Pathfinders
Page 15
We now come to the more sophisticated issue of treating zero as a number in its own right, which would necessarily bring together the ideas of the Greeks and Babylonians. Many historians have argued that this did not happen until relatively recently. Even al-Khwārizmi avoided ever having to equate his algebraic quantities to zero. Instead, he would always have non-zero quantities on both sides of the equation. To state this symbolically, he would never set up an equation like x2 + 3x − 10 = 0, but rather as x2 + 3x = 10. In both cases, the value of x is 2 and the difference between these two notations is a trivial rearranging of the number 10 (according to rules laid down by al-Khwārizmi himself). But the first equation would have been quite alien to him since the ‘zero’ was not yet considered a number that other numbers could combine together to give.
Despite this, it is commonly agreed that the symbol for zero arrived in Baghdad from India as part of the package with decimal notation. Certainly it seems that the Hindus looked upon zero as a real number, rather than just a symbol, as early as 505 CE.15 Until that time, the Hindu numerals contained separate symbols for 10, 11 and so on, and it was the invention of zero that did away with the need for separate symbols for any numbers above 9. Soon, Indian mathematicians were carrying out arithmetical operations involving zero. Brahmagupta correctly stated in 628 CE that zero multiplied by a finite number gives zero, and he described the impossibility of the division of a number by zero. It was only much later, towards the end of the twelfth century, that another Indian, Bhaskara, argued correctly that the value of any finite number divided by zero is infinity.
But it is the use of zero in algebra that seems to have been missing in the work of Islamic mathematicians. For instance, consider the relation x2 = 2x. This is an example of a simple quadratic equation and as such has two possible roots (x can have two values). Clearly, these are 0 and 2 – both values ‘work’ if you replace x by them in the equation. But the early Islamic mathematicians would not have recognized x = 0 as one of the solutions. This quite subtle conceptual leap did not take place until the seventeenth century with the work of the French mathematician Albert Girard.16
Finally, as for the origin of the word ‘zero’, this can be traced back to the Indian word sunya, meaning the ‘void’, which was translated into Arabic as sifr, or ‘nothing’, and the word is still in use in the Arabic language today. Around the early thirteenth century its Latinized form became zephirum and this gradually evolved into the word ‘zero’. But in Western Europe, a further detour was made: the Arabic sifr became the Latin cifra and the English cipher. And instead of denoting zero it was taken to mean any of the Hindu-Arabic numerals. Later, it came to signify a secret symbol, obscure way of writing, or the key to unlocking it (hence the word decipher). Because of this confusion, the English eventually adopted the Italian word zero.
*
The Book of Chapters on Hindu Arithmetic (Kitab al-Fusūl fi al-Hisāb al-Hindī) is the earliest known text on arithmetic still extant in Arabic and contains the earliest known use of the decimal point. The only known copy of this remarkable book was written in Damascus in 952 and is kept in the Yeni Gami Library in Istanbul. It is hugely important in the history of mathematics, though probably not widely enough known. Apart from a few missing pages and three unfinished chapters (one on the description of a computing board for the blind), it comprises 230 pages of clearly written text and mathematical calculations. But its author is hardly a well-known figure. Indeed there is no mention of him in the usually reliable and comprehensive Fihrist of Ibn al-Nadīm, the tenth-century historian and biographer of Arabic scholars. His name was Abū al-Hassan al-Uqlīdisi, whose title refers to Uqlīdis (Euclid), implying an association with the great Greek mathematician. But it would appear that this connection was not so much from al-Uqlīdisi’s great expertise and mastery of Euclid’s geometry (although no one can be sure, of course) as from the way he is thought to have earned his living by making and selling copies of the Arabic translation of Euclid’s Elements.
