The 13th Century Italian Leonardo Pisano was perhaps the most talented European mathematician of the Middle Ages. But we little know about his real name, as he is best known by his nickname Fibonacci. This would come as a surprise to Leonardo himself, that he has been immortalized in the famous sequence – 0, 1, 1, 2, 3, 5, 8, 13, … – rather than for what is considered his far greater mathematical achievement – helping to popularize our modern number system in the Latin-speaking world.

As I mentioned earlier, his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa, Italy, the city with the famous Leaning Tower, about 1175 AD. There are a couple of explanations for the meaning of Fibonacci (pronounced fib-on-arch-ee) that we know now: Fibonacci is a shortening of the Latin “filius Bonacci”, which means “the son of Bonaccio”. His father’s name was Guglielmo Bonaccio. Fi’-Bonacci is like the English names of Robin-son and John-son. But in Italian Bonacci is also the plural of Bonaccio; therefore, two early writers on Fibonacci: Boncompagni and Milanesi regard Bonacci as his family name (as in “the Smiths” for the family of John Smith). Fibonacci himself wrote both “Bonacci” and “Bonaccii” as well as “Bonacij”; the uncertainty in the spelling is partly to be ascribed to this mixture of spoken Italian and written Latin, common at that time. However he did not use the word “Fibonacci”. Others also think Bonacci may be a kind of nick-name meaning “lucky son” (literally, “son of good fortune”).

Pisa in its day, was an important commercial town and had links with many Mediterranean ports. Leonardo’s father, Guglielmo, was a kind of customs officer in the present-day Algerian town of Béjaïa, formerly known as Bugia or Bougie, where wax candles were exported to France. They are still called “bougies” in French. So Leonardo grew up with a North African education under the Moors and later sent on business trips to Egypt, Syria, Greece, Sicily and Provence. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realized the many advantages of the “Hindu-Arabic” system over all the others.

The Roman Empire left Europe with the Roman numeral system which we still see, amongst other places, in the copyright notices after films and TV programs. The Roman numerals were not displaced until the mid 13th Century AD, and Leonardo’s book, *Liber Abaci* (which means “The Book of Calculations”), was one of the first Western books to describe their eventual replacement. In 1200 he returned to Pisa and used the knowledge he had gained on his travels from the Mediterranean coast, to write *Liber Abaci* which was published in 1202 in which he introduced the Latin-speaking world to the decimal number system.

The first chapter of part 1 says:

These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.

Italy at the time was made up of small independent towns and regions and this led to use of many kinds of weights and money systems. Merchants had to convert from one to another whenever they traded between these systems. Leonardo wrote *Liber Abaci* especially for these merchants, filled with practical problems and worked examples demonstrating how simply commercial and mathematical calculations could be done with this new number system compared to the clumsy system of Roman numerals then in use in Europe. Despite its obvious advantages, uptake of the system in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify than Roman numerals. However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century, making the Roman system obsolete. The horizontal bar notation for fractions was also first used in this work (although following the Arabic practice of placing the fraction to the left of the integer). The impact of Leonardo’s book as the beginning of the spread of decimal numbers was his greatest mathematical achievement indeed.

The book’s influence on medieval mathematics is undeniable, and it does also include discussions of a number of other mathematical problems such as the Chinese Remainder Theorem, perfect numbers and prime numbers, formulas for arithmetic series and for square pyramidal numbers, Euclidean geometric proofs, and a study of simultaneous linear equations along the lines of Diophantus and Al-Karaji. He also described the lattice (or sieve) multiplication method of multiplying large numbers, a method – originally pioneered by Islamic mathematicians like Al-Khwarizmi – algorithmically equivalent to long multiplication.

Neither was *Liber Abaci* Fibonacci’s only book, although it was his most important one. His *Liber Quadratorum* (which means “The Book of Squares”), for example, is a book on algebra, published in 1225 in which appears a statement of what is now called Fibonacci’s identity – sometimes also known as Brahmagupta’s identity after the much earlier Indian mathematician who also came to the same conclusions – that the product of two sums of two squares is itself a sum of two squares e.g. (1^{2} + 4^{2})(2^{2} + 7^{2}) = 26^{2} + 15^{2} = 30^{2} + 1^{2}.

