From al-Khwārizmī to Steve Jobs

The sixth and last in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, the December 19 entry “The golden age of Arabic mathematics” for the third, the December 25 entry “al-Khwārizmī” for the fourth, and the December 30 entry “What is algebra good for?” for the fifth.

JANUARY 5, 2012. History tends to focus on key individuals, when in fact most advances are the cumulative affect of the contributions of many. In the case of algebra, claims that al-Khwārizmī invented algebra are not sustainable. As I have explained in previous articles in this short series, the chain leading to algebra goes back at least to the ancient Babylonians, and to modern eyes Diophantus’s book Arithmetica was clearly a book on algebra. Nevertheless, al-Khwārizmī does deserve the credit for establishing algebra as a major collection of intellectual tools.

He can’t be credited with establishing it as a branch of mathematics, howver, since the mathematicians of the Arabic period did not view the methods they developed as anything other than a set of very valuable practical tools. (Likewise, Diophantus viewed his work as a sophisticated form of arithmetic, as the title of his famous work suggests.) Viewing algebra as a discipline in its own right came later.

Al-Khwārizmī’s greatness is in the same category as Euclid nine centures earlier, as Leonardo of Pisa four hundred years later, or as Steve Jobs in our own time: their impact on society and thence the course of history. None of these four were the original inventors or discoverers of the seminal developments we associate with their names. Their greatness was not one of original discovery – though both Euclid and al-Khwārizmī may well have contributed some of the methods they described in their seminal books and we know from his other works besides Liber abbaci that Leonardo was a first-rate, original mathematician. Rather, all four had the highly unusual ability to take a collection of powerful new ideas and package and present them to society in a manner that made them acceptable to – indeed eagerly sought-after by – a wide range of people. In our present-day society we tend to focus on the priority of discovery and invention, as epitomised by the status we award Nobel Laureates, but initial discovery would be of little value were others not able to take the new knowledge and use it to change society.

Each of Euclid, al-Khwārizmī, and Leonardo (and Steve Jobs) were followed by many others who carried the torch forwards, and they too deserve credit.

Among the hundreds of Arabic mathematicians who helped to develop and spread algebraic knowledge after al-Khwārizmī, several stand out as worthy of special mention. I’ll list a few.

Abū Kāmil The Egyptian born Abū Kāmil Shujāʿ ibn Aslam ibn Muḥammad ibn Shujā (c. 850 – c. 930) was the first major Arabic algebraist after al-Khwārizmī. By all accounts he was a prolific author. There are references to works with the titles Book of fortune, Book of the key to fortune, Book of the adequate, Book on omens, Book of the kernel, Book of the two errors, and Book on augmentation and diminution. None of these have survived. Works that did survive include the Book on algebra, the Book of rare things in the art of calculation, Inheritance by means of algebra, and the Book on surveying and geometry.

The Book on algebra (Kitāb fi al-jabr wa al-muqābala) is arguably Abū Kāmil’s most influential work. It expanded on al-Khwārizmī’s Algebra. Whereas the latter was aimed at the general public, Abū Kāmil wrote more for other mathematicians, assuming familiarity with Euclid’s Elements. He extended the range of polynomials studied beyond al-Khwārizmī to include 8^th powers.

Al-Karajī A century after Abū Kāmil did his work, around 1000 C.E., another major advance in algebra was made by the Persian mathematician and engineer Abū Bakr ibn Muḥammad ibn al Ḥusayn al-Karajī, who lived from c. 953 to c. 1029. His three major works were Al-Badi’ fi’l-hisab (Wonderful on calculation), Al-Fakhri fi’l-jabr wa’l-muqabala (Book of al-Fakhri on the Art of Algebra), and Al-Kafi fi’l-hisab (Sufficient on calculation).

