Alex's Adventures in Numberland - Alex Bellos [75]
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. In it, he explains two tich, iques for solving arithmetical problems. Al-Khwarizmi wrote out his problems rhetorically, but here, for ease of understanding, they are expressed in modern symbols and terminology.
Consider the equation A = B – C.
Al-Khwarizmi described al-jabr, or restoration, as the process by which the equation becomes A + C = B. In other words, a negative term can be made positive by resetting it on the other side of the equal sign.
Now, consider the equation A = B + C.
Reduction is the process that turns the equation into A – C = B.
Thanks to modern notation, we can now see that both restoration and reduction are examples of the general rule that whatever you do to one side in an equation, you must do to the other as well. In the first equation we added C to both sides. In the second equation we subtracted C from both sides. Because by definition the expressions on either side of an equation are equal, they must continue to be equal when another term is simultaneously added to or subtracted from either side. It follows that if we multiply one side by an amount, we must multiply the other by the same amount, and the same applies for division and other operations.
The equals sign is like a picket fence separating the gardens of two very competitive families. Whatever the Joneses do to their garden, the Smiths next door will do exactly the same.
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.
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