### MODERN MATHEMATICS

### INTRODUCTION

The digits 0-9 that we use today have been developed from a system known as 'Arabic-Hindu numerals,' so named because of their development through a number of different Middle Eastern and Indian systems of language. They emerged originally from Brahmi and Sanskrit, developing into Eastern and Western Arabic forms, and were used in Europe from around the eleventh century onwards. The words 'zero' and 'cipher' actually derive from the Arabic word zephirum.

### THE HOUSE OF WISDOM AND GREEK TRANSLATION

Originally the Europeans of the Middle Ages ascribed this numeral system entirely to the Arabs, although we now know this to be inaccurate. The reason for this is that it appeared to European scholars and historians to emanate almost entirely from a particular source, the House of Wisdom in Baghdad. This centre of learning was set up by the ruler al-Mamun in the eighth century, and was comparable to the great centres of learning in Alexandria in ancient Greece.

More than anything, this school was concerned with translation of the mathematical and philosophical texts that were available in other languages at the time, among which were great works by the Indian mathematician Brahmagupta, and the texts of Greek thinkers such as Aristotle and Euclid.

The translation of Euclid’s works, which were written around 300 BCE, was particularly important for present day mathematics. Some of his texts, such as The Division of Figures, have not survived in the original Greek; they would not therefore be known to us were it not for the Baghdad-based translation movement.

The most important of Euclid’s works was the book Elements, which can today be considered quite simply the most important maths textbook to have ever been compiled. It is actually thought to contain little entirely original work, but in it the author set out very clearly all the most complex mathematical ideas of the day, and this is what ensured its longevity.

[For further reading see: Carl Boyer, A History of Mathematics. London: John Wiley and Sons, 1968; and Howard Eves, An Introduction to the History of Mathematics. Saunders, 1983.]

### AL-KHOWARIZMI AND THE DEVELOPMENT OF ALGEBRA

The most important mathematician working in Baghdad was a man called al-Khowarizmi, who died around 850 CE. It is largely because of his books that the Arabic-Hindu numeral system came to be thought of as an entirely Arabic innovation. In fact, the numbers 0-9 were for a while collectively known by the name 'algorism,' which derives from al-Khowarizmi’s name - and, of course, is closely related to the English term 'algorithm,' meaning a series of numerical instructions.

Along with the Greek mathematician Diophantus, al-Khowarizmi can claim to be the 'father of algebra.' His influence comes from a book titled Al-jabr wa’l muqabalah, in which a whole host of algebraic problems are dealt with. Among other things, the book tackles equations involving square roots, the squares of numbers, and general numerical problems.

The actual meaning of the words in the title, al-jabr wa’l muqabalah, is still disputed today. It is generally thought that the first section (al-jabr) means 'restoration,' while the second (wa’l muqabalah) means 'balancing' - both terms perhaps referring to the balancing of the two sides to an equation. Today, however, al-jabr has taken on an entirely different meaning which is much broader: 'algebra.'

### ARABIC-HINDU NUMERALS IN EUROPE

Three figures in particular helped bring Arabic-Hindu numerals into use in Europe after the Dark Ages: the Frenchman Alexandre de Villedieu; an English schoolmaster called John of Halifax; and the most famous, the Italian Leonardo of Pisa, who is better known today as Fibonacci.

Fibonacci, who died in 1250 CE, was the son of a merchant, and he travelled widely through Egypt, Syria and Greece. His father assigned him a Muslim teacher, and as a result he became well-versed in the Arabic-Hindu numeral system, and the works of al-Khowarizmi and his predecessors.

He is best known today for his work the Liber Abaci, or "Book of Abacus," which was a treatise on algebraic methods. It is important to us today because it showed European mathematicians why using the 0-9 system was useful; it employed the simple digits to solve incredibly advanced problems for the time.

The most famous section of the book today by a long way is not, in fact, about fractions at all. It is a question about 'rabbits,' and it reads:

"How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?"

The answer to this problem, written in today’s algebra, is this: un = un-1 + un-2. Written sequentially in straight digits this little formula comes out as follows: 1, 1, 2, 3, 5, 8, 13, 21, and it is today known in English as the "Fibonacci sequence." Each number in this sequence is the sum of the last two numbers.

The Fibonacci sequence has been discovered in older texts than the Liber Abaci; its true origins are unclear. It may not look like much, but this number sequence has helped scientists and mathematicians understand all manner of things, from the patterning of leaves and organic growth to the science of predicting outcomes. It has influenced many of the most famous mathematicians of the last five centuries, including Paccioli, Cardano and Kepler.

Picture 1: Arabic Numerals, Wikipedia

Picture 2: Talhoffer Thott, Wikipedia

Picture 3: Al-Khwarizmi, Wikipedia