History & Background

• 50 AD: While studying the volume of a section of a pyramid, Heron of Alexandria was led to the computation of √(81-114). He deemed this calculation as impossible and made no more efforts towards solving the problem. (History of Complex Numbers)


• 665 AD: Brahmagupta, who was an Indian native from Bhinmal, authored two early works on mathematics and astronomy. In his book Khandakhadyaka, quadratic equations were solved and the possibility of negative solutions was allowed. (Brahmagupta,2015)


• 1484: Nicolas Chuquet, a French mathematician who made significant progress in the realm of exponents and negative numbers, showed that some equations in fact lead to imaginary solutions but dismissed the idea. (Biggus, 2010).


• 1545: Around this time, imaginary numbers began to be recognized more heavily by society. Building from advancements made by Nicolo Fontana in 1535 where he found a general method for solving all cubic equations, Girolamo Cardano extended this idea to discover what is now known as Cardan's Formula. With this formula however, several examples were noted where one would have to take the square root of a negative number. Cardano acknowledged the existence of these imaginary numbers as solutions; nonetheless, he highly disliked the idea of working with imaginary numbers stating that such work is "as subtle as it is useless" as well as "mental torture". (History) (Biggus, 2010) (Nahin, 1988)


• 1572: Using a common technique known today as conjugation, Rafael Bombelli expanded upon a previous idea that one can use the square roots of negative numbers to find real solutions. While not highly accepted, Bombelli was a firm believer in what we now know as "complex numbers". (Biggus, 2010) (History)


• 1637: Rene Descartes, a renowned mathematician, came up with the term "imaginary" to represent expressions that require the square root of a negative number. This term was coined originally as a derogatory term since most people saw the use of these numbers as being fictitious or useless. He also created the standard form for a complex number as being (a + bi) with a being a real part and b being the imaginary part. While he made significant progress, he also disliked working with complex numbers and stated that equations with these qualities are impossible to solve. (Biggus, 2010).


• 1673: John Wallis made improvements helping the idea of complex numbers to be accepted. He was one of the first people to come up with a way to represent complex numbers geometrically. This representation uses a graph where the X axis represents real numbers and the Y axis represents imaginary numbers. Wallis had some important contributions to this idea of graphing, but this representation was not widely accepted until Caspar Wessel publishes the similar idea of graphing in a plane. (Biggus, 2010) (History)


• 1747: Beginning around this time, Leonhard Euler had some notable contributions that helped society begin to accept the use of imaginary numbers. Euler developed the symbol i to represent √-1 which helped people better understand the difficult concept of complex numbers. (Biggus, 2010)


• 1797: As mentioned previously, Caspar Wessel, who was a Danish Mathematician was ultimately credited for the geometrical interpretation of complex numbers. In a published paper, he was able to describe complex numbers as points in a two dimensional complex plane which is referred to today as the "Gaussian" plane. (Biggus, 2010)


• 1831: Carl Freidrich Gauss made some monumental advancements in the realm of complex numbers. He developed the term "complex" to represent Descarte's notation of (a+bi) as well as constructed the algebra for this set of numbers. Augustin Louis Cauchy also made progress with the algebra, but his work around this same time, extended mostly towards further developments in calculus. (Biggus, 2010)


• 1835: William Hamilton is credited for the advancement of complex numbers being represented as pairs. Even though Gauss originally discovered this idea, the results were never published which led to Hamilton being credited with this discovery as well as the geometric discovery of (x+iy) being represented with coordinates (x,y). (Biggus, 2010)

While the previous list is far from comprehensive, it contains some of the pivotal moments throughout history that motivated the development of imaginary numbers and extending them towards complex numbers. Apparent from the timeline, the development of complex numbers was a long process where mathematicians continued to build off of previously developed ideas. You will notice as you dive into the explanation of complex numbers that many of the same algebraic properties hold when dealing with these numbers. The discoveries of these properties and other qualities of complex numbers were determined over time since the development of complex numbers. Even in the current day, mathematicians are able to advance the concept of complex numbers to find more applications from this truly "complex" idea.