Fibonacci NumbersThe Fibonacci numbers were first discovered by a man named LeonardoPisano. He was known by his nickname, Fibonacci. The Fibonacci sequence is asequence in which each term is the sum of the 2 numbers preceding it.
The first10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbersare obviously recursive. Fibonacci was born around 1170 in Italy, and he died around 1240 inItaly. He played an important role in reviving ancient mathematics and madesignificant contributions of his own. Even though he was born in Italy he waseducated in North Africa where his father held a diplomatic post. He did a lotof traveling with his father.
He published a book called Liber abaci, in 1202,after his return to Italy. This book was the first time the Fibonacci numbershad been discussed. It was based on bits of Arithmetic and Algebra thatFibonacci had accumulated during his travels with his father. Liber abaciintroduced the Hindu-Arabic place-valued decimal system and the use of Arabicnumerals into Europe. This book, though, was somewhat contraversial because itcontradicted and even proved some of the foremost Roman and GrecianMathematicians of the time to be false. He published many famous mathematicalbooks.
Some of them were Practica geometriae in 1220 and Liber quadratorum in1225. The Fibonacci sequence is also used in the Pascal trianle. The sum ofeach diagnal row is a fibonacci number. They are also in the right sequence:1,1,2,5,8. .
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. . Fibonacci sequence has been a big factor in many patterns of things innature. One has found that the fractions u/v representing the screw-likearrangement of leaves quite often are members of the fibonacci sequence.
On manyplants, the number of petals is a Fibonacci number: buttercups have 5 petals;lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89petals. Fibonacci nmbers are also used with animals. The first problem Fibonaccihad wehn using the Fibonacci numbers was trying to figure out was how fastrabbits could breed in ideal circumstances. Using the sequence he was ale toapproximate the answer. The Fibonacci numbers can also be found in many other patterns.
The diagrambelow is what is known as the Fibonacci spiral. We can make another pictureshowing the Fibonacci numbers 1,1,2,3,5,8,13,21,. . if we start with two smallsquares of size 1, one on top of the other.
Now on the right of these draw asquare of size 2 (=1+1). We can now draw a square on top of these, which hassides 3 units long, and another on the left of the picture which as side 5. Wecan continue adding squares around the picture, each new square having a sidewhich is as long as the sum of the latest two squares drawn. If we take the ratio of two successive numbers in Fibonacci’s series, (1 1 2 3 58 1 3. .
) we find:1/1=1; 2/1=2; 3/2=1. 5; 5/3=1. 666. .
. ; 8/5=1. 6; 13/8=1. 625;It is easier to see what is happening if we plot the ratios on a graph:Greeks called the golden ratio and has the value 1. 61803. It has someinteresting properties, for instance, to square it, you just add 1.
To take itsreciprocal, you just subtract 1. This means all its powers are just wholemultiples of itself plus another whole integer (and guess what these wholeintegers are? Yes! The Fibonacci numbers again!) Fibonacci numbers are a bigfactor in Math, The Golden Ratio, The Pascal Triangle, the production of manyspecies, plants, and much much more.
