Algebra

Algebra is the generic term for the maths of equations, in which numbers and operations are written as symbols. The word itself has a curious history. In medieval Spain, barbershops displayed signs saying Algebrista y Sangrador. The phrase means ‘Bonesetter and Bloodletter’, two trades that used to be part of a barber’s repertoire. (This is why a barber’s pole has red and white stripes—the red symbolizes blood, and the white symbolizes the bandage.)

The root of algebrista is the Arabic al-jabr, which, in addition to referring to crude surgical techniques, also means restoration or reunion. In ninth-century Baghdad, Muhammad ibn Musa al-Khwarizmi wrote a maths primer entitled Hisab al-jabr w’al-muqabala, or Calculation by Restoration and Reduction.

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Al-Khwarizmi wasn’t the first person to use restoration and reduction—these operations could also be found in Diophantus; but when Al-Khwarizmi’s book was translated into Latin, the al-jabr in the title became algebra. Al-Khwarizmi’s algebra book, together with another one he wrote on the Indian decimal system, became so widespread in Europe that his name was immortalized as a scientific term: Al-Khwarizmi became Alchoarismi, Algorismi and, eventually, algorithm.

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Between the fifteenth and seventeenth centuries mathematical sentences moved from rhetorical to symbolic expression. Slowly, words were replaced with letters. Diophantus might have started letter symbolism with his introduction of  for the unknown quantity, but the first person to effectively popularize the habit was François Viète in sixteenth-century France. Viète suggested that upper-case vowels—A, E, I, O, U—and Y be used for unknown quantities, and that the consonants B, C, D, etc., be used for known quantities.

Within a few decades of Viète’s death, René Descartes published his Discourse on Method. In it, he applied mathematical reasoning to human thought. He started by doubting all of his beliefs and, after stripping everything away, was left with only certainty that he existed. The argument that one cannot doubt one’s own existence, since the process of thinking requires the existence of a thinker, was summed up in the Discourse as I think, therefore I am. The statement is one of the most famous quotations of all time, and the book is considered a cornerstone of Western philosophy. Descartes had originally intended it as an introduction to three appendices of his other scientific works. One of them, La Géométrie, was equally a landmark in the history of maths.

In La Géométrie Descartes introduces what has become standard algebraic notation. It is the first book that looks like a modern maths book, full of as, bs and cs and xs, ys and zs. It was Descartes’s decision to use lower-case letters from the beginning of the alphabet for known quantities, and lower-case letters from the end of the alphabet for the unknowns. When the book was being printed, however, the printer started to run out of letters. He enquired if it mattered if x, y or z was used. Descartes replied not, so the printer chose to concentrate on x since it is used less frequently in French than y or z. As a result, x became fixed in maths—and the wider culture—as the symbol for the unknown quantity. That is why paranormal happenings are classified in the X-Files and why Wilhelm Röntgen came up with the term X-ray. Were it not for issues of limited printing stock, the Y-factor could have become a phrase to describe intangible star quality and the African-American political leader might have gone by the name Malcolm Z.

With Descartes’ symbology, all traces of rhetorical expression had been expunged.

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In 1621, a Latin translation of Diophantus’s masterpiece Arithmetica was published in France. The new edition rekindled interest in ancient problem-solving techniques, which, combined with better numerical and symbolic notation, ushered in a new era of mathematical thought. Less convoluted notation allowed greater clarity in describing problems. Pierre de Fermat, a civil servant and judge living in Toulouse, was an enthusiastic amateur mathematician who filled his own copy of Arithmetica with numerical musings. Next to a section dealing with Pythagorean triples—any set of natural numbers a, b and c such that a²+ b² = c², for example 3, 4 and 5—Fermat scribbled some notes in the margin. He had noticed that it was impossible to find values for a, b and c such that a³ + b³= c³. He was also unable to find values for a, b and c such that 

a⁴+ b⁴ = c⁴. Fermat wrote in his Arithmetica that for any number n greater than 2, there were no possible values a, b and c that satisfied the equation aⁿ + bⁿ = cⁿ. ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain,’ he wrote.

