Short answers, pulled from the story.
Gerolamo Cardano published the book Ars Magna in the year 1545, which contained the first known solution to cubic equations. This publication introduced the concept of complex numbers through the square root of negative numbers.
René Descartes coined the symbol i, known as the imaginary unit, in 1637 to distinguish these numbers from real ones. Descartes used the term imaginary to stress their unreal nature, believing they existed only in the mind of the mathematician.
Caspar Wessel published his memoir in the Proceedings of the Copenhagen Academy in 1799. This work proposed that complex numbers could be represented as points in a plane, though it went largely unnoticed at the time.
The defining property of i is that it satisfies the equation i squared equals negative one. This property allows every complex number to be expressed in the form a plus bi, where a and b are real numbers.
Carl Friedrich Gauss published his proof of the fundamental theorem of algebra in 1799. This theorem states that every non-constant polynomial equation with real or complex coefficients has at least one complex solution.
Carl Friedrich Gauss published his treatise on complex numbers as points in the plane in 1831. This work largely established modern terminology and overcame his initial doubts about the true metaphysics of the square root of negative one.