Posts Tagged 'algebra'

From al-Khwārizmī to Steve Jobs

Published January 5, 2012 algebra 2 Comments
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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.

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 8th 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 16th century.

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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?

Published December 30, 2011 algebra 2 Comments
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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.

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

to

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.

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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

Published December 25, 2011 algebra 2 Comments
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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.

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 12th 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 19th 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.

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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.

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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.

The golden age of Arabic mathematics

Published December 19, 2011 algebra 6 Comments
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The third in a series. See the November 20 entry, “What is algebra?” for the first and the December 13 entry “When did algebra begin?” for the second.

On 14 September 786, Harun al-Rashid became the fifth Caliph of the Abbasid dynasty. From his court in the capital city of Baghdad, Harun ruled over the vast Islamic empire, stretching from the Mediterranean to India. He brought culture into his court and encouraged the widespread pursuit of learning.

Al-Rashid had two sons, the elder al-Amin, the younger al-Mamun. Harun died in 809 and there was an armed conflict between the brothers. Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad.

Al-Mamun continued the patronage of learning started by his father. With his encouragement, scholars of the time set about collecting and writing down in books all available practical knowledge, much of which had hitherto been transmitted only orally, including mathematics and folk astronomy. They translated into Arabic works of Greek and Indian science.

Many of the works collected and created may have been housed in a library called the House of Wisdom, though there is no evidence to support the commonly repeated claims that (1) it was massive, (2) it was founded by al-Mamun, or (3) translations were carried out there.

The tradition of learning, writing, and translation begun by al-Rashid and al-Mamun continued for the next quarter century, making the Islamic civilization the center of world knowledge. The aristocracy and other wealthy groups within Muslim society supported the appropriation of all practical and scientific knowledge they could acquire. They employed scholars to translate into Arabic works by Indian, Sasanian, and especially Greek authors, and mathematicians recorded on paper all that was known of arithmetic, algebra, and mensuration, which had hitherto been communicated orally by traders. In addition to the mathematical sciences (arithmetic, geometry, optics, mathematical astronomy, etc.), they also translated texts on geography, astrology, philosophy, medicine, agriculture, alchemy, and even falconry.

Greek works formed the bulk of the material translated. In addition, the more scientifically oriented mathematicians adopted the Greek tradition of definitions, axioms, and propositions with rigorous proof, and astronomers embraced the Greek idea of geometric models of planetary motion. Within this framework, Indian techniques were incorporated into this new Arabic/Islamic mathematics.

In addition to the translations, scholars wrote commentaries and criticisms of the ancient mathematics and made their own original contributions. For example, in the 9th century, Thābit ibn Qurra (d. 901) translated several works of Archimedes, wrote commentaries on Euclid’s Elements and Ptolemy’s Almagest, critiqued Euclid’s definition for the composition of ratios of numbers, and derived and proved new formulas for volumes of solids of revolution.

When the sources of Greek and other foreign texts was finally exhausted, scholars continued to produce new results in all branches of mathematics. For instance, in the 11th century, Ibn al-Haytham made major contributions to optics and geometry, and at the start of the 12th century, al-Khāyyamī wrote his book on algebra.

Over a thousand mathematical manuscripts from the period have survived, about half of them dating before the 15th  century.

Al-Khwārizmī, who may have studied and worked in the House of Wisdom, was one of the earliest contibutors to this vast undertaking, and arguably had the most impact of all the mathematicians involved. But his books – he wrote one on Hindu arithmetic in addition to the one on algebra – should be viewed as part of this larger movement.

At the time, algebra was viewed primarily as a practical, numerical problem solving technique, not the autonomous branch of mathematics it became later. Indeed, the greatest contribution of Arabic mathematical work to society was its development as a set of practical tools.

Three systems of practical calculation were taught and practiced in the medieval Islamic world: finger reckoning, Hindu arithmetic, and the base 60 system of the astronomers. Merchants preferred finger-reckoning, which worked for numbers up to 10,000. Finger reckoning was used to solve problems by various methods, such as double false position and algebra. Al-Khwārizmī is known to have written a work, now lost, called Book of Adding and Subtracting, in the early 9th century, which was probably devoted to the use of finger reckoning. (If so, it was probably the earliest written text on the subject.)

