Arabic Mathematics, Indian Mathematics and zero

1. Arabic Mathematics, Indian Mathematics and zero

2. Al-Khwarizmi • Born: about 780 in Baghdad (now in Iraq)Died: about 850 CE • We know few details of Abu Ja'far Muhammad ibn Musa al-Khwarizmi's life. perhaps we should call him Al for short • One unfortunate effect of this lack of knowledge seems to be the temptation to make guesses based on very little evidence.

3. al-Khwarizmi • Having introduced the natural numbers, al-Khwarizmi introduces the main topic of the first section of his book, namely the solution of equations. • His equations are linear or quadratic and are composed of units, roots and squares. • Known now as the solution by radicals • For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. • However, although we shall use the now familiar algebraic notation in this presentation to help us understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.

4. al-Khwarizmi • He first reduces an equation (linear or quadratic) to one of six standard forms: • 1. Squares equal to roots.2. Squares equal to numbers.3. Roots equal to numbers.4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39.5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x.6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2.

5. al-Khwarizmi • The reduction is carried out using the two operations of al-jabr and al-muqabala. • Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. It is where we get the word algebra from: Restoration and equivalence • For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. • The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. • For example, two applications of "al-muqabala" reduces • 50 + 3x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots).

6. al-Khwarizmi • Al-Khwarizmi then shows how to solve the six standard types of equations. • He uses both algebraic methods of solution and geometric methods. • How can we solve a quadratic quation geometrically? • For example to solve the equation x2 + 10 x = 39 he writes

7. al-Khwarizmi • ... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

8. al-Khwarizmi • The geometric proof by completing the square follows. • Al-Khwarizmi starts with a square of side x, which therefore represents x2 (see Figure to follow). • To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (see Figure again). • The figure has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 × 5/2 = 25/4. • Hence the outside square in the Figure has area • 4 × 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

9. al-Khwarizmi

10. al-Khwarizmi and Euclid • These geometrical proofs are a matter of disagreement between experts. • The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid’s Elements. • We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid’s work.

11. al-Khwarizmi and Euclid • In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid’s Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom. • This would support the comments of a mathematical historian- • ... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid’s "Elements".

12. Al-Khwarizmi • Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects. • For example he shows how to multiply out expressions such as

13. Al-Khwarizmi • (a+bx)(c+dx) • although again please remember that al-Khwarizmi uses only words to describe his expressions, and no symbols are used. • Rashed (a historian of mathematics) sees a remarkable depth and novelty in these calculations by al-Khwarizmi which appear to us, when examined from a modern perspective, as relatively elementary.

14. Al-Khwarizmi's • Al-Khwarizmi's concept of algebra can now be grasped with greater precision: it concerns the theory of linear and quadratic equations with a single unknown, and the elementary arithmetic of relative binomials and trinomials. ... The solution had to be general and calculable at the same time and in a mathematical fashion, that is, geometrically founded. ... The restriction of degree, as well as that of the number of unsophisticated terms, is instantly explained. From its true emergence, algebra can be seen as a theory of equations solved by means of radicals, and of algebraic calculations on related expressions...

15. Al-Khwarizmi • Al-Khwarizmi's algebra is regarded as the foundation and cornerstone of the sciences. • In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus himself is primarily concerned with the theory of numbers and the integer solution to equations

16. al-Khwarizmi • The next part of al-Khwarizmi's Algebra consists of applications and worked examples. • He then goes on to look at rules for finding the area of figures such as the circle and also finding the volume of solids such as the sphere, cone, and pyramid. • This section on mensuration certainly has more in common with Hindu and Hebrew texts than it does with any Greek work.

17. al-Khwarizmi • The final part of the book deals with the complicated Islamic rules for inheritance but require little from the earlier algebra beyond solving linear equations.

18. al-Khwarizmi • Al-Khwarizmi also wrote a treatise on Hindu-Arabic numerals. • The Arabic text is lost but a Latin translation, Algoritmi de numero Indorum in English Al-Khwarizmi on the Hindu Art of Reckoning gave rise to the word algorithm deriving from his name in the title. • Unfortunately the Latin translation (which has been translated into English) is known to be much changed from al-Khwarizmi's original text (of which even the title is unknown). • The work describes the Hindu place-value system of numerals based on 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0.