Now let’s talk about Leonardo’s the most popular work: the famous sequence. In *Liber Abaci*, chapter 12, he introduces the following problem:

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.

He then continues on to solve and explain the solution:

Because the above written pair in the first month bore, you will double it; there will be two pairs in one month.

One of these, namely the first, bears in the second month, and thus there are in the second month 3 pairs;

of these in one month two are pregnant and in the third month 2 pairs of rabbits are born, and thus there are 5 pairs in the month;

…

there will be 144 pairs in this [the tenth] month;

to these are added again the 89 pairs that are born in the eleventh month; there will be 233 pairs in this month.

To these are still added the 144 pairs that are born in the last month; there will be 377 pairs, and this many pairs are produced from the above written pair in the mentioned place at the end of the one year.You can indeed see in the margin how we operated, namely that we added the first number to the second, namely the 1 to the 2, and the second to the third, and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above written sum of rabbits, namely 377, and thus you can in order find it for an unending number of months.

Now imagine that there are pairs of rabbits after months. The number of pairs in month will be (in this problem, rabbits never die) plus the number of new pairs born. But new pairs are only born to pairs at least 1 month old, so there will be new pairs. So we have . which is simply the rule for generating the Fibonacci numbers: add the last two to get the next.

Now the question is: The work Leonardo was famous for, did he really invent the sequence. Mathematician D. E. Knuth adds in his monumental work *The Art of Computer Programming: Volume 1: Fundamental Algorithms* errata to second edition:

Before Fibonacci wrote his work, the sequence had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having beats altogether is ; therefore both Gospala (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, … explicitly.

Actually, the sequence, which had actually been known to Indian mathematicians since the 6th Century, has many interesting mathematical properties, and many of the implications and relationships of the sequence were not discovered until several centuries after Leonardo’s death. For instance, the sequence regenerates itself in some surprising ways: every third F-number is divisible by 2 (F_{3} = 2), every fourth F-number is divisible by 3 (F_{4} = 3), every fifth F-number is divisible by 5 (F_{5} = 5), every sixth F-number is divisible by 8 (F_{6} = 8), every seventh F-number is divisible by 13 (F_{7} = 13), etc. The numbers of the sequence has also been found to be ubiquitous in nature: among other things, many species of flowering plants have numbers of petals in the Fibonacci Sequence; the spiral arrangements of pineapples occur in 5s and 8s, those of pinecones in 8s and 13s, and the seeds of sunflower heads in 21s, 34s, 55s or even higher terms in the sequence; etc.

In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater accuracy the higher the terms, a ratio of approximately 1 : 1.6180339887 (it is actually an irrational number equal to ^{(1 + √5)}⁄_{2} which has since been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ). Essentially, two quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The Golden Ratio itself has many unique properties, such as ^{1}⁄_{φ} = φ – 1 (0.618…) and φ^{2} = φ + 1 (2.618…), and there are countless examples of it to be found both in nature and in the human world.

Many artists and architects throughout the history (dating back to ancient Egypt and Greece, but particularly popular in the Renaissance art of another Leonardo: Leonardo da Vinci and his contemporaries) have proportioned their works approximately using the Golden Ratio and Golden Rectangles, which are widely considered to be innately aesthetically pleasing. An arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral. The Golden Ratio and Golden Spiral can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human bodies to storm systems to complete galaxies.

It should be remembered, though, that the Fibonacci Sequence was actually only a very minor element in *Liber Abaci* – indeed, the sequence only received Fibonacci’s name in 1877 when Eduouard Lucas decided to pay tribute to him by naming the series after him – and that Leonardo himself was not responsible for identifying any of the interesting mathematical properties of the sequence, its relationship to the Golden Mean and Golden Rectangles and Spirals, etc. Nonetheless, the legacy of Leonardo Pisano, aka Fibonacci, lies in the heart of every flower, as well as in the heart of our number system.

The explanation is very complete and easy to understand .. perhaps it is also true that ..

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