The Arabic mathematicians referred to the numerals 1, 2, 3, etc., as Hindī numerals, because they acquired the system from India. These numerals were already in use in the Middle East by the 7th century CE. The earliest known Arabic text describing the system is al-Khwārizmī’s Book on Hindī Reckoning, written in the early 9th  century, which survives only in Latin translation. The original algorithms for calculating in this system were devised for use on a dust board, where erasing is easy. In the middle of the 10th century, al-Uqlīdisī introduced new algorithms for use with pen and paper. The Arabic mathematicians introduced the concept of decimal fractions, wihich al-Uqlīdisī described for the first time.

Unlike Diophantus, most of the Arabic authors, including al-Khwārizmī, wrote their algebra almost entirely in words. For example, where we would write down the symbolic equation  x + 1 = 2, they might write “The thing plus one equals two” (and very occasionally “The thing plus 1 equals 2”). This is generally known as the rhetorical form, and remained in common use right up to the 16th  century. This is, however, a notational distinction, not one of content. Commentators who refer to “rhetorical algebra” as being a form of algebra distinct from “literal algebra” are in error. For, although the Arabic authors wrote their books rhetorically, with no notation even for numbers, they did not solve problems rhetorically. Throughout most of Arabic algebra, problems were worked out on some ephemeral surface, by writing the coefficients and numbers in Hindu form. For example, they would write

1  2  1

to mean x2 + 2x + 1. Later, Arabic scholars in the Maghreb developed a truly algebraic notation, with symbols for the words representing the powers of the unknown, but even they they would resort to rhetorical text to communicate the result of a calculation.

Symbolic algebra, where full symbolism is used, is generally credited in the first instance to the French mathematician François Viète (1540 –1603), followed by René Descartes (1596 – 1650), though traces can be discerned in the writings of some Arabic mathematicians as early as the 13th century.

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In my next two articles in this short series, I’ll say a bit about  al-Khwārizmī and take a look at the contents of his seminal book on algebra. In particular, I’ll give his answer to that perennial student question, “What is algebra good for?”

I use terms like “Arabic mathematics” in the standard historical fashion to refer to the mathematics done where and when the primary language for scholastic texts was Arabic. Mathematics, like all of science, belongs to the world.

When did algebra begin?

Published December 13, 2011 algebra 11 Comments
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The second in a series. See the November 20 entry, “What is algebra?” for the first.

Two key features of algebra as we understand the word today are:

1. Reasoning about numbers by recognizing patterns across numbers;

2. Solving a problem by introducing a term for an unknown and then, starting with what is known, reasoning to determine its value.

We first see the emergence of both features of algebra in the mathematics of ancient Babylonia, around 2,000 BCE.

Several hundred of the many thousands of Babylonian’s cuneiform-inscribed clay tablets that have been found are devoted to mathematics. They show that those ancient mathematicians had systematic procedures for solving geometric problems involving the determination of lengths and areas of figures. Today, we would solve those kinds of problems using linear and quadratic equations and indeterminate systems of linear equations. Their methods amounted to a form of geometric algebra that could be applied to solve problems beyond overtly geometric examples such as calculating the perimeters or areas of various plane figures or the volumes of solid objects: arithmetic problems arising in trade and commerce, for example, and other financial transactions such as inheritance. In addition, the Babylonians considered problems that seemed to have had no practical application, pursuing them purely for recreation. Although they described their procedures in terms of specific lengths and areas, they did so in a way that made it clear they applied in general, and in that sense they were starting to think algebraically, by recognizing patterns across quantities.

Moreover, some of their writings show the second characteristic feature of algebra, namely introducing an unknown and then reasoning to find its value. In their case, however, the unknown was not numeric but geometric – an unknown line on which they performed geometrical operations to get the answer.

In reasoning with unknown quantities, the Babylonians went further than other early civilizations with a mathematical tradition, such as the Egyptians, the Chinese, and the early Greeks, all of the first millennium BCE. Our knowledge of the mathematics of those peoples comes from works such as the Rhind papyrus, The Nine Chapters of the Mathematical Art, and Euclid’s Elements, respectively. The approach described in those documents was, like that of the Babylonians, fundamentally geometric and exhibited reasoning about patterns of quantities, but we do not find the introduction of an unknown followed by an argument to determine its value.