19. al-Khwarizmi • The first use of zero as a place holder in positional base notation was probably due to al-Khwarizmi in this work. • Methods for arithmetical calculation are given, and a method to find square roots is known to have been in the Arabic original although it is missing from the Latin version.

20. al-Khwarizmi • The Indian text on which al-Khwarizmi based his treatise was one which had been given to the court in Baghdad around 770 as a gift from an Indian political mission. • There are two versions of al-Khwarizmi's work which he wrote in Arabic but both are lost. • In the tenth century al-Majriti made a critical revision of the shorter version and this was translated into Latin by Adelard.

21. al-Khwarizmi • There is also a Latin version of the longer version and both these Latin works have survived. • The main topics covered by al-Khwarizmi in the Sindhind zij are calendars; calculating true positions of the sun, moon and planets, tables of sines and tangents; spherical astronomy; astrological tables; parallax and eclipse calculations; and visibility of the moon.

22. al-Khwarizmi • Although his astronomical work is based on that of the Indians, and most of the values from which he constructed his tables came from Hindu astronomers, al-Khwarizmi must have been influenced by Ptolemy’s work too:- • It is certain that Ptolemy’s tables, in their revision by Theon of Alexandria, were already known to some Islamic astronomers; and it is highly likely that they influenced, directly or through intermediaries, the form in which Al-Khwarizmi's tables were cast.

23. Al-Khwarizmi • Al-Khwarizmi wrote a major work on geography which give latitudes and longitudes for 2402 localities as a basis for a world map. • The book, which is based on Ptolemy’s Geography, lists with latitudes and longitudes, cities, mountains, seas, islands, geographical regions, and rivers. • The manuscript does include maps which on the whole are more accurate than those of Ptolemy.

24. Al-Khwarizmi • In particular it is clear that where more local knowledge was available to al-Khwarizmi such as the regions of Islam, Africa and the Far East then his work is considerably more accurate than that of Ptolemy, but for Europe al-Khwarizmi seems to have used Ptolemy’s data.

25. al-Khwarizmi • A number of minor works were written by al-Khwarizmi on topics such as the astrolabe, on which he wrote two works, on the sundial, and on the Jewish calendar. • He also wrote a political history containing horoscopes of prominent persons.

26. astrolabe • An astrolabe (Greek: ἁστρολάβον astrolabon 'star-taker') is a historical astronomical instrument used by classical astronomers, navigators, and astrologers. • Its many uses include locating and predicting the positions of the Sun, Moon, planets, and stars; determining local time given local latitude and vice-versa; and surveying.

27. Arabic mathematics • Recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. • Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier.

28. Arabic mathematics • In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks. • There is a widely held view that, after a brilliant period for mathematics when the Greeks laid the foundations for modern mathematics, there was a period of stagnation before the Europeans took over where the Greeks left off at the beginning of the sixteenth century.

29. Arabic mathematics • The common perception of the period of 1000 years or so between the ancient Greeks and the European Renaissance is that little happened in the world of mathematics except that some Arabic translations of Greek texts were made which preserved the Greek learning so that it was available to the Europeans at the beginning of the sixteenth century.

30. Arabic mathematics • That such views should be generally held is of no surprise. • Many leading historians of mathematics have contributed to the perception by either omitting any mention of Arabic/Islamic mathematics in the historical development of the subject or with statements such as that made by Duhem (historian) • ... Arabic science only reproduced the teachings received from Greek science.

31. Arabic mathematics • Before proceeding it is worth trying to define the period that we are covering and give an overall description to cover the mathematicians who contributed. • The period covered is easy to describe: it stretches from the end of the eighth century to about the middle of the fifteenth century. • Giving a description to cover the mathematicians who contributed, however, is much harder. There are works on "Islamic mathematics", detailing "Muslim contribution to mathematics".

