Numbers: Their Tales, Types, and Treasures.

Chapter 6: Number Explorations

 

6.1.THE FIBONACCI NUMBERS IN EUROPE

Pingala's numbers A(n), introduced in chapter 5section 4, in order to describe the number of verse meters with a given duration, are known today as Fibonacci numbers and are usually represented in an algebraic context with the symbol Fn representing the nth Fibonacci number. They are perhaps one of the most ubiquitous number sequences in mathematics. For historical reasons, one has Fn = A(n–1) starting with n = 2, and the definition (which parallels the definition of the A(n) in section 5.5) now reads

F1 = 1, F2 = 1.
Fn = Fn–1 + Fn–2, for all natural numbers n greater than 2.

In other words, beginning with the numbers 1 and 1, each succeeding number is the sum of the two preceding numbers. They are named after Fibonacci, whose real name was Leonardo da Pisa and who is famous for promoting the use of the Hindu-Arabic numeral system in Europe during the early thirteenth century (see chapter 3section 8). The numbers Fn appeared for the first time in the Western world in Fibonacci's book Liber Abaci, published in 1202. In chapter 12 of that book, he posed the following famous problem concerning the regeneration of rabbits:

Fibonacci's problem:

A man had one pair of newborn rabbits together in a certain closed place. He wishes to know how many pairs of rabbits will be created from the pair in one year, making the following assumptions concerning the nature of the rabbits:

·      For a newborn pair, it takes two months to mature and afterward give birth to a new pair of rabbits.

·      Mature pairs bear a new pair every month.

·      No rabbit will die.

The page from Liber Abaci that contains the solution to this problem is shown in figure 6.1.

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Figure 6.1: The page from Liber Abaci explaining the rabbit problem. (From Wikimedia Commons, user Otfried Liberknecht. Original from Fibonacci, Liber Abaci, 1202, located at Biblioteca Nazionale di Firenze, Florence, Italy.)

The situation described in Fibonacci's problem is shown in table 6.1. Notice that we will be counting pairs and not individual rabbits. Thus, we begin with a single pair, born at the beginning of the year. It needs two months to mature and reproduce. Thus, in the third month, there will be the first baby pair. At the end of this month, we then have two pairs. In the fourth month, the young pair is still growing, while the older one again gives birth to a baby pair. At the end of the fourth month, we therefore have three pairs. In month number five, these three pairs are still there because it is the assumption that the rabbits will not die. While the newborn pair from the fourth month has to wait to mature, two of the three pairs are now adults, and, hence, there will be two additional pairs in the fifth month.

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Table 6.1: Proliferation of Fibonacci rabbits showing babies, young rabbits, and adults.

We see that the number of pairs obviously grows according to a rule that we call the Fibonacci sequence. The explanation is this: If the number of rabbit pairs in month n is denoted by Fn, then this number consists of the number (Fn–1) of rabbits in month n – 1, plus precisely Fn–2 newborn pairs (children of all the rabbits that existed in month n – 2 because those are the adults in month n). This can be summarized as Fn = Fn–1 + Fn–2, which (together with the initial condition) defines the Fibonacci sequence.

Now, we can easily compute the number of pairs produced in one year. It is F12 = 144. This is the solution of Fibonacci's famous problem. But this is just the beginning, as an incredible explosion of the population of rabbits would follow using this scheme. After two years, the number of pairs would be F24 = 46,368. After one hundred months (a little over eight years), the number of rabbits would be

F100 = 354,224,848,179,261,915,075.

The animals shown in table 6.1 are, in fact, not rabbits, but copies of Albrecht Dürer's famous painting Young Hare from the year 1502 CE. If these animals had reproduced according to Fibonacci's assumptions for rabbit reproduction, then today the number of pairs—about five hundred years, or six thousand months, later—would be described by a number with 1,254 digits:

 

F6000 =

377,013,149,387,799…(1,224 digits omitted)…

   

475,233,419,592,000.

The whole mass of the observable universe, if converted into hares, would by far not be sufficient to create that many pairs. Obviously, the rules defined by Fibonacci for the proliferation of rabbits are not, in the long run, very realistic. But, of course, it was not Fibonacci's goal to give a realistic model of population growth; he just wanted to provide the reader with an intellectually stimulating and entertaining mathematics problem.