I have been teaching mathematics in an Australian High School since 1982, and I am a contributing author to mathematics text books.
Leonardo Pisano (nicknamed Leonardo Fibonacci) was a well-known Italian mathematician.
He was born in Pisa in 1170 AD and died there around 1250 AD.
Fibonacci travelled widely, and in 1202 he published Liber abaci, which was based on his knowledge of arithmetic and algebra developed during his extensive travels.
One investigation described in Liber abaci refers to how rabbits might breed.
Fibonacci simplified the problem by making several assumptions.
Start with one newly-born pair of rabbits, one male, one female.
Each rabbit will mate at the age of one month and that at the end of its second month a female will produce a pair of rabbits.
No rabbit dies, and the female will always produce one new pair (one male, one female) every month from the second month on.
This scenario can be shown as a diagram.
The sequence for the number of pairs of rabbits is
1, 1, 2, 3, 5, ….
If we let F(n) be the nth term, then F(n) = F(n - 1) + F(n - 2), for n > 2.
That is, each term is the sum of the two preceding terms.
For example, the third term is F(3) = F(2) + F(1) = 1 + 1 = 2.
Using this implicit relationship, we can determine as many terms of the sequence as we like. The first twenty terms are:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765
The ratio of consecutive Fibonacci numbers approaches the Golden Ratio, represented by the Greek letter, Φ. The value of Φ is approximately 1.618034.
This is also referred to as the Golden Proportion.
The convergence to the golden ratio is clearly seen when the data is plotted.
The ratio of the length and width of a Golden Rectangle produces the Golden Ratio.
Two of my videos illustrate the properties of the Fibonacci sequence and some applications.
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Explicit form and the exact value of Φ
The drawback in using the implicit form F(n) = F(n - 1) + F(n - 2) is its recursive property. To determine a particular term, we need to know the two preceding terms.
For instance, if we want the value of the 1000th term, the 998th term and the 999th term are required. To avoid this complication, we obtain the explicit form.
Let F(n) = xn be the nth term, for some value, x.
Then F(n) = F(n - 1) + F(n - 2) becomes xn = xn-1 + xn-2
Divide each term by xn-2 to obtain x2 = x + 1, or x2 – x – 1 = 0.
This is a quadratic equation which can be solved for x to get
The first solution, of course, is our Golden Ratio, and the second solution is the negative reciprocal of the Golden Ratio.
So we have for our two solutions:
The explicit form can now be written in the general form.
Solving for A and B gives
Let’s check this. Suppose we want the 20th term, which we know is 6765.
The Golden Ratio is pervasive
Fibonacci numbers exist in nature, such as in the number of petals in a flower.
We see the Golden Ratio in the ratio of the two lengths on the body of a shark.
Architects, craftsmen and artists incorporate the Golden Ratio. The Parthenon and the Mona Lisa use golden proportions.
I have provided a glimpse of the properties and use of Fibonacci numbers. I encourage you to explore this famous sequence further, especially in its real-world setting, such as in stock-market analysis and the ‘rule of thirds’ used in photography.
When Leonardo Pisano postulated the number sequence from his study of the population of rabbits, he could not have foreseen the versatility of his discovery can be used and how it dominates many aspects of Nature.