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Fibonacci ’ s Rabbits & Fibonacci Numbers

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Fibonacci’s Rabbits& Fibonacci Numbers

Notes 10 - Sections 9.1 & 9.2

- Students will understand the mathematical significance of Fibonacci’s Rabbits.
- Students will be able to solve problems involving Fibonacci Numbers.

Leonardo Fibonacci

In 1202 a young Italian named Leonardo Fibonacci published a book titled LiberAbaci(which roughly translated from Latin means “The Book of Calculation”).

Although not an immediate success, Liber Abaci turned out to be one of the mostimportant books in the history of Western civilization.

“The Book of Calculation”

Liber Abaci was a remarkable book full of wonderful ideas and problems, but ourstory in this chapter focuses on just one of those problems – a purely hypotheticalquestion about the growth of a very special family of rabbits.

Here is thequestion, presented in Fibonacci’s own (translated) words:

Fibonacci’s Rabbits

A man puts one pair of rabbits in a certain place entirely surrounded by a wall. How many pairs of rabbits can be produced from that pair in a year if the nature of these rabbits issuch that every month each pair bears a new pair which fromthe second month on becomes productive?

Fibonacci’s Rabbits

We will call P1 the number of pairs of rabbits in thefirst month, P2 the number of pairs of rabbits in the second month, P3 the number ofpairs of rabbits in the third month, and so on.

With this notation the question askedby Fibonacci (...how many pairs of rabbits can be produced from [the original] pairin a year?) is answered by the value P12 (the number of pairs of rabbits in month 12).

Fibonacci’s Rabbits

For good measure we will add one more value, P0, representing the original pairof rabbits introduced by “the man” at the start.

Let’s see now how the number of pairs of rabbits grows month by month.

We start with the original pair, which we will assume is a pair of young rabbits.

Fibonacci’s Rabbits

In the first month we still have just the original pair (for convenience, let’s callthem Pair A), so P1 = 1.

By the second month the original pair matures, becomes“productive,” and generates a new pair of young rabbits.

Thus, by the second monthwe have the original mature Pair A plus the new young pair we will call Pair B, soP2 = 2.

Fibonacci’s Rabbits

By the third month Pair B is still too young to breed, but Pair A generatesanother new young pair, Pair C, so P3 = 3.

By the fourth month Pair C is still young,but both Pair A and Pair B are mature and generate a new pair each (Pairs Dand E). It follows that P4 = 5.

We could continue this way, but our analysis can be greatly simplified by thefollowing two observations:

Fibonacci’s Rabbits

1.In any given month (call it month N) the number of pairs of rabbits equalsthe total number of pairs in the previous month (i.e., in month N – 1) plusthe number of mature pairs of rabbits in month N (these are the pairs thatproduce offspring – one new pair for each mature pair).

2.The number of mature rabbits in month N equals the total number of rabbitsin month N – 2(it takes two months for newborn rabbits to become mature).

Fibonacci’s Rabbits

Observations 1 and 2 can be combined and simplified into a single mathematical formula:

PN= PN – 1 + PN – 2

The number of pairs of rabbits in any givenmonth (PN) equals the number of pairs of rabbits the previous month (PN – 1) plusthe number of pairs of rabbits two months back (PN – 2).

Fibonacci’s Rabbits

P5= P4 + P3 = 5 + 3 = 8

P6= P5 + P4 = 8 + 5 = 13

P7= P6 + P5 = 13 + 8 = 21

P8= P7 + P6 = 21 + 13 = 34

P9= P8 + P7 = 34 + 21 = 55

P10= P9 + P8 = 55 + 34 = 89

P11= P10 + P9 = 89 + 55 = 144

P12= P11 + P10 = 144 + 89 = 233

It follows, in order, that

Fibonacci’s Rabbits

So there is the answer to Fibonacci’s question: In one year the man will haveraised 233 pairs of rabbits.

This is the end of the story about Fibonacci’s rabbits and also the beginningof a much more interesting story about a truly remarkable sequence of numberscalled Fibonacci numbers.

Fibonacci’s Numbers

The sequence of numbers shown above is called the Fibonacci sequence, and theindividual numbers in the sequence are known as the Fibonacci numbers.

THE FIBONACCI SEQUENCE

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

You should recognize these numbers as the number of pairs ofrabbits in Fibonacci’s rabbit problem as we counted them from one month to thenext.

The Fibonacci sequence is infinite, and except for the first two 1s, eachnumber in the sequence is the sum of the two numbers before it.

Fibonacci Sequence

We will denote each Fibonacci number by using the letter F (for Fibonacci)and a subscript that indicates the position of the number in the sequence.

In otherwords, the first Fibonacci number is F1= 1, the second Fibonacci number isF2= 1, the third Fibonacci number is F3= 2, the tenth Fibonacci number isF10= 55.

Fibonacci Number

We may not know (yet) the numerical value of the 100th Fibonaccinumber, but at least we can describe it as F100.

