; accounting; quadratic and cube roots; quadratic equations; binomials; proportion; rules of algebra; checking calculations by casting out nines; progressions; and applied algebra.
The algebra in the Book of Calculation was principally influenced by work published by the mathematicians Al-Khwarizmi from Persia; Abu-Kamil from Egypt; and Al-Karaji from Baghdad.
See the Pen Roman & Arabic Numeral Converter by Fabien Laurent Patrice Egot (@equant_org) on CodePen.
The Fibonacci Sequence
Fibonacci also famously considered the rabbit problem, which gave rise to the Fibonacci Sequence.
The Fibonacci sequence appears in nature more frequently than one might expect; the branching of trees, arrangement of leaves on a stem, the fruitlets of a pineapple, even the spiral galaxies all exhibit Fibonacci patterns.
The Problem
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair, which from the second month on becomes productive?
The Solution
The month-by-month solution to the problem became known as the Fibonacci Sequence. It involves adding the preceding two terms to one another to generate the next term:
This remarkable sequence, which was already known in Indian mathematics, occurs repeatedly in mathematics and also in the natural world, where, for example, the scales of pine cones run in spirals arranged in ratios determined by the Fibonacci Sequence.
Even in art the Fibonacci Sequence is prominent. If you divide one term in the sequence by the previous term, the result gets closer and closer to the golden ratio – loved by artists and architects – as the terms get larger.
See the Pen Animated Fibonacci Spiral by Tim Inman (@thehack) on CodePen.
A Great Mathematician
Fibonacci did not merely copy the works of the Greeks, Indians, and Arabs. He was a brilliant mathematician in his own right.
His fame spread to Frederick II, the Holy Roman emperor, whose own mathematicians were unable to solve a number of problems, so he challenged Fibonacci. Fibonacci published his solutions to the challenges in his 1225 book Flos (Flower).
Completing the Basic Notation of Modern Arithmetic
After Fibonacci had introduced modern numbers to the West, a number of symbols still had to be introduced to transform arithmetic into its modern notation. These were:
- The plus (+) and minus (-) signs introduced by the German mathematician Johannes Widmann in 1489.
- The equals sign (=) introduced by the Welsh mathematician Robert Recorde in 1557.
- The multiplication sign (x) introduced by the English mathematician William Oughtred in 1631.
- The division sign (÷) introduced by the Swiss mathematician Johann Rahn in 1659 in his book Teutsche Algebra. (It is possible the sign was actually introduced by the book’s editor John Pell, an English mathematician.)
Golden ratio
Golden ratio is based on Fibonacci numbers. Even though Fibonacci introduced these numbers to the Western world, they were actually discovered by Indian mathematicians hundreds of years earlier. The poet Pingala used them to count the syllables of Sanskrit poetry around 200 B.C., and the method for calculating them was formulated by the Indian mathematician Virahanka (Devanagari: विरहाङ्क) around 800 years later. Interesting how mathematics owes so much to India.
The Mona Lisa is a famous painting by Leonardo da Vinci, a Renaissance artist and scientist. The painting depicts a woman with a mysterious smile and a landscape background. One of the reasons why the Mona Lisa is so admired is because of its use of the golden ratio, a mathematical proportion that is considered to be aesthetically pleasing and harmonious.
The golden ratio, also known as the divine proportion, is a ratio of approximately 1.618 that can be found in nature, art, architecture, music, and geometry. It is expressed by the Greek letter phi (Φ) and can be derived from the Fibonacci sequence.
The golden ratio can be seen in the Mona Lisa in several ways. For example, if a rectangle bounds the face and this rectangle is divided by drawing a line across her eyes, it creates another golden rectangle. The result is that the ratio of the length of Mona Lisa’s head to her eyes is also that of the golden ratio. Another example is that if a triangle is drawn from her chin to her eyes and to the top of her head, it forms a golden triangle, which has angles of 36°, 72°, and 72°. The golden triangle can also be found in other parts of the painting, such as the shape of her nose and mouth.
Da Vinci was fascinated by the golden ratio and its connection to nature, art, and geometry. He illustrated a book called The Divine Proportion by Luca Pacioli, a mathematician and friend of Da Vinci. The book explores the mathematical and artistic properties of the golden ratio and its applications in various fields. Da Vinci also used the golden ratio in some of his other works, such as The Last Supper and The Vitruvian Man.
The use of the golden ratio in the Mona Lisa shows Da Vinci’s skill and knowledge as an artist and a scientist. He was able to create a realistic and harmonious portrait that captures the beauty and mystery of the human face. The Mona Lisa is not only a masterpiece of art, but also a testament to the power and elegance of mathematics.
Fibonacci’s Other Work
Fibonacci’s most famous work by far is his Liber Abaci (Book of Calculation). The main purpose of this book was to encourage everyone to abandon Roman numerals and use the Indian system of numbers; it was a general book of mathematics. He also wrote other books, some of which were solely for pure mathematicians. He established the Tuscan school of mathematicians and wrote:
In 1223: Practica Geometriae (Practical Geometry) – a mixture of pure mathematics, theorems, proofs, and practical applications of geometry, such as using similar triangles to calculate the heights of tall objects.
Before 1225: Epistola and Magistrum Theodorum (A Letter to Master Theodore) – a letter to Frederick II’s philosopher Theodorus Physicus solving three problems in mathematics.
In 1225: Flos (Flower) – solutions to problems in algebra
In 1225: Liber Quadratorum (The Book of Squares) – A highly mathematical number theory book dealing with solutions to Diophantine equations – in this work we see just how accomplished a mathematician Fibonacci truly was.
Date unknown: Di Minor Guisa (A Smaller Manner) a book on commercial arithmetic. (No copies exist today.)
Date unknown: Commentary on Book X of Euclid’s Elements (No copies exist today.)
Death and Legacy
Little is known about the end of Fibonacci’s life. We know he was alive in 1240, because his achievements were recognized by his home city of Pisa, which granted him a salary for his work. He would have been about 70 years old at this time.
In addition to "Liber Abaci," Fibonacci authored several other books on mathematical topics ranging from geometry to squaring numbers (multiplying numbers by themselves). The city of Pisa (technically a republic at that time) honoured Fibonacci and granted him a salary in 1240 for his help in advising Pisa and its citizens on accounting issues. Fibonacci died between 1240 and 1250 in Pisa.
Fibonacci is famous for his contributions to number theory.
- In his book, "Liber Abaci," he introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.
- He introduced the bar that is used for fractions today; previous to this, the numerator had quotations around it.
- The square root notation is also a Fibonacci method.
It has been said that the Fibonacci Numbers are nature's numbering system and that they apply to the growth of living things, including cells, petals on a flower, wheat, honeycomb, pine cones, and much more.
Further Reading
R. E. Grimm
The Autobiography of Leonardo Pisano
Fibonacci Quarterly, Vol 11, 1973, pp.99-104
Leonardo Pisano Fibonacci and L. E. Sigler
The Book of Squares
Academic Press, February 11, 1987
Karen Hunger Parshall
The Art of Algebra from Al-Khwarizmi to Viète
History of Science, Vol. 26, No. 72, June 1988, pp.129-164