Al-Uqlīdisi is the very first mathematician we know of to use decimal fractions, and to suggest a symbol for the decimal point (an oblique dash over the number).17 In the preface to his book, al-Uqlīdisi explains that he has taken great pains to describe the best arithmetical methods of manipulating and computing fractions of all previous writers. This makes it very hard to determine whether the notion of decimal fractions, or even the notation used, is his discovery or that of an earlier mathematician.18 But since this was not something the Islamic world inherited from the Indians it can be said that, unlike decimal numbers or the zero, decimal fractions were almost certainly an invention of Arabic mathematicians.
Some modern historians have tried to downplay the impact of al-Uqlīdisi’s work, arguing that, apart from certain quite specific examples in which he used decimals,19 he did not fully appreciate just how powerful and important they are in arithmetic. It has even been suggested that poor al-Uqlīdisi’s use of decimal fractions was no more than intuitive and accidental, and nowhere near as comprehensive in their application as that of later mathematicians. This is rather harsh, but it is true that much more accomplished mathematicians than al-Uqlīdisi were to follow, such as al-Samaw’al (c. 1130–c. 1180),20 the son of a Moroccan rabbi and a mathematical genius, who wrote a book on algebra at the age of 19 and developed the concept of proof by mathematical induction as well as an important contribution to the binomial theorem. For other historians, it is the great Persian mathematician al-Kāshi and his encyclopedic treatise The Calculators’ Key (Miftāh al-Hussāb)21 five centuries later, who is the first to write comprehensively about, and use, decimal fractions.22
The first use of a decimal point seen here in The Book of Chapters on Hindu Arithmetic by al-Uqlīdisi, written in the mid-tenth century CE. The box on the left shows the magnified text with his notation of a dash above the digit 9, implying that the digits to its right are decimal places. The number itself is written today as 179.685 and is arrived at when al-Uqlīdisi carries out the operation of adding a number (135) to its tenth, then the result to its tenth, etc., three times – in other words, multiplying 135 by the cube of 11/10.
Just to muddy the water further, it has also been argued that Chinese mathematicians, such as the great Lui Hui in the third century, were in fact the first to use decimal fractions, although there is no evidence that al-Uqlīdisi was familiar with Chinese mathematics. In any case, the most prominent of the Chinese mathematicians known to have worked with decimal fractions was Yang Hui in the thirteenth century (two centuries after al-Uqlīdisi), although he did not use a symbol for the decimal point.23
What is interesting is that al-Uqlīdisi’s notation was very close to the one we use for the decimal point today.24 He wrote the number 179.685 as , with a dash over the unit number.25 This contrasts with the later work of al-Kāshi, who would either write the decimal part of a number in a different colour ink or write his numbers within a table with the decimal part in a separate column. Even more cumbersome is the notation used by the sixteenth-century Flemish mathematician Simon Stevin (1548–1620), who would have written al-Uqlīdisi’s number above as , where the numbers in circles denote the units, tenths, hundredths and so on. Slightly earlier than Stevin, in fifteenth-century European texts, we also see the use of a stroke to separate units from decimals as 179|685. But I much prefer al-Uqlīdisi’s very early notation.
The decimal system as a whole, we now see, must be jointly credited to both Indian and Arabic mathematicians. The Indians were the first to use the base-ten positional system of nine ciphers, or digits, plus the zero symbol, instead of the accumulated strokes of the Egyptians, Babylonians and Romans. However, they did not extend this system to fractions. Since that important addition came from the Arabic mathematicians (both Arabs and Persians who wrote in Arabic), we quite properly refer to these numbers as Hindu-Arabic. What is important, I feel, is to stress that the addition of ‘Arabic’ to the naming is more than simply a reference to the fact
that the Indian numerals arrived in Europe via the Muslim world.
8
Algebra
Suppose that a man, in his illness, emancipates two slaves, the price of one being three hundred dirhems and that of the other five hundred dirhems; the one for three hundred dirhems dies, leaving a daughter; then the master dies, leaving a daughter likewise; and the slave leaves property to the amount of four hundred dirhems. With how much must everyone ransom himself?
