Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.
Image: ubuntu2204
Leonardo Pisano, or Fibonacci
1. Biography.
Leonardo Fibonacci Pisano (Leonard, fils de Bonacci, of Pisa), was born in Pisa around 1170, and died around 1240, likely also in Pisa[WikiFib] , [MTFib]. He grew up in what is now Algeria, where his father Guilielmo Bonacci had been posted as a consul.
There he was introduced to the Hindu-Arabic system for writing numbers, which he realised was a great improvement for doing effective arithmetic over the Roman numerals in common use in Europe at the time.
As a young man, he travelled widely around the Mediterranean, learning about the mathematics used by the merchants of the region. This period of traveling may have led to his other name, Leonardo Bigollo Pisano: Leonard "the traveller" from Pisa.
He returned to Pisa around 1200, where it is believed that he stayed. He wrote a number of books (see below) and earned a reputation as a mathematician. He was recognised internationally, being presented to the court of Holy Roman Emperor Frederick II when the court visited Pisa in 1225, and engaging in mathematics with the court scholars.
He was awarded a salary by Pisa in 1240, in recognition of his contributions to accounting, it seems. By 1250 he had passed away, likely still in his home town of Pisa.
No contemporaneous images of Fibonacci are known. The image[HPP] below shows a statue from Pisa erected in his honour in 1853.
2. Fibonacci's Contributions to Discrete Mathematics.
A. The Liber Abaci and Hindu-Arabic numerals
Fibonacci's 1202 treatise Liber Abaci , or "Book of Calculation," introduced the notion of the Hindu-Arabic numerals, and the into Europe, where typically Roman numerals were still in use. These numbers had been in use in North Africa and the middle east since the 900s, and originated in India in the 1st to 4th centuries[HAN].
When my father, who had been appointed by his country as public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it[MTFib].
The Liber Abaci described a number of algebraic and calculational techniques intended for commercial purposes, such as currency conversions and calculations of interest on a loan, as well as algorithms for calculating remainders under integer division, formulae for arithmetic series, methods for making approximations (for example, of irrational numbers) and geometric proofs.
It is not clear what amount of the content of the Liber Abaci is due to Fibonacci; a certain amount would have been collected from material he came across in his travels. For example, mathematical historians[Abu] draw a direct line from the Islamic mathematician Abu Kamil Shuja to Fibonacci:
... his [Abu Kamil's] work was the basis of Fibonacci's books. So not only is Abu Kamil important in the development of Arabic algebra, but, through Fibonacci, he is also of fundamental importance in the introduction of algebra into Europe.
B. The Liber Quadratorum
Fibonacci is known to be the author of other texts including Practica geometriae (1220), Flos (1225), and Liber quadratorum (?), as well as lost treatises including Di minor guisa.
Liber quadratorum, or the book of squares, deals with number theory[LQ], and among other results includes the observation that squares can be decomposed into the sum of odd numbers. Fibonacci uses this to construct Pythagorean triples. It also contains the result known as Fibonacci's identity, which says that the set of sums of integers is closed under multiplication.
While it contains many interesting and apparently original results, it appears that it was mostly ignored or simply unknown for hundreds of years that followed[LQ].
C. The Fibonacci sequence
An example in the Liber Abaci introduces a problem in the growth of a rabbit population which involves the use of a recurrence relation to generate the next generation's size from the existing population. The sequence thus obtained: has become known as the Fibonacci sequence in his honour.
The sequence was, however, already described by the Indian teacher Virahanka[Singh] around the 6th to 8th centuries in the context of poetic meters, and may have been suggested by Indian musicians even earlier.
3. Reflections on the constructed nature of the digital world.
On the one hand, there wasn't a definitive statement on Fibonacci's date of death... anywhere from 1240, but probably by 1250. But overall, I managed to find sources which seem reliable, inasmuch as they agree on the dates and facts of Fibonacci's life, and on the important contributions he made in his life.
On the other hand, I could find webpages which referred to astrological connections of Fibonacci numbers with the zodiac... which did not give any sources, and which I am pretty confident are not reliable; and quite possibly just something someone threw together based on a few ideas they came up with that morning.
I imagine that if I had been researching a different person - or perhaps different type of person (not a mathematician), the constructed nature of the digital world would be even more apparent.
References.
[WikiFib] The Wikipedia page Fibonacci, accessed April 2023.
[MTFib] O'Connor, John J.; Robertson, Edmund F., "Leonardo Pisano Fibonacci", MacTutor History of Mathematics Archive, University of St Andrews
[HPP] The photo was taken in 2006, by Hans Peter Postel. Its use is licensed under the Creative Commons Attribution 2.5 Generic license.
[Abu] O'Connor, John J.; Robertson, Edmund F., "Abu Kamil Shuja ibn Aslam ibn Muhammad ibn Shuja", MacTutor History of Mathematics Archive, University of St Andrews.
[Singh] Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44.
[HAN] The Wikipedia page Hindu–Arabic numeral system, accessed April 2023.
[LA] The Wikipedia page Liber Abaci, accessed April 2023.
[LQ] The Wikipedia page Liber Quadratorum, accessed April 2023.
[WFS] The Wikipedia page Fibonacci Sequence, accessed April 2023.