Fermat never produced a proof—marvellous or otherwise—of his proposition even when unconstrained by narrow margins. His jottings in Arithmetica may have been an indication that he had a proof, or he may have believed he had a proof, or he may have been trying to be provocative. In any case, his cheeky sentence was fantastic bait to generations of mathematicians. The proposition became known as Fermat’s Last Theorem and was the most famous unsolved problem in maths until the Briton Andrew Wiles cracked it in 1995. Algebra can be very humbling in this way—ease in stating a problem has no correlation with ease in solving it. Wiles’s proof is so complicated that it is probably understood by no more than a couple of hundred people.

Evrenin Dili

Modern ilmî metodolojinin benimsediği araştırma usûlüne göre matematik; ilmî tespitler için "objektif" bir usûl olmasının yanında, elde edilen neticelerin umumîleştirilmesinde de en objektif vasıtadır. Bilim ve teknolojnin arka plânında Kudret-i Sonsuz'un ilminin bir ifadesi sayılan ve çoğunlukla gözden kaçırılan matematik vardır. Orta Çağ'da Müslüman ilim adamlarının fark ettiği bu riyazî düşünce ve matematiğe ait hususiyetler Gazzalî'den Birûnî'ye, Nasiruddin Tûsî'den Hucendî'ye ve Harizmî'ye kadar yüzlerce ilim adamının eserinde vurgulanmıştır.

İslâm âlimlerinin yolunda yürüyen ve modern bilimin öncülerinden sayılan Galileo, 1623'te basılan ikinci kitabı Saggiatore'de şöyle yazmıştı: "Öncelikle kâinattaki geçerli dil öğrenilmedikçe ve sonra da onda yazılı karakterler okunmadıkça kâinat anlaşılamaz. Kâinat, matematik dilinde yazılmıştır ve insan olarak onda yazılan kelimeleri matematik olmaksızın anlamamız imkansızdır." 

Galileo'nun bu sözü, önemli bir hakikate işaret etmekle birlikte; kâinattaki nizam ve cereyan eden hâdiseler çok kompleks olduğundan, bugüne kadar geliştirilen matematikle son derece girift olan bu mükemmelliği kısmen açıklasak bile, bütün kâinatı ifade edebilen matematik sistem ve formülleri anlamada henüz yetersiz kaldığımız görülmektedir. Bilim tarihine bakıldığında; kâinatın varlık yapısı ve işleyiş özellikleri, matematik kullanılarak kısmen ifade edilebilmiştir. Bu kısmî anlaşılma kâinattaki her şeyin bir matematikî açıklaması olduğunu veya matematikle çelişmediğini gösterirken, varlığın izahında mevcut matematik bilgilerinin yetersiz kalan bir boyutunun olduğunu da göstermektedir. Fizikçiler, maddenin yapısını ve tabiattaki kuvvetleri açıklayan denklemler yazarlar. Sun'î kalb tasarlayan bir mühendis, kanın damarlarda nasıl aktığını ifade eden denklemleri dikkate alır. NASA'daki bir astronom, bir uydunun veya uzay gemisinin yörüngesini ifade eden denklemleri kullanır.

Sızıntı Dergisi

Mayıs 2005

Math and Islamic History

In his work on algebra, al-Khwarizmi worked with both what we now call linear equations – that is, equations that involve only units without any squared figures – and quadratic equations, which involve squares and square roots. His advance was to reduce every equation to its simplest possible form by a combination of two processes: al-jabr and al-muqabala.

Al-jabr means ‘completion’ or ‘restoration’ and involves simply taking away all negative terms. Using modern symbols, al-jabr means simplifying. Al-muqabala means ‘balancing’, and involves reducing all the postive terms to their simplest form. 

In developing algebra, al-Khwarizmi built on the work of early mathematicians from India, such as Brahmagupta, and from the Greeks such as Euclid, but it was al-Khwarizmi who turned it into a simple, all-embracing system, which is why he is dubbed the ‘father of algebra’. The very word algebra comes from the title of his book, al-Kitab al-mukhtasar fi hisab al-jabr wa’l muqabala or The Compendious Book on Calculating by Completion and Balancing.

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Beyond al-Khwarizmi, many other Arabic-speaking scholars explored mathematics. Indeed, it was fundamental to so many things, from calculating tax and inheritance to working out the direction of Mecca, that it is hard to find a scholar who did not at some time or other work in mathematics. But it wasn’t just practical applications that fascinated many of them, and they began to push mathematics to its limits.