Al-Khwārizmi
The above quotation comes from al-Kitab al-Mukhtasar fi Hisāb al-Jebr wal-Muqābala. That’s easy for you to say, you might be thinking. The full translation of this title is: The Compendium on Calculation by Restoration and Balancing, and for reasons that will soon become clear, it is admissible to abbreviate this mouthful to just al-Jebr. Its author is that stalwart of al-Ma’mūn’s House of Wisdom, Ibn Mūsa al-Khwārizmi, and in it he sets out for the first time the subject of algebra as a mathematical discipline in its own right rather than a branch of arithmetic or geometry. Indeed, the word ‘algebra’ originates from the al-jebr in the title.
Al-Khwārizmi came to Baghdad in the early ninth century from a region of Central Asia just south of the Aral Sea. He was originally a Zoroastrian who we think converted to Islam. On the very first page of al-Jebr, he begins with the line Bism-Illāh al-Rahmān al-Rahīm (‘In the name of God, the most Gracious and Compassionate’), with which all books written by Muslims begin, even to this day. But it could of course be that al-Khwārizmi was simply following tradition and did not wish to offend the caliph whose patronage he enjoyed. We have already seen how al-Khwārizmi was one of the central characters in al-Ma’mūn’s circle of scholars. In producing his famous Picture of the Earth treatise, in which he tabulated the coordinates of hundreds of cities in the known world and gave instructions for drawing a new map of the world, he secured his legacy as the first geographer of Islam. And by overseeing the astronomical work at the Shammāsiyya observatory in Baghdad and then producing a highly influential zīj, he marked himself out as one of its great astronomers. But he is primarily known as a mathematician, and his treatise on Hindu numerals introduced the Muslim world to the decimal number system. Yet all these achievements pale alongside his greatest claim to fame, which is without doubt his book on algebra. Interestingly, and unlike his famous contemporary, al-Kindi, he never ventured into philosophy; nor was he involved in translations, and had no knowledge of the Greek language.
It is not known in what year al-Khwārizmi completed his al-Jebr, but on the very first page he wrote a dedication to his patron, al-Ma’mūn. It is from these early passages that we learn of his motivation for writing it: ‘That fondness for science, by which God has distinguished the Imam al-Ma’mūn, the Commander of the Faithful … has encouraged me to compose a short work on calculating by (the rules of) completion and reduction.’1 And here lies part of the real value of his work, for what al-Khwārizmi did was to bring together obscure mathematical rules, known only to the few, and turn them into an instruction manual for solving mathematical problems that crop up in a wide range of everyday situations.
Before delving into the details of his book, it might be useful to explore exactly what is meant by algebra. We all learn at school how to solve problems involving ‘unknown’ quantities, usually labelled as x and y. It is quite straightforward to demonstrate why algebra is so useful in solving many different kinds of problems in mathematics, science, engineering, finance and so on. As a quick refresher, let us begin with a trivial example. Writing the equation x − 4 = 2 means that there is a number, currently designated by the letter x, that has a value such that if we subtract 4 from it the answer will be 2. It is obvious then that x must be 6, and I could have dispensed with the trouble of writing a mathematical equation involving the symbol x and simply stated in words: what is the number that, if we subtract 2 from it, leaves 4?
But how about another problem, where a knowledge of algebra and its rules can come in useful (even though the problem itself is no more than a simple brainteaser)? Here it is. You and I each have a basket of eggs, but we do not know how many eggs either basket contains. We are informed that if I give you one of my eggs then we shall both have the same number. If, on the other hand, you were to give me one of your eggs then I would have twice as many as you. How many eggs does this mean we each had originally? Try thinking through this puzzle before reading on.
The standard response from most people when set this problem is to resort to ‘trial and error’, testing out pairs of numbers to see if they satisfy the two criteria. First, you should quickly surmise that I must have two more eggs than you, so that by giving you one, we end up with the same number (I lose one but you gain one). But this does not give us a unique answer, for I could have twelve and you ten, or I could have 150 and you 148. The second piece of information now needs to be taken into account, but without algebra you would just be trying pairs of numbers until you hit upon the correct combination. In fact, the answer is that I have seven eggs and you have five – giving you one of mine means we both end up with six, but you giving me one of yours results in you being left with four while I have eight – twice as many as you.
