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October 29, 2019

What are some facts about Leonardo Fibonacci and the Fibonacci sequence that everyone wants to know?

Italian Mathematician Leonardo Pisano( born in 1175 and died around 1250) also known as Fibonacci is mostly famous for his Fibonacci sequence. His name got originated from a misreading on a manuscript of “filius Bonacci”(son of Bonaccio).

He also played an important role in establishing the Hindu-Arabic numeral system in Europe.What Fibonacci did in his Book “Liber Abaci” in 1202,is that he played a major role in introducing the numbers which we now use to replace Roman numerals.The concept of Fibonacci sequence was referred to by him in a problem about breeding rabbits which is discussed later.He was considered to be the most talented western Mathematician of the Middle age. He introduced the mind blowing concept of Fibonacci sequence.He is also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano (“Leonardo the Traveler from Pisa”).He had written a book known as “Liber Abaci”, which is translated as “The book of Calculation” and the book was published in 1202. He brought focus on the famous Hindu-Arabic Numeral System on his book Liber Abaci.The book has several applications related to above mentioned topic which includes conversion of weights and measures, money changing ,interest calculation and many other practical applications.The 1228 edition of the book includes methods for converting various numeral systems into Hindu-Arabic numerals.Popular Mathematical topic Abacus is also mentioned in the book .These facts have played a vital role in making calculations smoother and quicker thus aiding in the development if banking and other economic terms in Europe.Topic like prime numbers and irrational numbers are also mentioned in the book. In short he evolved the concept of Number theory .

Now discussing on his exceptional work on Fibonacci Sequence.The name “Fibonacci sequence” was first applied by Theorist Edouard Lucas in the 19th century.

In the field of Mathematics, Fibonacci numbers denoted as \(F_n\).The sequence states that that each number is the sum of the two preceding numbers starting from 0 followed by 1.

The general term of the sequence

\(F_n\) = \(F_{n-1}\) + \(F_{n−2}\) where \(F_0\) =0 and \(F_1\)=1 for all \(n>1\)

Thus the sequence becomes

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Outside India, the Fibonacci sequence was first termed in Liber Abaci as mentioned above by Fibonacci.This concept was actually used to estimate the growth of rabbit population.

Fibonacci discovered a very interesting concept of the rabbit population.Rabbits usually never die and they are able to reproduce at the end of its second month.

.Now if a male and a female rabbit that is a newly born pair of rabbits are placed in a field then they will always produce a new pair at the end of each month starting from the second month.This way the following observations were made.

  1. By the end of first month there is only one pair. (\(F_1\)=1)
  2. By the end of second\d month, a new pair is born thus amounting to 2 pairs \(F_2\) =2)
  3. By the end of third month a new pair is born from the original pair thus amounting to 3 pairs ( \(F_3\) = \(F_2\) + \(F_1\) = 2+1 = 3)
  4. By the end of fourth month again a new pair is born from the original pair and another pair is born from the first female produced by the original female amounting to 5 pairs ( \(F_4\) = \(F_3\) + \(F_2\) = 3+2 = 5)

We can conclude from the above mentioned facts that by the end of n month, the number of pairs will be

\(F_n\) = \(F_{n-1}\) + \(F_{n−2}\) , which is the Mathematical generalised expression of the Fibonacci Sequence.

Now mentioning a few applications of the Fibonacci Sequence.

  1. The Fibonacci numbers are vital in analyzing Euclid’s Algorithm to determine the greatest common factor of two integers.
  2. Every positive integer can be expressed as a sum of Fibonacci numbers provided any one number is used at most once thus resulting in a complete sequence.
  3. Sculpture and Painter, Mario Merz included Fibonacci sequence in his works in 1970s.
  4. Fibonacci numbers also has its applications in Physics. In optics the number of different beam paths when a ray of light shines at an angle through two different transparent plates of different refractive index and material,there are k reflections ,for k>1 and k is the Fibonacci number.
  5. This sequence plays a very essential role in Computer Programming as as well.
  6. It is widely used in the filed of Botany.

Another very interesting fact about the Fibonacci number is that number of petals on flower Daisy is always a Fibonacci number (21, 34, 55 being most common numbers).

As recorded 1597, was the last year that was a Fibonacci number and the next will be 2584.

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