It is with the work of the Greek mathematician Diophantus  (ca. 210–290 CE) that we first find clearly recognizable algebra, where the unknowns represent numbers whose values are to be determined. Around 250 CE, Diophantus, who lived in Alexandria in Egypt, wrote a multi-volume work, Arithmetica, which its title notwithstanding was an algebra book. Its author used letters (literals) to denote the unknowns and to express equations, but that is a purely notational distinction. He also was one of the first mathematicians to use negative numbers in calculations. He showed how to solve equations by using two techniques called restoration and confrontation. In modern terms, these correspond more or less (but not precisely) to (1) adding a quantity to both sides of an equation to eliminate a negative term on one side, and (2) eliminating like terms from both sides. He used these techniques to solve polynomial equations involving powers up to 6.

Almost four hundred years later, the Indian mathematician Brahmagupta (598–668 CE) likewise displayed recognizable algebra, in his book Brahmasphutasiddhanta, where he described the first complete arithmetic solution (including zero and negative solutions) to quadratic equations.

Following Diophantus and Brahmagupta, the next major step in the development of algebra – and it was huge – took place in the period generally referred to as “Arabic mathematics” or “Muslim mathematics”, a significant outpouring of mathematical activity stretching from the 8th century to the end of the 16th. Indeed, the word algebra itself comes from the Arabic word al-jabr, which occurs in the title of a highly influential book by the Persian mathematician al-Khwārizmī, completed around 830: al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqābala. The phrase al-jabr wa’l-muqābalah translates literally as “restoration and confrontation,” but more loosely means “solving an equation.”

That period will be the focus of my next article on algebra.

What is algebra?

Published November 20, 2011 algebra 30 Comments
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We hear a lot about the importance that all children master algebra before they graduate from high school. But what exactly is algebra, and is it really as important as everyone claims? And why do so many people find it hard to learn?

Answering these questions turns out to be a lot easier than, well, answering a typical school algebra question, yet surprisingly, few people can give good answers.

First of all, algebra is not “arithmetic with letters.” At the most fundamental level, arithmetic and algebra are two different forms of thinking about numerical issues. (I should stress that in this article I’m focusing on school arithmetic and school algebra. Professional mathematicians use both terms to mean something far more general.)

Let’s start with arithmetic. This is essentially the use of the four numerical operations addition, subtraction, multiplication, and division to calculate numerical values of various things. It is the oldest part of mathematics, having its origins in Sumeria (primarily today’s Iraq) around 10,000 years ago. Sumerian society reached a stage of sophistication that led to the introduction of money as a means to measure an individual’s wealth and mediate the exchange of goods and services. The monetary tokens eventually gave way to abstract markings on clay tablets, which we recognize today as the first numerals (symbols for numbers). Over time, those symbols acquired an abstract meaning of their own: numbers. In other words, numbers first arose as money, and arithmetic as a means to use money in trade.

It should be noticed that counting predates numbers and arithmetic by many thousands of years. Humans started to count things (most likely family members, animals, seasons, possessions, etc.) at least 35,000 years ago, as evidenced by the discovery of bones with tally marks on them, which anthropologists conclude were notched to provide what we would today call a numerical record. But those early humans did not have numbers, nor is there any evidence of any kind of arithmetic. The tally markers themselves were the record; the marks referred directly to things in the world, not to abstract numbers.

Something else to note is that arithmetic does not have to be done by the manipulation of symbols, the way we are taught today. The modern approach was developed over many centuries, starting in India in the early half of the First Millennium, adopted by the Arabic speaking traders in the second half of the Millennium, and then transported to Europe in the 13th Century. (Hence its present-day name “Hindu-Arabic arithmetic.”) Prior to the adoption of symbol-based, Hindu-Arabic arithmetic, traders performed their calculations using a sophisticated system of finger counting or a counting board (a board with lines ruled on it on which small pebbles were moved around). Arithmetic instruction books described how to calculate using words, right up to the 15th Century, when symbol manipulation began to take over.

