The story of Fibonacci series

All of you must have heard about the term “Fibonacci series”. Many of you (including me) might have written programs in different programming languages to generate a “Fibonacci series”. It is quite interesting to see that tutorials of all programming languages would contain a problem to generate fibonacci series upto “N” terms.  Well, this made me a little curious. What is so important with this particular series? In mathematics, there may be a 100 or 1000 types of series – but why “Fibonacci” is learned by everyone?

Now let’s get into the story. Fibonacci series is – 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377… it moves to infinity. This series is named after the famous Italian mathematician – Leonardo of Pisa (aka Fibonacci). His period of life is assumed to be in between 1175 AD to 1250 AD.  Leonardo’s father was a merchant in the port of Pisa. As a young boy, Leonardo traveled extensively with his father across the mediterranean sea shores. The Mediterranean sea  (known in latin as “middle of land”) connects together 3 continents – Africa, Europe and Asia. Trade had been happening between major ports of all these continents and Leonardo actively involved in these trades with his father. Leonardo and his people who belonged to Europe, was using Roman numerals for trade calculations. The Europeans were completely unaware of the “Hindu Arabic” system which was used by Arab traders in Asian/African countries (this is the presently known decimal number system ). The basic difference between a Roman numeral system and “Hindu Arabic system” is shown below.

Roman Numerals  has no symbol for zero. They dont have a symbol for 2,3,4,7,8 and 9. Instead they were depending upon the symbols for 1,5,10,50,100,500 and 1000 for their calculations.  See the table below.

So to represent 1910 – Romans would write – MDCCCCX. The interesting thing is you can write this in any order – means – CCCCDMX – would also read 1910. This is because the number is read finally by calculating M+D+C+C+C+C+X. It was very difficult to add or subtract two numbers using the roman number system.

The “Hindu Arabic” system was quiet different. They only had symbols for numbers from 0 to 9. This is our decimal number system. Higher numbers were represented in this system by repeated addition or repeated subtraction. This means 1910 in “Hindu Arabic” is  1000 one times + 100 nine times + 10 one time + 1 zero times.

Note: History says that this system originated in India. Arabs learned this from Indians (as part of their Eastern trade) and they called this “Hindu Numerals”. Later westerns learned this from Arabs and they called this system “Arab Numerals”

Leonardo learned about various methods used by merchants for calculations and he found the “Hindu-Arabic” system very easy and convenient to perform mathematical calculations compared to the cumbersome “Roman numerals” method. He later introduced this “Hindu Arabic” system in Europe and he documented the system and how to make calculations using this system in a book called “Liber Abaci” (1202 AD). This “new” system later influenced many European mathematicians.

The Fibonacci series…

This series is invented by Leonardo in an attempt to solve a real life problem. A rabbit farmer wanted to know how many rabbits he can grew in a year from one pair. The problem is quiet an interesting one. A pair of rabbits bear another new pair in a single month. This new born pairs can bear another pair after the first month.

Now he explains the problem as:-

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 montth, 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 abovewritten pair in the mentioned place at the end of the one year.

So that is the real origin of Fibonacci series. This series is invented in an attempt to solve a real life business problem of – “how many rabbits can be made from a single pair in an year” .  

I said, “business problem” because Leonardo  was basically involved in trade and this problem of rabbits might have raised to him by one of his business contact!

Other interesting facts:-

After the eighth sequence of calculations, there are constant relationships that can be derived from the series. For example, if you divide the former number by the latter, it yields 0.618.

• 34/55 is approx = 0.618
• 55/89 is approx = 0.618

And, if you divide the latter number by the former, it gives 1.618.

• 144/89 = 1.6179 approximately 1.618
• 233/144 = 1.6180 approximately 1.618

Note: The series is named “Fibonacci series” by French Mathematician Edouard Lucas. He invented many other applications of the series and also discovered another series which is closely related to Fibonacci series, known as the Lucas numbers (2, 1, 3, 4, 7, 11, 18, 29, 47)

If you are interested in further reading, you can refer the following articles:-

1. Biography and contributions of Fibonacci  

2. Introduction to analysis of Algorithms 

3.  Wiki of Fibonacci