# Fibonacci numbers

The "nacci" in "Fibonacci" rhymes with "blotchy." He was also known as Leonardo of Pisa.

In his book *Liber abaci*, which was written in 1202, Fibonacci posed this problem:
*How many pairs of rabbits will be produced in a year, beginning with a single
pair, if in every month each pair bears a new pair which becomes productive
from the second month on?*

There are two kinds of pairs: productive ones
and young ones. At first we have one productive pair. The next month we have one productive
pair and one young pair. The next month we have two productive pairs and one young pair.
We can make a table of what happens.

Productive pairs | 1 | 1 | 2 | 3 | 5 |
8 | 13 | 21 | 34 | 55 |
89 | 144 | 233 | 377 |

Young pairs | 0 | 1 | 1 | 2 | 3 |
5 | 8 | 13 | 21 | 34 |
55 | 89 | 144 | 233 |

The number of young pairs is equal to the number of productive pairs in the
preceding month (see the two 8's, the two 13's, the two 21's, and so on?).
The number of productive pairs is equal to the total number of
pairs the preceding month because the young pairs become productive and the productive
pairs remain productive (the 13 in the top row is equal to the sum of the 8 and
the 5 in the preceding column).

What is the answer to Fibonacci's problem?

The numbers in either of the two rows of the table are called *Fibonacci numbers*.
The first Fibonacci numbers is 1, the fourth is 3, the seventh is 13.
The number 0 is sometimes called the *zeroth* Fibonacci number.

To generate the Fibonacci numbers, start with the numbers 0 and 1,

0 1

then write their sum next to them

0 1 1

then write the sum of the last two numbers, 1 and 1, next to that

0 1 1 2

and so on:

0 1 1 2 3 5 8 13 21
34 55 89 144 233 377

Each number is the sum of the two preceding numbers: 0 + 1 = 1, 1 + 1 = 2,
1 + 2 = 3, 2 + 3 = 5. Continuing in this way, you can extend this sequence of
Fibonacci numbers indefinitely. What are the next two Fibonacci numbers after 377?
The number 1 appears twice in the sequence, but no other number does. Note that the
Fibonacci numbers 2, 3, 5, 13, 89, and 233 are primes, the number 144 = 12·12 is
a square, and the numbers 21 = (7·6)/2 and 55 = (11·10)/2 are triangle numbers.

### Fibonacci fractions

No two consecutive Fibonacci numbers have a common factor. So the fractions

0/1 1/1 1/2 2/3 3/5 5/8 8/13 13/21
21/34 34/55

are all in lowest terms. If you like to play with fractions, which few people do,
you can get from each fraction to the next one using the formula
For example, if you calculate

you get 8/13 (multiply the numerator and denominator by 8).

Can you work out what this Fibonacci fraction is?

How about this one?

### The golden mean

There is a pattern to the Fibonacci fractions that becomes apparent if we write them
as decimals. Here they are to four decimal places.

1/1 | = | 1.0000 |

1/2 | = | 0.5000 |

2/3 | = | 0.6667 |

3/5 | = | 0.6000 |

5/8 | = | 0.6250 |

8/13 | = | 0.6154 |

13/21 | = | 0.6190 |

21/34 | = | 0.6176 |

34/55 | = | 0.6182 |

55/89 | = | 0.6180 |

89/144 | = | 0.6181 |

144/233 | = | 0.6180 |

233/377 | = | 0.6180 |

As we go further and further along the sequence of Fibonacci fractions, their decimal
digits settle down. If we kept track of twenty decimal digits, and went out a lot
further out in the sequence, we would find that
the number settles down to 0.61803398874989484820.

What is this number 0.61803...?
It is one of the most amazing numbers in the universe. We can see it in the
mystical pentagram which we construct by taking five equally spaced points around a circle
and joining them in all possible ways:

The red lines have one length, the green
lines another. The length of a red line divided by the length of a green line
is 0.61803....

See Ron Knott's Fibonacci
home page

Fibonacci calculations with multiprecision Java