32. Arabic mathematics • Other authors try the description "Arabic mathematics“. • However, certainly not all the mathematicians we included were Muslims; some were Jews, some Christians, some of other faiths. • Nor were all these mathematicians Arabs, but for convenience we will call our topic "Arab mathematics".

33. Arabic mathematics • The regions from which the "Arab mathematicians" came was centred on Iran/Iraq but varied with military conquest during the period. • At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China.

34. Arabic mathematics • The background to the mathematical developments which began in Baghdad around 800 AD is not well understood. • Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important. • There began a remarkable period of mathematical progress with al-Khwarizmis work and the translations of Greek texts

35. Arabic mathematics • This period begins under the Caliph Harun al-Rashid, the fifth Caliph of the Abbasid dynasty, whose reign began in 786. • He encouraged scholarship and the first translations of Greek texts into Arabic, such as Euclid’s Elements by al-Hajjaj, were made during al-Rashid's reign.

36. Arabic mathematics • The next Caliph, al-Ma'mun, encouraged learning even more strongly than his father al-Rashid, and he set up the House of Wisdom in Baghdad which became the centre for both the work of translating and of of research. • Al-Kindi (born 801) and the three Banu Musa brothers worked there

37. Arabic mathematics • One should emphasise that the translations into Arabic at this time were made by scientists and mathematicians such as those previously named above, not by language experts ignorant of mathematics, and the need for the translations was stimulated by the most advanced research of the time. • It is important to realise that the translating was not done for its own sake, but was done as part of the current research effort.

38. Arabic mathematics • Most of the important Greek mathematical texts were translated and list of these exist • Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects". • It gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject.

39. Arabic mathematics • Another important aspect of the introduction of algebraic ideas was that it allowed mathematics to be applied to itself in a way which had not happened before. One commentary states

40. Arabic mathematics • Al-Khawarizmi’s successors undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

41. Arabic mathematics • Let us follow the development of algebra for a moment and look at Al-Khawarizmi’s successors. • About forty years after Al-Khawarizmi’s is the work of al-Mahani (born 820), who conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. • Abu Kamil (born 850) forms an important link in the development of algebra between Al-Khawarizmi and al-Karaji

42. Arabic mathematics • Despite not using symbols, but writing powers of x in words, he had begun to understand what we would write in symbols as xn.xm = xm+n. • Let us remark that symbols did not appear in Arabic mathematics until much later. • Ibn al-Banna and al-Qalasadi used symbols in the 15th century and, although we do not know exactly when their use began, we know that symbols were used at least a century before this.

43. Arabic mathematics • Omar Khayyam (born 1048) gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. • Khayyam also wrote that he hoped to give a full description of the algebraic solution of cubic equations in a later work:- • If the opportunity arises and I can succeed, I shall give all these fourteen forms with all their branches and cases, and how to distinguish whatever is possible or impossible so that a paper, containing elements which are greatly useful in this art will be prepared.

44. Indian Mathematics • It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. • What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them.

45. Indian Mathematics • We shall examine the contributions of Indian mathematics now, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. • Laplace put this with great clarity:-

46. Laplace: Indian mathematics • The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius

47. Indian mathematics • We shall look briefly at the Indian development of the place-value decimal system of numbers later. • First, however, we go back to the first evidence of mathematics developing in India. • Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita.

48. Indian mathematics • Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times.

49. Indian numerals

50. Indian numerals • There is no problem in understanding the symbols for 1, 2, and 3. However the symbols for 4, ... , 9 appear to us to have no obvious link to the numbers they represent. There have been quite a number of theories put forward by historians over many years as to the origin of these numerals. Ifrah (historian) lists a number of the hypotheses which have been put forward. 1 The Brahmi numerals came from the Indus valley culture of around 2000 BC. 2 The Brahmi numerals came from Aramaean numerals. 3 The Brahmi numerals came from the Karoshthi alphabet. 4 The Brahmi numerals came from the Brahmi alphabet. 5 The Brahmi numerals came from an earlier alphabetic numeral system, possibly due to Panini. 6. The Brahmi numerals came from Egypt. • So there is much debate