Al-Fakhri is regarded as one of the key works on the path that led to the final separation of algebra from geometry as a discipline in its own right. Al-Karajī gave a systematic treatment of reducible higher-degree equations. He studied the algebra of exponents, and was the first to state explicitly that the sequence x, x², x³, … could be extended indefinitely, and likewise the reciprocals 1/x, 1/x², 1/x³, …

Omar Khayyám Shortly after al-Karajī died, another famous Arab scholar came onto the scene: Omar Khayyám. Although in the West he is better known today as a poet, he was a first rate mathematician.

Al-Khayyám, more fully Ghiyath al-Din Abu’l-Fath Umar ibn Ibrahim al-Nisaburi al-Khayyámi, was born on 18 May, 1048 in Nishapur, Persia (now Iran), and died there on 4 December, 1131. As a young man he studied philosophy, and went on to be an outstanding mathematician and philosopher. By the time he was 25, he had written several books, covering arithmetic, geometry, algebra, and music. His major work in algebra was an analysis of polynomial equations titled Treatise on the Proofs of Algebra Problems.

Al-Khayyám approached mathematics primarily as a geometer, firmly rooted in the Greek tradition. Whereas Abū Kāmil and al-Karajī presented algebra as a method for numerical problem-solving, al-Khayyám viewed it as a tool for theoretical geometers.

al-Samawʾal Several further advances in algebra were made around the mid-twelfth century by a teenager (yes, that’s right, a teenager) called Ibn Yaḥyā al-Maghribī al-Samawʾal, who was born around 1130 in Baghdad. His parents were Jewish, his father a literature scholar and Rabbi from Morocco, his mother from Basra, in Iraq.

Although his initial interest as a child was to become a doctor, al-Samawʾal proved to be a child prodigy in mathematics, and the study of medicine was soon relegated to second place (but not abandoned). He began to study the Hindu methods of calculation when he was thirteen or so. Rapidly finding himself ahead of his teachers, he continued on his own, reading the works of Abū Kāmil, al-Karajī, and others. By the time he was eighteen years old he had read almost all the available mathematical literature. He wrote his most famous treatise, al-Bahir fi’l-jabr (The brilliant in algebra), when he was just nineteen years old.

Mathematicians before al-Samaw’al had begun to develop what contemporary historians have called the “arithmetization of algebra”. Al-Samaw’al was perhaps the first to give this development a precise description, writing that it involved “operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.” This can be regarded as a significant step toward the development of modern algebra.

In all, al-Samaw’al is reported to have written 85 books or articles, though most have not survived. He died in Maragha, Iran, around 1180.

Further advances in algebra were made in the Maghreb in the twelfth to fifteenth century, by a highly organized teacher-student network linked to mosque and madrasah teaching. The Maghrebs used abbreviations for both unknowns and their powers and for operations; another innovation in the chain that culminated in the development of modern symbolic algebra in Europe in the 16^th century.

* * *

For the next episode in the development and growth of algebra, when the ideas found their way to Europe, see my recent book The Man of Numbers: Fibonacci’s Arithmetic Revolution. (And for a comparison between Fibonacci’s role and that of Steve Jobs, see the companion e-book Leonardo and Steve.)

Acknowledgement
I am greatful to Professor Jeffrey Oaks of the University of Indianapolis for his assistance in the prepartion of the essays in this series. In particular, he supplied me with preprints of his forhtcoming articles for Springer Verlag’s upcoming Encyclopedia of Sciences and Religions (2012): “Mathematics and Islam”, “Arithmetic and Islam”, “Algebra and Islam”, and “Geometry and Islam”, which I drew on heavily. He also commented in detail on a more substantial work from which these essays were abridged.

What is algebra good for?

The fifth in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, the December 19 entry “The golden age of Arabic mathematics” for the third, and the December 25 entry “al-Khwārizmī” for the fourth.