Fibonacci Number

A generic Fibonacci number is usually written as FN(where N represents ageneric position).

If we want to describe the Fibonacci number that comes before FNwe write FN – 1 ; the Fibonacci number two places before FN is FN – 2, and so on.

Fibonacci Number

Clearly, this notation allows us to describe relations among the Fibonacci numbersin a clear and concise way that would be hard to match by just using words.

Fibonacci Number

The rule that generates Fibonacci numbers – aFibonacci number equals the sumof the two preceding Fibonaccinumbers – is called a recursive rule because it definesa number in the sequence using earlier numbers in the sequence.

Fibonacci Number

Using subscript notation, the above recursive rule can be expressed by the simple and concise formula:

FN = FN – 1+ FN – 2.

Fibonacci Number

There is one thing still missing.

The formula FN = FN – 1+ FN – 2 requires twoconsecutive Fibonacci numbers before it can be used and therefore cannot beapplied to generate the first two Fibonacci numbers, F1 and F2.

Fibonacci Number

For a completedefinition we must also explicitly give the values of the first two Fibonaccinumbers, namely F1 = 1 andF2 = 1.

These first two values serve as “anchors” forthe recursive rule and are called the seeds of the Fibonacci sequence.

Fibonacci Number

FIBONACCI NUMBERS (RECURSIVE DEFINITION)

■F1 = 1, F2 = 1 (the seeds)

■FN = FN – 1+ FN – 2 (the recursive rule)

How could one find the value of F100?

With a little patience (and a calculator)we could use the recursive definition as a “crank” that we repeatedly turn toratchet our way up the sequence.

Example – Cranking Out Large Fibonacci Numbers

From the seeds F1 and F2 we compute F3, thenuse F3 and F4 to compute F5, and so on.

If all goes well, after many turns of thecrank (we will skip the details) you will eventually get to

F97 = 83,621,143,489,848,422,977

Example – Cranking Out Large Fibonacci Numbers

and then to

F98 = 135,301,852,344,706,746,049

one more turn of the crank gives

F99 = 218,922,995,834,555,169,026

Example – Cranking Out Large Fibonacci Numbers

and the last turn gives

F100 = 354,224,848,179,261,915,075

converting to dollars yields

$3,542,248,481,792,619,150.75

Example – Cranking Out Large Fibonacci Numbers

$3,542,248,481,792,619,150.75

How much money is that?

If you take $100 billion for yourself and then dividewhat’s left evenly among every man, woman, and child on Earth (about 6.7 billionpeople), each person would get more than $500 million!

Example – Cranking Out Large Fibonacci Numbers

In 1736,Leonhard Euler discovered a formula for the Fibonacci numbers that does not relyon previous Fibonacci numbers.

The formula was lost and rediscovered 100 yearslater by French mathematician and astronomer Jacques Binet, who somehowended up getting all the credit, as the formula is now known as Binet’s formula.

Leonard Euler

BINET’S FORMULA

You can use the following shortcut of Binet’s formulato quickly find the Nth Fibonacci number for large values of N:

Using a Programmable Calculator

give you FN.

Using a Programmable Calculator

Step 1Store in the calculator’s memory.

Step 2Compute AN.

Step 3Divide the result in step 2 by

Step 4Round the result in Step 3 to the nearest whole number. This will give you FN.

Use the shortcut to Binet’s formula with a programmable calculator to compute F100.

Example – Cranking Out Large Fibonacci Numbers: Part 2

Step 1Compute The calculator should give something like: 1.6180339887498948482.

Step 2Using the power key, raise the previous number to the power 100. The calculator should show 792,070,839,848,372,253,127.

Step 3Divide the previous number by The calculator should show354,224,848,179,261,915,075.

Step 4The last step would be to round the number in Step 3 to the nearestwhole number. In this case the decimal part is so tiny that the calculatorwill not show it, so the number already shows up as a whole number andwe are done.

Example – Cranking Out Large Fibonacci Numbers: Part 2

We find Fibonacci numbers when we count the number of petals in certain varietiesof flowers: lilies and irises have 3 petals; buttercups and columbines have 5 petals;cosmos and rue anemones have 8 petals; yellow daisies and marigolds have 13 petals;English daisies and asters have 21 petals; oxeye daisies have 34 petals, and thereare other daisies with 55 and 89 petals

Why Fibonacci Numbers Are Special

Fibonacci numbers also appear consistently in conifers, seeds, and fruits. The bracts in a pinecone, forexample, spiral in two different directions in 8 and 13 rows; the scales in a pineapplespiral in three

Why Fibonacci Number Are Special

different directions in8, 13, and 21 rows; the seeds in the center of asunflower spiral in 55 and 89 rows.

Is it all a coincidence?

p. 351: 1, 3, 5, 7, 8, 9, 11, 14,

51, 52a, 53