History of Complex Numbers

Abstract

The problem of complex numbers dates back to the 1st century, when Heron of Alexandria (about 75 AD) attempted to find the volume of a frustum of a pyramid, which required computing the square root of 81-144 (though negative numbers were not conceived in the Hellenistic world). We also have the following quotation from Bhaskara Acharya (working in 486 AD), a Hindu mathematician: “The square of a positive number, also that of a negative number, is positive: and the square root of a positive number is two-fold, positive and negative; there is no square root of a negative number, for a negative number is not square.” Later, around 850 AD, another Hindu mathematician, Mahavira Acharya, wrote: “As in the nature of things, a negative (quantity) is not a square (quantity), it has therefore no square root.” In 1545, the Italian mathematician, physician, gambler, and philosopher Girolamo Cardano (1501-76) published his Ars Magna (The Great Art), in which he described algebraicmethods for solving cubic and quartic equations. This book was a great event in mathematics. In fact, it was the first major achievement in algebra in 3000 years, after the Babylonians showed how to solve quadratic equations. Cardano also dealt with quadratics in his book.