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: 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 pairs11235 813213455 89144233377
Young pairs01123 58132134 5589144233

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
1
1 + x
For example, if you calculate
1
1 + 
 5 
 8 

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

Can you work out what this Fibonacci fraction is?
1
1 + 
1
1 + 1


How about this one?
1
1 + 
 
 
1
1 + 
1
1 + 1

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.

1/11.0000
1/20.5000
2/30.6667
3/50.6000
5/80.6250
8/130.6154
13/210.6190
21/340.6176
34/550.6182
55/890.6180
89/1440.6181
144/2330.6180
233/3770.6180

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