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Fibonacci, also called Leonardo Pisano, English Leonardo of Pisa, original name Leonardo Fibonacci, (born c. 1170, Pisa?—died after 1240), medieval Italian mathematician who wrote Liber abaci (1202; “Book of the Abacus”), the first European work on Indian and Arabian mathematics.

Little is known about Fibonacci’s life beyond the few facts given in his mathematical writings. During Fibonacci’s boyhood his father, Guglielmo, a Pisan merchant, was appointed consul over the community of Pisan merchants in the North African port of Bugia (now Bejaïa, Algeria). Fibonacci was sent to study calculation with an Arab master. He later went to Egypt, Syria, Greece, Sicily, and Provence, where he studied different numerical systems and methods of calculation.

When Fibonacci’s Liber abaci first appeared, Hindu-Arabic numerals were known to only a few European intellectuals through translations of the writings of the 9th-century Arab mathematician al-Khwārizmī. The first seven chapters dealt with the notation, explaining the principle of place value, by which the position of a figure determines whether it is a unit, 10, 100, and so forth, and demonstrating the use of the numerals in arithmetical operations. The techniques were then applied to such practical problems as profit margin, barter, money changing, conversion of weights and measures, partnerships, and interest. Most of the work was devoted to speculative mathematics— proportion (represented by such popular medieval techniques as the Rule of Three and the Rule of Five, which are rule-of-thumb methods of finding proportions), the Rule of False Position (a method by which a problem is worked out by a false assumption, then corrected by proportion), extraction of roots, and the properties of numbers, concluding with some geometry and algebra. In 1220 Fibonacci produced a brief work, the Practica geometriae (“Practice of Geometry”), which included eight chapters of theorems based on Euclid’s Elements and On Divisions.

The Liber abaci, which was widely copied and imitated, drew the attention of the Holy Roman emperor Frederick II. In the 1220s Fibonacci was invited to appear before the emperor at Pisa, and there John of Palermo, a member of Frederick’s scientific entourage, propounded a series of problems, three of which Fibonacci presented in his books. The first two belonged to a favourite Arabic type, the indeterminate, which had been developed by the 3rd-century Greek mathematician Diophantus. This was an equation with two or more unknowns for which the solution must be in rational numbers (whole numbers or common fractions). The third problem was a third-degree equation (i.e., containing a cube), x 3 + 2x 2 + 10x = 20 (expressed in modern algebraic notation), which Fibonacci solved by a trial-and-error method known as approximation; he arrived at the answer

in sexagesimal fractions (a fraction using the Babylonian number system that had a base of 60), which, when translated into modern decimals (1.3688081075), is correct to nine decimal places.

Contributions to number theory

For several years Fibonacci corresponded with Frederick II and his scholars, exchanging problems with them. He dedicated his Liber quadratorum (1225; “Book of Square Numbers”) to Frederick. Devoted entirely to Diophantine equations of the second degree (i.e., containing squares), the Liber quadratorum is considered Fibonacci’s masterpiece. It is a systematically arranged collection of theorems, many invented by the author, who used his own proofs to work out general solutions. Probably his most creative work was in congruent numbers—numbers that give the same remainder when divided by a given number. He worked out an original solution for finding a number that, when added to or subtracted from a square number, leaves a square number. His statement that x 2 + y 2 and x 2 − y 2 could not both be squares was of great importance to the determination of the area of rational right triangles. Although the Liber abaci was more influential and broader in scope, the Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the 17th-century French mathematician Pierre de Fermat.