DECEMBER 30, 2011. Modern algebra is generally acknowledged to have begun with the appearance around 830 CE of al-Khwārizmī’s book al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. What better source can there be to find the answer to that perennial student question, “What is this stuff good for?” In the introduction to his seminal work, al-Khwārizmī stated that its purpose was to explain:

… what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

It doesn’t get much more practical and useful than that! Either in 830 or today! Throughout history, the nations that led the world in mathematics led the world in commerce, industry, and science. In the 9th century, Baghdad was the commercial, industrial, and scientific center of world. In the 13th century, the leadership role crossed the Mediterranean to Italy, then over the ensuing centuries continued gradually westwards through Europe, crossing the Atlantic to the East coast of the US in the middle third of the 20th century, arriving in California in the 1980s, and likely to cross the Pacific (back) to China within the next couple of decades.

Al-Khwārizmī’s strong emphasis on practical applications typified Arabic texts of the time, every bit as much as the intense focus on applications of mathematics and science you find in today’s Silicon Valley.

The book was divided into three parts. The first part was devoted to algebra, giving the rules together with 39 worked problems, all abstract. Then came a short section on the rule of three and mensuration. Two mensuration problems dealing with surveying were solved with algebra. Finally, al-Khwārizmī presented a long section on inheritance problems solved by algebra.

The term al-jabr (“restoration” or “completion”) in al-Khwārizmī’s title refers to a procedure whose modern counterpart is eliminating negative terms from a (linear or quadratic) equation by adding an appropriate qantity to both sides of the equation. For example, using one of al-Khwārizmī’s own examples (but expressed using modern symbolic notation), al-jabr transforms

x² = 40x – 4x²

into

5x² = 40x.

The other key term in the title, al-muqābala (“confrontation”) refers to the process of eliminating identical quantities from the two sides of the equation. For example, (again in modern notation) one application of al-muqābala simplifies

50 + 3x + x² = 29 + 10x

21 + 3x + x² = 10x

and a second application simplifies that to

21 + x² = 7x.

Procedurally (but not conceptually) these are the methods we use today to simplify and solve equations. Hence, a meaningful, modern English translation for Hisâb al-Jabr wa’l-Muqābala would be, simply, “Calculation with Algebra.”

The symbolic notation is not the only difference between medieval algebra and its present-day counterpart. The medieval mathematicians did not acknowledge negative numbers. For instance, they viewed “ten and a thing” (10 + x) as a composite expression (it entails two types of number: “simple numbers” and “roots”), but they did not see “ten less a thing” (10 – x) as composite. Rather, they thought of it as a single quantity, a “diminished” 10, or a 10 with a “defect” of x. The 10 retained its identity, even though x had been taken away from it. When an x was added to both sides of an equation, the diminished 10, (10 – x), was restored to its rightful value. Hence the terminology.

The first degree unknown, our x, was usually called shay’ (“thing”), but occasionally jidhr (“origin” or “base”, also “root” of a tree, giving rise to our present-day expression “root of an equation”). The second power, our x², was called māl (a sum of money/property/ wealth). Units were generally counted in dirhams, a denomination of silver coin, occasionally simply “in number”. For example, al-Khwārizmī’s (rhetorical) equation “a hundred ten and two māls less twenty-two things equals fifty-four dirhams” corresponds to our symbolic equation 110 + 2x² – 22x = 54.

Arabic authors typically explained the methods of algebra in two stages. First they provided an explanation of the names of the powers, described six simplified forms of equations and their solutions, and gave rules for operating on polynomials and roots. They then followed this introduction by a collection of solved problems which illustrated the methods.

Their solutions followed a standard template:

Stage 1: an unknown quantity was named (usually referred to as a “thing”), and an equation was set up.

Stage 2: the equation was simplifed to one of six canonical types.

Stage 3: the appropriate procedure was applied to arrive at the answer.

Because they allowed only positive coefficients, they had to consider six equation types, rather than the single template ax² + bx + c = 0 we use today:

(1) māls equals roots (in modern terms, ax² = bx),

(2) māls equals numbers (ax² = c),

(3) roots equals numbers (bx = c),

(4) māls and roots equals numbers (ax² + bx = c),

(5) māls and numbers equals roots (ax² + c = bx),

(6) māls equals roots and numbers (ax² = bx + c).

