AL-KHWARIZMI (800-847)

820 – Baghdad, Iraq

Portrait of AL-KHWARIZMI

AL-KHWARIZMI

The man often credited with the introduction of ‘Arabic’ numerals was al-Khwarizmi, an Arabian mathematician, geographer and astronomer. Strictly speaking it was neither invented by al-Khwarizmi, nor was it Middle Eastern in origin.

786 – Harun al-Rashid came to power. Around this time al-Khwarizmi born in Khwarizm, now Khiva, in Uzbekistan.

813 – Caliph al-Ma’mun, the patron of al-Khwarizmi, begins his reign in Baghdad.

Arabic notation has its roots in India around 500 AD, thus the current naming as the ‘Hindu-Arabic’ system. al-Khwarizmi, a scholar in the Dar al-ulum (House of Wisdom) in Baghdad in the ninth century, is responsible for introducing the numerals to Europe. The method of using only the digits 0-9, with the value assigned to them determined by their position, as well as introducing a symbol for zero, revolutionised mathematics.

al-Khwarizmi explained how this system worked in his text ‘Calculation with Hindu numerals‘. He was clearly building upon the work of others before him, such as DIOPHANTUS and BRAHMAGUPTA, and on Babylonian sources that he accessed through Hebrew translations. By standardizing units, Arabic numerals made multiplication, division and every other form of mathematical calculation much simpler. His text ‘al-Kitab al-mukhtasar- fi hisab al-jabr w’al-muqabala’ (The Compendious Book on Calculating by Completion and Balancing) gives us the word algebra. In this treatise, al-Khwarizmi provides a practical guide to arithmetic.

In his introduction to the book he says the aim of the work is to introduce ‘what is easiest and most useful in mathematics, such as men constantly require in cases of inheritance, legacies, partition, lawsuits and trade, and in all their dealings with one another, or when measuring lands, digging canals and making geometrical calculations.’ He introduced quadratic equations, although he described them fully in words and did not use symbolic algebra.
It was in his way of handling equations that he created algebra.

The two key concepts were the ideas of completion and balancing of equations. Completion (al-jabr) is the method of expelling negatives from an equation by moving them to the opposite side

4x2 = 54x – 2x2  becomes  6x2 = 54x

Balancing (al-muqabala) meanwhile, is the reduction of common positive terms on both sides of the equation to their simplest forms

x2 + 3x + 22 = 7x + 12  becomes  x2 + 10 = 4x

Thus he was able to reduce every equation to simple, standard forms and then show a method of solving each, showing geometrical proofs for each of his methods – hence preparing the stage for the introduction of analytical geometry and calculus in the seventeenth century.

The name al-Khwarizmi also gives us the word algorithm meaning ‘a rule of calculation’, from the Latin title of the book, Algoritmi de numero Indorum.

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LEONARDO FIBONACCI (c.1170-c.1250)

Also known as Leonardo Pisano. Published ‘Liber Abaci’ in 1202.

1202 – Italy

image of statue of Leonardo Fibonacci ©

FIBONACCI

Picture of a statue of Leonardo Pisano

FIBONACCI

‘A series of numbers in which each successive term is the sum of the preceding two’

For example:   1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144….

The series is known as the Fibonacci sequence and the numbers themselves as the Fibonacci numbers.

The Fibonacci sequence has other interesting mathematical properties – the ratio of successive terms ( larger to smaller;   1/1, 2/1, 3/2, 5/3, 8/5…. ) approaches the number 1.618
This is known as the golden ratio and is denoted by the Greek letter Phi.

Phi was known to ancient Greeks.
Greek architects used the ratio 1:Phi as part of their design, the most famous example of which is the Parthenon in Athens.

Fibonacci sequence in flower petals. flowers often have a Fibonacci number of petals - link to <http://pinterest.com/mcvjfly/fibonacci/>

Fibonacci sequence in flower petals

Phi also occurs in the natural world.
Flowers often have a Fibonacci number of petals.

      

During his travels in North Africa, Fibonacci learned of the decimal system of numbers that had evolved in India and had been taken up by the Arabs.
In his book Liber Abaci he re-introduced to Europe the Arabic numerals that we use today, adhering roughly to the recipe ‘the value represented must be proportional to the number of straight lines in the symbol’.

Following the Arabs, Fibonacci ( ‘son of the simpleton’ euph. or ‘son of the innocent’ ) introduced the place–value concept, with each position representing a different power of ten and these arranged in ascending order from right to left.

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