Algebra is the branch of mathematics where letters and other widely recognized symbols replace numbers in formulae and equations. In this way, algebra is able to establish general axioms (i.e. statements that are taken to be true). By then substituting the letters and symbols with numbers relevant to a particular problem, it becomes possible to use the axiom to solve specific sums. Whereas arithmetic is only possible where all the values are known, algebra is a way of solving problems in which some values may be unknown (such values being called variables).
Algebra works on the principle that both sides of an equation (i.e. the terms to the left and right of the ‘=’ sign) are balanced. Take, for instance, perhaps the most famous equation in all of mathematics and science: Einstein’s E=mc2. Here, the ‘E’ denotes kinetic energy, the ‘m’ stands for mass and the ‘c’ for the speed of light in a vacuum. The speed of light is a known constant (that is to say, it remains the same whatever the numbers for ‘E’ and ‘m’ are), so it thus becomes possible to find out the value of ‘E’ if we know what ‘m’ is, or vice versa, by a simple process of ‘balancing’ the equation.
The Persian mathematician Omar Khayyam established himself as a pioneer of the discipline with the Treatise on Demonstrations of Problems of Algebra (c. 1070), translations of which were key to the dissemination of algebra in Europe. By the 17th century, several Europeans were themselves extending the discipline – among them François Viète (who helped lay the groundwork for modern algebra) and René Descartes, who was responsible for introducing much modern algebraic notation. Seki Kwa in Japan and Gottfried Leibniz also made important contributions, while the birth of abstract algebra followed in the late 19th century. While algebra may remain the bane of many a school pupil’s day, its applications in the modern world are myriad – in fields as disparate as engineering, computing, medicine and financial services.
Ancient cultures and algebra
The ancient Babylonians are the first culture known to have used algebra, employing it to establish general arithmetical proofs, while the ancient Greeks developed a system of geometric algebra. Credit for first using the word algebra is believed to rest with the Persian mathematician Muhammad ibn Musa al-Khwarizmi – it being Arabic for ‘the reuniting of broken parts’. His Compendious Book on Calculation by Completion and Balancing, written around 820 CE, did much to establish algebra as a mathematical discipline distinct from geometry and arithmetic.