To set up the problem algebraically, we would begin by saying: let the larger number of eggs be x and the smaller number y. We can now generate two equations: x − 1 = y + 1 and x + 1 = 2(y − 1). We would then have to know the rules of algebraic manipulation (rearranging and reorganizing the letters and numbers in the equations) in order to arrive at the answer: x = 7, y = 5. It was this set of rules that al-Khwārizmi describes in his al-Jebr, and he is therefore widely hailed as the ‘father of algebra’.
But the matter turns out to be rather more complicated than this. We should be careful not to credit al-Khwārizmi with inventing a discipline just because the name we use for it today originated from the title of his book. After all, I did not credit Jābir ibn Hayyān with the title ‘the father of chemistry’ on the basis of etymology; or, more specifically, credit him with the discovery of alkalis just because that particular word has Arabic origins; for alkalis, known by various names, were in use many hundreds of years before Jābir. The same should apply to al-Khwārizmi, and such a distinction as the founder of a discipline will need to be backed up with a more careful investigation into the mathematical legacy he inherited.
This issue was highlighted for me several years ago when I gave a public lecture at the Royal Society in London on the contributions to science from the golden age of Islam. I glibly, and without really backing up my claim, credited al-Khwārizmi with the invention of algebra. At the end of the lecture, a member of the audience approached me and argued indignantly that in fact algebra went back long before al-Khwārizmi and that if anyone deserved the title of ‘the father of algebra’ then it was a Greek mathematician by the name of Diophantus. Not being an expert on this matter at that time, I had no strong counter-argument. Had I been too hasty in my praise of al-Khwārizmi? Worse still, had I been guilty of intellectually lazy bias towards the scholars of Islam by downplaying some of the great achievements of the ancient Greeks – something I unfortunately encounter regularly and which I was determined to avoid? I undertook to look into the matter more carefully and describe in what follows what I managed to uncover. It turns out to be a quite fascinating subject, one that I do not believe has been carefully explored outside academic circles.
One cannot help but admire the concerted effort on the part of the Arabic mathematicians who came after al-Khwārizmi to promote and seal his reputation, a PR job that was helped further because of the impact his book then had in Europe. It was translated into Latin in the twelfth century, not once but twice, by the Englishman Robert of Chester and the Italian Gerard of Cremona. His work was also known to Fibonacci, the greatest European mathematician of the Middle Ages, who quotes al-Khwārizmi in his Liber Abaci of 1202. In it he refers to the Modum Algebre et Almuchabale and its author Maumeht, the Latinized version of al-Khwārizmi’s first name
, Muhammad.
But given the ubiquity of mathematical problems that needed to be solved algebraically, whether involving working out areas of land for agriculture, or financial problems to do with inheritance or taxes, or purely for solving puzzles for recreational purposes, it is hardly surprising that some form of algebra existed long before Islam. The question is whether it really qualifies as algebra. An ancient Babylonian problem found in a cuneiform text reads: ‘What is the number, when added to its reciprocal, gives a known number?’ The way we would formulate this puzzle algebraically today is to write the unknown number as x and the known number as b. We can then express the problem as an equation:
This equation can be recast in the form x2 − bx + 1 = 0, which is what is known as a quadratic equation (one in which the highest power of the unknown quantity is 2, i.e. x2). Similarly, a cubic equation is one in which the highest power is x3, a quartic, x4, and so on. The solution of the quadratic equation written above is given by a formula that every schoolchild gets drummed into them (even if only to be forgotten later in life). For this particular example it is of the form
This means that, if given the value of b, you can work out x. The Babylonians knew this formula, as did the Greeks. They did not write a general equation as I have done above, but instead solved particular examples for particular values of the known quantity b.