In the early 11th century in Cairo, Hassan ibn al-Haitham, for instance, laid many of the foundations for integral calculus, which is used for calculating areas and volumes. Half a century later, the brilliant poet/mathematician Omar Khayyam found solutions to all thirteen possible kinds of cubic equations – that is, equations in which numbers are cubed. He regretted that his solutions could only be worked out geometrically rather than algebraically. ‘We have tried to work these roots by algebra, but we have failed’, he says ruefully. ‘It may be, however, that men who come after us will succeed.’

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Omar Khayyam is one of the most extraordinary figures in Islamic science, and tales of his mathematical brilliance abound. In 1079, for instance, he calculated the length of the year to 365.24219858156 days. That means that he was out by less than the sixth decimal place – fractions of a second – from the figure we have today of 365.242190, derived with the aid of radio telescopes and atomic clocks. And in a highly theatrical demonstration involving candles and globes, he is said to have proved to an audience that included the Sufi theologian al-Ghazali that the earth rotates on its axis.

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Trigonometry was first developed in ancient Greece, but it was in early Islam that it became an entire branch of mathematics, as it was aligned to astronomy in the service of faith. Astronomical trigonometry was used to help determine the qibla, the direction of the Ka’bah in Mecca. Modern historians such as David King have discovered that the Ka’bah itself is astronomically inclined. On one side it points towards Canopus, the brightest star in the southern sky. The axis that is perpendicular to its longest side points towards midsummer sunrise.

Mecca’s significance is such that when a deceased person is to be buried, contemporary Islamic tradition determines that his or her body must face Mecca. When the famous call to prayer is announced, it must be done facing Mecca. And when animals are slaughtered, slaughtermen must also turn in the direction of the holy city. Islamic-era astronomers began to compute the direction of Mecca from different cities from around the 9th century. One of the earliest known examples of the use of trigonometry (sines, cosines and tangents) for locating Mecca can be found in the work of the mathematician al-Battani, which, according to David King, was in use until the 19th century.

Numbers From India

One of al-Khwarizmi’s greatest contributions was to provide a comprehensive guide to the numbering system which originated in India about 500 CE. It is this system, later called the Arabic system because it came to Europe from al-Khwarizmi, that became the basis for our modern numbers. It was first introduced to the Arabic-speaking world by al-Kindi, but it was al-Khwarizmi who brought it into the mainstream with his book on Indian numerals, in which he describes the system clearly.

The system, as explained by al-Khwarizimi, uses only ten digits from 0 to 9 to give every single number from zero up to the biggest number imaginable. The value given to each digit varies simply according to its position. So the 1 in the number ‘100’ is 10 times the 1 in the number ‘10’ and 100 times the 1 in the number ‘1’. An absolutely crucial element of this system was the concept of zero.

This was a significant advance on previous numbering systems, which were often cumbersome with any large numbers. The Roman system, for instance, needs seven digits to give a number as small as, for example, 38: XXXVIII. Arabic numbering can give even very large numbers quite compactly. Seven digits in Arabic numerals can, of course, be anything up to 10 million. What’s more, by standardising units, Arabic numerals made multiplication, division and every other form of mathematical calculation simpler.

This system quickly caught on, and has since spread around the world to become a truly global ‘language’. Along with the numbers, English also gained another word, ‘algorithm’, for a logical step-by-step mathematical process, based on the spelling of al-Khwarizmi’s name in the Latin title of his book, Algoritmi de numero Indorum. The new numbers took some time to embed themselves in the Islamic world, however, as many people continued with their highly effective and fast method of finger-reckoning.

Numbers

In many areas of science, the contribution of early Islam is sometimes open to interpretation and shifts of opinion, but when it comes to numbers and mathematics the legacy is immense and indisputable. The very numbers in use in our world every day for everything from buying food to calculating the spin on an atomic particle are called Arabic numerals, because they came to the West from scholars who wrote in Arabic. What’s more, with al-Khwarizmi’s algebra, these scholars provided us with the single most important mathematical tool ever devised, and one that underpins every facet of science, as well as more everyday processes.

Prejudice

Names such as al-Khwarizmi and ibn al-Haitham are as integral to the history of science and technology as are Newton and Archimedes, James Watt and Henry Ford, but the Arabic-sounding names somehow became lost in the myth of the Dark Ages. The reasons for this are the subject of an intense debate, which is as much about the relationship between the West and Islam as it is about the history of science and technology.