Except for his role in spreading the use of the Hindu-Arabic numerals, Fibonacci’s contribution to mathematics has been largely overlooked. His name is known to modern mathematicians mainly because of the Fibonacci sequence (see below) derived from a problem in the Liber abaci:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (Fibonacci himself omitted the first term), in which each number is the sum of the two preceding numbers, is the first recursive number sequence (in which the relation between two or more successive terms can be expressed by a formula) known in Europe. Terms in the sequence were stated in a formula by the French-born mathematician Albert Girard in 1634: un + 2 = un + 1 + un, in which u represents the term and the subscript its rank in the sequence. The mathematician Robert Simson at the University of Glasgow in 1753 noted that, as the numbers increased in magnitude, the ratio between succeeding numbers approached the number α, the golden ratio, whose value is 1.6180…, or (1 + Square root of √ 5 )/2. In the 19th century the term Fibonacci sequence was coined by the French mathematician Edouard Lucas, and scientists began to discover such sequences in nature; for example, in the spirals of sunflower heads, in pine cones, in the regular descent (genealogy) of the male bee, in the related logarithmic (equiangular) spiral in snail shells, in the arrangement of leaf buds on a stem, and in animal horns.

Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, .

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The next number is found by adding up the two numbers before it.

  • The 2 is found by adding the two numbers before it (1+1)
  • The 3 is found by adding the two numbers before it (1+2),
  • And the 5 is (2+3),
  • and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, .

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

The Rule

The Fibonacci Sequence can be written as a „Rule“ (see Sequences and Series).

First, the terms are numbered from 0 onwards like this:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 .
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 .

So term number 6 is called x6 (which equals 8).

Example: the 8th term is
the 7th term plus the 6th term:

So we can write the rule:

Example: term 9 is calculated like this:

Golden Ratio

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio „φ“ which is approximately 1.618034.

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

1.666666666. 1.618055556. 1.618025751.

Note: this also works when we pick two random whole numbers to begin the sequence, such as 192 and 16 (we get the sequence 192, 16, 208, 224, 432, 656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, . ):

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

The answer always comes out as a whole number, exactly equal to the addition of the previous two terms.

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033. A more accurate calculation would be closer to 8.

Try it for yourself!

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: 8 × φ = 8 × 1.618034. = 12.94427. = 13 (rounded)

Some Interesting Things

Here is the Fibonacci sequence again:

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 .
xn = 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 .

There is an interesting pattern:

  • Look at the number x3 = 2. Every 3rd number is a multiple of 2 (2, 8, 34, 144, 610, . )
  • Look at the number x4 = 3. Every 4th number is a multiple of 3 (3, 21, 144, . )
  • Look at the number x5 = 5. Every 5th number is a multiple of 5 (5, 55, 610, . )

And so on (every nth number is a multiple of xn).

1/89 = 0.011235955056179775.

Notice the first few digits (0,1,1,2,3,5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap, like this:

. etc .
0.011235955056179775. = 1/89

Terms Below Zero

The sequence works below zero also, like this:

n = . -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 .
xn = . -8 5 -3 2 -1 1 0 1 1 2 3 5 8 .

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- . pattern. It can be written like this:

Which says that term „-n“ is equal to (в€’1) n+1 times term „n“, and the value (в€’1) n+1 neatly makes the correct 1,-1,1,-1. pattern.


Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

„Fibonacci“ was his nickname, which roughly means „Son of Bonacci“.

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits „1, 1, 2, 3“ which is part of the sequence. So next Nov 23 let everyone know!

¿Qué es la sucesión de Fibonacci?

Digital Vision./Digital Vision/Thinkstock

¿Alguna vez escuchaste hablar acerca de la sucesión de Fibonacci? ¿Imaginas una ecuación capaz de explicar matemáticamente todo en el universo? ¿Crees que semejante cosa realmente sería posible?

Bueno, de las tantas sucesiones matemáticas que existen, ninguna es tan famosa, tan interesante y tan asombrosa como la que inventó Fibonacci. A lo largo de los años, hombres de ciencia, artistas de todo tipo y arquitectos, la han utilizado para trabajar, a veces a propósito y otras de forma inconsciente, pero siempre con resultados majestuosos. Te invito a conocer la historia detrás de todo este asunto y a que hoy aprendamos qué es la sucesión de Fibonacci.

