# Ann Varela: Great Mathematicians

An Interview with Ann Varela: Great Mathematicians

Michael F. Shaughnessy –

- I would like to ask you about Leonardo of Pisa, also known as Leonardo Fibonacci, who apparently was born in 1170. What do we know about this person’s birth and early years and education?

Leonardo of Pisa was born in Pisa, Italy. He was also known as “Fibonacci” since he took on the nickname in his youth to honor his father whose surname was Bonacci. In Italian, Fibonacci means “son of Bonacci.”

He was the son of a wealthy customs official and traveled extensively, mainly in Arabic countries, taking in the mathematical knowledge of the Islamic world. As a youth, Fibonacci went to school and studied grammar and logic, along with astronomy, geometry, arithmetic, and music. During his adolescent years, he was educated in North Africa due to his father’s employment as a government official. Fibonacci was taught mathematics in Bougie, Algeria. He travelled to other destinations with his father including Egypt, Provence, Sicily, Constantinople, Syria, and Greece.

2. It is apparently well known that this person introduced something called the Modus Inodrum (method of the Indians). Can you clarify what exactly this means and why is it important to mathematicians?

The Modus Inodrum (methods of the Indians) is known to us today as Hindu-Arabic numerals. We refer to these numbers as Arabic numbers. These numbers include the digits 0-9 and are used along with place value. The Arabic number system is important because it allows us to write all other numbers. Fibonacci also believed that the Arabic number system streamlined extremely difficult calculations.

3. What were some of the applications of his ideas or forays into math? (For example, weights and measures, interest and money changing?)

In the second section of *Liber Abaci*, Fibonacci included several problems relating to the cost of goods, computing profit on transactions, converting among currencies, and conversion of weights and measures, to demonstrate to merchants and others in commerce how easy and less awkward it was to use the Arabic number system. Imagine using an abacus to perform one of the aforementioned conversions. Being able to write down the calculations performed using the Arabic number system would make for more accurate and rapid record-keeping as well.

4. Apparently he was also involved with the mathematics of numbers- can you give us some examples of his contributions- for example- Fibonacci numbers, the Fibonacci sequence and the golden ratio?

In the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc., each term in the sequence is obtained by adding the two previous terms.

The terms themselves are called Fibonacci numbers. We can take each of the Fibonacci numbers individually and consecutively, and divide each by the previous number in the sequence and obtain the following quotients:

1/1=1, 2/1=2, 3/2=1.5, 5/3=**1.66**, 8/5=1.6, 13/8=**1.625**, 21/13=**1.61538**…

Starting with the fourth ratio, we see these values tend to round to about **1.6**, which we call the golden ratio. The golden ratio is thought to be the most appealing ratio to the eye; consequently, frequently used in architecture and graphic design. See Figure.

5. It seems well documented that he wrote a book , *Liber Abaci* around 1202, and delved into both irrational and prime numbers. Was he the first to investigate and write about irrational and prime numbers, and why are these two concepts so critical to mathematicians around the world?

As a result of his travels, Fibonacci witnessed and scrutinized the arithmetical systems used in the markets of the different countries he visited. He was able to distinguish the vast benefits of the Hindu-Arabic decimal systems, with its positional notation, which we now call place value, and the use of zero, in comparison to the Roman system still used in his home country. Once Fibonacci returned to Pisa in 1202, he wrote his famous *Liber Abaci*, which translates to “Book of Counting.” In this manuscript, he explained the decimal number system and its merits. The *Liber Abaci *included much of the arithmetical knowledge of Fibonacci’s time. He also included fundamental analyses for the contents of this manuscript. Fibonacci’s *Liber Abaci* was not very popular at the time perhaps because it was too complex to comprehend, but nonetheless came into print in the 1800’s.

Although ancient Greek mathematicians were among the first to discover prime numbers and irrational numbers, Fibonacci used prime numbers in his work and related irrational numbers to geometric structures.

6. There are often stories about mathematicians- odd, strange, bizarre or idiosyncratic behaviors- or sometimes just funny stories- what does history tell us about this person?

One story I read about Fibonacci said that he would sometimes sign his name as Fibonacci Bigollo, which could translate as either “traveler” or “blockhead” just to show the European world what a “blockhead” could do.

7. What have I neglected to ask about this mathematician and his contributions?

What is fascinating to me about the Fibonacci sequence is how it is found in nature and so frequently in other mathematical circumstances. Evidently, there are so many instances of this sequence in mathematics, that there is a journal, *Fibonacci Quarterly*, devoted to their study.

A few examples of where the Fibonacci sequence occurs include the following: the arrangement of seeds of a sunflower, the growth pattern of leaves around a stem in plants, the specialized leaf of a pinecone, the scales of a pineapple, the spiral pattern found in the shell of the chambered Nautilus, and surprisingly the height of the central incisor is in the Golden Proportion to the width of the two central incisors, as was discovered by an oral surgeon in California, Dr. Stephen Marquardt.