We see how al-Khwārizmī used the two simplification steps in Stage 2, al-jabr wa’l-muqābala, (“restoration and confrontation”) in his solution to a quadratic equation, which he described in these words:

If some one say: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.

The American scholar Jeffrey Oaks has translated this (fairly literally) as follows, adding headings to assist the reader:

Enunciation
If [someone] said, ten: you divided it into two parts. You multiplied one of the parts by itself, which is the same as eighty-one times the other.

Setting up and simplifying the equation
The rule for this is that you say ten less a thing by itself is a hundred and a mal less twenty things [which] equal eighty-one things. Restore the hundred and a mal by the twenty things and add them to the eighty-one [things]. This yields: a hundred and a mal equal a hundred roots and a root.

Solving the simplified equation
So halve the roots, which yields fifty and a half, and multiply it by itself, which yields two thousand five hundred fifth and a fourth. Subtract from it the hundred, leaving two thousand four hundred fifty and a fourth. Take its [square] root, which is forty-nine and a half. Subtract it from half the roots, which is fifty and a half. There remains one, which is one of the two parts.

Using modern notation, and substituting the letter x for “thing”, al-Khwārizmī was solving the equation

(10 − x)² = 81x

which can be written in the equivalent form

x² + 100 = 101x

Al-Khwārizmī did not state the equation

(10 − x)² = 81x

Rather, he set up the equation

100 + x² – 20x = 81x.

Nothing like the equation (10 − x)² = 81x was ever stated in medieval algebra; the left side of such an expression entails what was then an unrealized operation. Medieval algebraists worked out all operations before stating equations, so al-Khwārizmī did not begin with (10 − x)² = 81x, as we would, rather he first worked out the multiplication.

Having demonstrated methods for solving linear and quadratic equations, al-Khwārizmī proceeded to examine how to manipulate algebraic expressions. For example he showed how to multiply out specific numerical instances

(a + bx) (c + dx)

expressing everything in words, not symbols.

He ended the first section of the book by presenting the solutions to 39 problems.

In the following section, al-Khwārizmī presented solutions to some mensuration problems, including rules for finding the area of figures such as the circle and for finding the volume of solids such as the sphere, cone, and pyramid.

The final part of the book dealt with the complicated Islamic rules for inheritance, which involved the solution of linear equations.

* * *

COMING UP: In the final article in this series I’ll summarize some of the amazing developments in algebra that were made in the Arabic period subsequent to al-Khwārizmī.

al-Khwārizmī

The fourth in a series. See the November 20 entry, “What is algebra?” for the first, the December 13 entry “When did algebra begin?” for the second, and the December 19 entry “The golden age of Arabic mathematics” for the third.

DECEMBER 25, 2011. Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī (c.780 – c.850 CE) was one of the most significant figures in the development of modern algebra. Yet we know virtually nothing about his life.

There is even some confusion in the literature as to his full name. Most present-day sources give it as Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī, which can be translated as “Father of ʿAbdallāh, Mohammed, son of Moses, native of the town of al-Khwārizmī”. References to Abū Jaʿfar Muḥammad ibn Mūsā al-Khwārizmī are erroneous in this context; that was a different person

Al-Khwārizmī wrote several books, two of which had a huge impact on the growth of mathematics, one focused on arithmetic, the other on algebra. He aimed both at a much wider audience than just his fellow scholars. As with Euclid and his Elements, it is not clear whether al-Khwārizmī himself developed some of the methods he desribed in his books, in addition to gathering together the work of others, though a later author, Abū Kāmil, suggested that his famous predecessor did develop some of the methods he presented in his books.