La sucesión de Fibonacci

La sucesión de Fibonacci, en ocasiones también conocida como secuencia de Fibonacci o incorrectamente como serie de Fibonacci, es en sí una sucesión matemática infinita. Consta de una serie de números naturales que se suman de a 2, a partir de 0 y 1. Básicamente, la sucesión de Fibonacci se realiza sumando siempre los últimos 2 números (Todos los números presentes en la sucesión se llaman números de Fibonacci) de la siguiente manera:

Fácil, ¿no? (0+1=1 / 1+1=2 / 1+2=3 / 2+3=5 / 3+5=8 / 5+8=13 / 8+13=21 / 13+21=34. ) Así sucesivamente, hasta el infinito. Por regla, la sucesión de Fibonacci se escribe así: xn = xn-1 + xn-2. Hasta aquí todo bien, pero de seguro estás preguntándote ¿quién fue Fibonacci?

¿Quién fue Fibonacci?

Bien, Fibonacci fue un matemático italiano del siglo XIII, el primero en describir esta sucesión matemática. También se lo conocía como Leonardo de Pisa, Leonardo Pisano o Leonardo Bigollo y ya hablaba de la sucesión en el año 1202, cuando publicó su Liber abaci. Fibonacci era hijo de un comerciante y se crió viajando, en un medio en donde las matemáticas eran de gran importancia, despertando su interés en el cálculo de inmediato.

Se dice que sus conocimientos en aritmética y matemáticas crecieron enormemente con los métodos hindúes y árabes que aprendió durante su estancia en el norte de África y luego de años de investigación, Fibonacci dió con interesantes avances. Algunos de sus aportes refieren a la geometría, la aritmética comercial y los números irracionales, además de haber sido vital para desarrollar el concepto del cero.

El espiral de Fibonacci

Ahora, ¿qué es lo asombroso de esta secuencia o sucesión matemática tan simple y clara? Que está presente prácticamente en todas las cosas del universo, tiene toda clase de aplicaciones en matemáticas, computación y juegos, y que aparece en los más diversos elementos biológicos.

Ejemplos claros son la disposición de las ramas de los árboles, las semillas de las flores, las hojas de un tallo, otros más complejos y aún mucho más sorprendentes es que también se cumple en los huracanes e incluso hasta en las galaxias enteras, desde donde obtenemos la idea del espiral de Fibonacci.

Un espiral de Fibonacci es una serie de cuartos de círculo conectados que se pueden dibujar dentro de una serie de cuadros regulados por números de Fibonacci para todas las dimensiones. Entre sí, los cuadrados encajan a la perfección como consecuencia de la naturaleza misma de la sucesión, en donde cualquier cifra es igual a la suma de las dos anteriores. El espiral o rectángulo resultante es conocido como el espiral dorado y el rectángulo de oro.

Cada uno de los números de Fibonacci se acerca mucho a la llamada proporción áurea, proporción dorada o número de oro (aproximádamente 1.618034). Cuanto mayor es el par de números de Fibonacci, más cerca de la proporción dorada estamos. Naturalmente, ésta cifra resulta más bella y más agradable a nuestra percepción y ya sea consciente o inconscientemente, artistas la han empleado a lo largo de toda la historia de la humanidad.

Desde arquitectos y escultores de la Antigua Grecia a pintores como Miguel Ángel y Da Vinci, a compositores como Mozart y Beethoven o, más próximo a nuestros días, las composiciones de artistas como Béla Bartók y Olivier Messiaen. La gloriosa banda de rock: Tool, también ha trabajado de forma conceptual con esta secuencia matemática de acuerdo a la sucesión de notas y estructuras musicales.

¿Ya no es tan fácil? Inténtalo con este vídeo:

¿Qué te parece? Realmente fascinante, ¿no? ¿Qué más sabes acerca de esta sucesión?

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