The development of complex numbers as a mathematical entity
The complex number was invented simply because the imaginary number is needed. There were ideas of imaginary numbers since ancient Greek era, but history of mathematics considers Italian mathematician Gerolamo Cardano (1501-1576) as a creator of complex numbers. In 1545, Cardano wrote a book titled Ars Magna, where he showed imaginary roots of quadratic and cubic equations. The story is here
Cardan noticed something strange when he applied his formula to certain cubics. When solving x3 = 15x + 4 he obtained an expression involving √-121. Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation. He wrote to Tartaglia on 4 August 1539 in an attempt to clear up the difficulty. Tartaglia certainly did not understand. In Ars Magna Cardan gives a calculation with 'complex numbers' to solve a similar problem but he really did not understand his own calculation which he says is as subtle as it is useless.
The lively narrative, and reluctance to accept their own ideas, is told on www.storyofmathematics.com, a site well worth exploring and linking in many topics.
The solution is outlined here:
L’Algebra, published in 1572 by another Italian mathematician Rafael Bombelli introduces a notation for √-1 and calls it “piu di meno” (translation: ‘plus of minus’). He set simple arithmetic rules for imaginary numbers.
In 1637, French mathematician Rene Decartes (1596–1650) had very first idea of geometric existence of imaginary number and Cartesian form. Derived from Decartes’ imaginary number in geometry, John Wallis (1616-1703) gave very first geometric interpretation of complex numbers, based on a line with a zero mark, and positive numbers being numbers at a distance from the zero point to the right, where negative numbers are a distance to the left of zero.
In the 18th century, complex numbers were applied into trigonometry and exponential. In 1722, Abraham de Moivre (1667-1754) introduced De Moivre’s formula, and Leonhard Euler (1707-1783) introduced for symbol for √-1 and linked exponential and trigonometric functions in the Euler’s formula.
The modern geometric expression of complex number on a complex number plane, was given by Norwegian-Danish mathematician Caspar Wessel (1745-1818). Wessel treated complex numbers as vector forms. In 1806, Jean Robert Argand (1768-1822) introduced Argand diagram. In 1831, Carl Friedrich Gauss (1777-1855) made Argand’s idea popular and also formally introduced the term complex number and the standard notation a + bi for complex numbers. Gauss established modern ideas and notations of complex numbers and became notable expansion where complex numbers were implied into mathematics, technical and natural sciences.
Complex Numbers
a short summary (Orlando Merino University of Rhode Island, 2006).
Lokenath, D. (2015) A Brief History of the Most Remarkable Numbers "e," "i" and "π" in Mathematical Sciences with Applications, International Journal of Mathematical Education in Science and Technology, Vol.46(6), pp.853-878.