Many people find arithmetic hard to learn, but most of us succeed, or at least pass the tests, provided we put in enough practice. What makes it possible to learn arithmetic is that the basic building blocks of the subject, numbers, arise naturally in the world around us, when we count things, measure things, buy things, make things, use the telephone, go to the bank, check the baseball scores, etc. Numbers may be abstract — you never saw, felt, heard, or smelled the number 3 — but they are tied closely to all the concrete things in the world we live in.

With algebra, however, you are one more step removed from the everyday world. Those x’s and y’s that you have to learn to deal with in algebra denote numbers, but usually numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction. Doing so requires quite a lot of effort and training.

The important thing to realize is that doing algebra is a way of thinking and that it is a way of thinking that is different from arithmetical thinking. Those formulas and equations, involving all those x’s and y’s, are merely a way to represent that thinking on paper. They no more are algebra than a page of musical notation is music. It is possible to do algebra without symbols, just as you can play and instrument without being ably to read music. In fact, traders and other people who needed it used algebra for 3,000 years before the symbolic form was introduced in the 16th Century. (That earlier way of doing algebra is nowadays referred to as “rhetorical algebra,” to distinguish it from the symbolic approach common today.)

There are several ways to come to an understanding of the difference between arithmetic and (school) algebra.

  • First, algebra involves thinking logically rather than numerically.
  • In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
  • Arithmetic involves quantitative reasoning with numbers; algebra involves qualitative reasoning about numbers.
  • In arithmetic, you calculate a number by working with the numbers you are given; in algebra, you introduce a term for an unknown number and reason logically to determine its value.

The above distinctions should make it clear that algebra is not doing arithmetic with one or more letters denoting numbers, known or unknown.

For example, putting numerical values for a, b, c  in the familiar formula

in order to find the numerical solutions to the quadratic equation

 is not algebra, it is arithmetic.

In contrast, deriving that formula in the first place is algebra. So too is solving a quadratic equation not by the formula but by the standard method of “completing the square” and factoring.

When students start to learn algebra, they inevitably try to solve problems by arithmetical thinking. That’s a natural thing to do, given all the effort they have put into mastering arithmetic, and at first, when the algebra problems they meet are particularly simple (that’s the teacher’s classification as “simple”), this approach works.

In fact, the stronger a student is at arithmetic, the further they can progress in algebra using arithmetical thinking. For example, many students can solve the quadratic equation  x2 = 2x + 15  using basic arithmetic, using no algebra at all.

Paradoxically, or so it may seem, however, those better students may find it harder to learn algebra. Because to do algebra, for all but the most basic examples, you have to stop thinking arithmetically and learn to think algebraically.

Is mastery of algebra (i.e., algebraic thinking) worth the effort? You bet — though you’d be hard pressed to reach that conclusion based on what you will find in most school algebra textbooks. In today’s world, most of us really do need to master algebraic thinking. In particular, you need to use algebraic thinking if you want to write a macro to calculate the cells in a spreadsheet like Microsoft Excel. This one example alone makes it clear why algebra, and not arithmetic, should now be the main goal of school mathematics instruction. With a spreadsheet, you don’t need to do the arithmetic; the computer does it, generally much faster and with greater accuracy than any human can. What you, the person, have to do is create that spreadsheet in the first place.  The computer can’t do that for you.

It doesn’t matter whether the spreadsheet is for calculating scores in a sporting competition, keeping track of your finances, running a business or a club, or figuring out the best way to equip your character in World of Warcraft, you need to think algebraically to set it up to do what you want. That means thinking about or across numbers in general, rather than in terms of (specific) numbers.

Of course, the need for algebra does not make it any easier to learn — though I think that spreadsheets can provide today’s students with more meaningful and fulfilling applications than problems about trains leaving stations or garden hoses filling swimming pools, that my generation had to endure. But in a world where our very national livelihood depends on staying ahead of the technology curve, it is crucial that we equip our students with the kind of thinking skills today’s world requires. Being able to use computers is one of those skills. And being able to use a computer to do arithmetic requires algebraic thinking.

In future postings I’ll describe the growth of algebra through the ages.

ABOUT ME

I’m Dr. Keith Devlin, an emeritus mathematician at Stanford University, an author, and was for many years “the Math Guy” on NPR’s Weekend Edition. Off duty, I’m an avid cyclist. (The header photo is me halfway up Mt. Baldy in Southern California.)

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