**Leonardo Pisano Fibonacci, 1170 - 1250, Italian**

**Mathematics, Golde ratio.**

His parents were wealthy merchants in the port city of Pisa, which was a center of trading from the Roman times.

*Leonard Fibonacci*

Leonard would often accompany his father on trips around the Mediterranean, and learn how to calculate the money exchange rates of the various currencies. When doing calculations, using the Roman system, the abacus was used. He introduced the decimal system, replacing the Roman numerals I, II, III, IV, V, etc

His early education was in North Africa, where he had a Muslim tutor who introduced him to algebra. He was also introduced to the "nine Indian figures", that is, the decimal digits "1,2,3, ... 9", and also zero "0".

It was Fibonacci who used decimals for the first time, keeping the accounts for his father. He was the first to introduce Indian figures into Europe.

He subsequently wrote a book, "Libre Abaci", showing how any number could be written using the Indian Figures, that is, the decimal system that is used around the world today.

His use of the decimal system in his father's business was not widely accepted. The other traders were suspicious of anyone who could use them. It took hundreds of years to be fully accepted.

Fibonacci was naturally drawn to mathematical problems, and one in particular concerning the breeding habits of rabbits. The problem was stated as follows:

*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 answer gave rise to the sequence now known as the Fibonacci numbers, that is 1, 1, 2, 3, 5, 8, 13, 21, ...

Each number is obtained by adding the 2 previous numbers. It is therefore called a recursive sequence.

If you divide each number by the preceding number, the answers will tend towards 1.618 033 988 ...for bigger numbers.

For example, 8/5= 1.6, 13/8 = 1.625, 21/13 = 1.615, etc.

This number is denoted by Φ, that is Φ = 1.618 033 988..., an irrational number.

There is an interesting, and totally unrelated way of dividing a line that arrives at the same number Φ.

Divide a straight line into 2 parts, one small part, and one large part.

Make the small part 1, and the large part x. The total length would then be x + 1.

Now if the total length divided by the larger part is equal to the larger part divided by 1, then in symbols,

(x + 1)/x = x/1, and simplifying, you get x² - x - 1 = 0, a simple quadratic with the solution x = (1 + √5)/2.

If you do the arithmetic, you will get the answer x = 1.618 033 988 .... that is Φ.

Φ is sometimes called the Golden ratio, the golden segment, or the divine proportion.

It has wide ramifications in diverse areas.

To draw the spiral, draw a small square and label it 1, then a 2x2 square, 3x3, 5x5, 8x8, etc.in the positions shown.

Then draw freehand a line connecting the corners of the squares, as illustrated.

The spiral is found in natural situations, for example in the structure of a marine shell.

It is even found in the shape of an astronomical galaxy.

The ratio is used by architects to position the windows relative to the ground, and by artists to position the mouth and nose relative to the top of the head, of a painting of a person,

Fibonacci is known principally because of the golden ratio, but he made other substantial contributions.

In 1240, he was awarded the "Republic of Pisa" prize for advising the city on accounting procedures and replacing the abacus with the decimal system.

The "Book of Calculations" was published 1202, also known as "Libre Abaci",

Later he wrote "The Practice of Geometry", which was a collection of theorems that dated back to the times of Euclid. He introduced the concept of using algebra to solve geometric problems, and also using geometry to find the solution to an algebraic problem.

The "Book of Squares" is considered to be his masterpiece, and was dedicated to the Diophantine Equation. It was the foundation book for work subsequently done in number theory.

Fibonacci introduced a number of other innovations, besides the decimal system.

He introduced a number of terms that are used universally, for example, words like **factors**, **numerators** and **denominators**, **square root**, to name a few.