A Short History of Complex Numbers

This is a compilation of historical information from various sources, about the number i = √ −1. The information has been put together for students of Complex Analysis who are curious about the origins of the subject, since most books on Complex Variables have no historical information (one exception is Visual Complex Analysis, by T. Needham). A fact that is surprising to many (at least to me!) is that complex numbers arose from the need to solve cubic equations, and not (as it is commonly believed) quadratic equations. These notes track the development of complex numbers in history, and give evidence that supports the above statement. 1. Al-Khwarizmi (780-850) in his Algebra has solution to quadratic equations of various types. Solutions agree with is learned today at school, restricted to positive solutions [9] Proofs are geometric based. Sources seem to be greek and hindu mathematics. According to G. J. Toomer, quoted by Van der Waerden, Under the caliph al-Ma’mun (reigned 813-833) al-Khwarizmi became a member of the “House of Wisdom” (Dar al-Hikma), a kind of academy of scientists set up at Baghdad, probably by Caliph Harun al-Rashid, but owing its preeminence to the interest of al-Ma’mun, a great patron of learning and scientific investigation. It was for al-Ma’mun that Al-Khwarizmi composed his astronomical treatise, and his Algebra also is dedicated to that ruler 2. The methods of algebra known to the arabs were introduced in Italy by the Latin translation of the algebra of al-Khwarizmi by Gerard of Cremona (1114-1187), and by the work of Leonardo da Pisa (Fibonacci)(1170-1250). About 1225, when Frederick II held court in Sicily, Leonardo da Pisa was presented to the emperor. A local mathematician posed several problems, all of which were solved by Leonardo. One of the problems was the solution of the equation x + 2x + 10x = 20 3. The general cubic equation x + ax + bx+ c = 0 can be reduced to the simpler form x + px+ q = 0