The first of al-Khwārizmī’s two most significant books, written around 825, described Hindu-Arabic arithmetic. Its original title is not known, and it may not have had one. No original Arabic manuscripts exist, and the work survives only through a Latin translation, which was most likely made in the 12^th century by Adelard of Bath. The original Latin translation did not have a title either, but the Italian bibliophile Baldassare Boncompagni gave it one when he published a printed edition in the 19^th century: Algoritmi de numero Indorum (“al-Khwārizmī on the Hindu Art of Reckoning”). The Latinized version of al-Khwārizmī’s name in this title (Algoritmi) gave rise to our modern word “algorithm” for a set of rules specifying a calculation. In English, the work is sometimes referenced as On the Calculation with Hindu Numerals, but it is most commonly referred to simply as “al-Khwārizmī’s Arithmetic.”

Al-Khwārizmī’s second pivotal book, completed around 830, was al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” or more loosely as “reducing (or solving) an equation.” The title of the book translates literally as “The Abridged Book on Calculation by Restoration and Confrontation”, but a more colloquial rendering would thus be “The Abridged Book on Algebra”. It is an early treatise on what we now call “algebra,” that name coming from the term al-jabr in the title. Scholars today usually refer to this book simply as “Al-Khwārizmī’s Algebra.” There are seven Arabic manuscripts known, not all complete. One complete Arabic copy is kept at Oxford and a Latin translation is kept in Cambridge. Two copies are in Afghanistan.

In Algebra, al-Khwārizmī described (but did not himself develop) a systematic approach to solving linear and quadratic equations, providing a comprehensive account of solving polynomial equations up to the second degree.

The Algebra was translated into Latin by Robert of Chester in 1145, by Gherardo of Cremona around 1170, and by Guglielmo de Lunis around 1250. In 1831, Frederic Rosen published an English language translation. In his preface, Rosen wrote:

ABU ABDALLAH MOHAMMED BEN MUSA, of Khowarezm, who it appears, from his preface, wrote this Treatise at the command of the Caliph AL MAMUN, was for a long time considered as the original inventor of Algebra. … … … From the manner in which our author [al-Khwārizmī], in his preface, speaks of the task he had undertaken, we cannot infer that he claimed to be the inventor. He says that the Caliph AL MAMUN encouraged him to write a popular work on Algebra: an expression which would seem to imply that other treatises were then already extant.

In fact, algebra (as al-Khwārizmī described it in his book) was being transmitted orally and being used by people in their jobs before he or anyone else started to write it down. Several authors wrote books on algebra during the ninth century besides al-Khwārizmī, all having the virtually identical title Kitāb al-ğabr wa-l-muqābala. Among them were Abū Hanīfa al-Dīnawarī, Abū Kāmil Shujā ibn Aslam, Abū Muḥammad al-ʿAdlī, Abū Yūsuf al-Miṣṣīṣī, ‘Abd al-Hamīd ibn Turk, Sind ibn ʿAlī, Sahl ibn Bišr, and Šarafaddīn al-Tūsī.

In addition to his two books on mathematics, al-Khwārizmī wrote a revised and completed version of Ptolemy’s Geography, consisting of a general introduction followed by a list of 2,402 coordinates of cities and other geographical features. Titled Kitāb ṣūrat al-Arḍ (“Book on the appearance of the Earth” or “The image of the Earth”), he finished it in 833. There is only one surviving Arabic copy, which is kept at the Strasbourg University Library. A Latin translation is kept at the Biblioteca Nacional de España in Madrid.

* * *

COMING UP NEXT: Al-Khwārizmī’s answer to that perennial student question, “What is algebra good for?” Plus a look at the contents of his seminal book, including an explanation of what exactly was being “restored” in the process for which al-Khwārizmī’s Arabic term was al-jabr.

* * *

Al-Khwārizmī on National Public Radio: I talked about al-Khwārizmī and the birth of algebra with host Scott Simon in my occasional “Math Guy” slot on NPR’s Weekend Edition on December 24.