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.
