Complex Numbers

The usual definition of complex numbers is all numbers of the form a+bi, where a and b are real numbers and i, the imaginary unit, is a number such that its square is -1. This gives no insight to where these came from nor why they were invented. In fact, the evolution of these numbers took about three hundred years.

In 1545 Jerome Cardan, an Italian mathematician, physician, gambler, and philosopher published a book called Ars Magna (The Great Art). In this he described an algebraic procedure for solving cubic and quartic equations. He also proposed a problem that dealt more with quadratics. He wrote:

If some one says to you, divide 10 into two parts, one of which multiplied into the other shall produce...40, it is evident that this case or question is impossible. Nevertheless, we shall solve it in this fashion.

Cardan essentially applied the method of completing the square to x + y = 10 and xy = 40 (x2 - 10x + 40 = 0) to get the numbers,

solutions to x<sup>2</sup> - 10x + 40 = 0

He multiplied these numbers and got 40. After that, he didn't do much else with this and concluded that the result was "as subtle as it is useless." Complex numbers did not come about from this example, but in connection with the solution to cubic equations.

The "cubic formula," usually known as Cardan's formula, gave solutions to the cubic x3 = ax + b and was given as

Cardan's cubic formula

An explanation of how this was derived can be found in "The Roots of Complex Numbers," Katz, Victor J., Math Horizons, Nov. 1995. When this was used to solve the classic example x3 = 15x + 4, the formula yields,
solution to cubic

When this happened, Cardan claimed the general formula wasn't applicable for this case (because of the square root of -121). More questions had to be answered. This cubic has real solutions,
solutions to cubic

The challenge of figuring this problem out was taken on by the hydraulic engineer Rafael Bombelli (1526-1572) almost thirty years after Cardan's work was published.

Bombelli justified the use of Cardan's formula by introducing complex numbers. He basically assumed that numbers of the form

a + (the square root of -b)

existed and applied the normal operational rules of algebra. In this case, Bombelli thought that since the radicands
2 radicands from above

differed only in sign, the same might be true of their cube roots. He set
setting radicands equal to quantities

and went about solving for a and b. He got a = 2 and b = 1, thus showing that
x, as defined above, shown to equal 4

as desired. The method gave him a correct solution to the equation and convinced Bombelli that there was some validity in his ideas about complex numbers.

Bombelli laid the groundwork for complex numbers. He went on to develop some rules for complex numbers. He also worked with examples involving addition and multiplication of complex numbers.

For years after Bombelli's work, many still thought complex numbers were a waste of time, but there were others who used complex numbers extensively and through their work, much more was discovered. Albert Girard suggested that an equation may have as many roots as its degree in 1620. René Descartes, who contributed the term "imaginary" for these numbers, said that even though one can imagine that every equation has as many roots as its degree, real numbers may not correspond to all of these imagined roots.

Gottfried Wilhelm von Leibniz (1646-1716) spent quite a bit of time trying to apply the laws of algebra to complex numbers. He and Johann Bernoulli used imaginary numbers as integration aids. Johann Lambert used complex numbers for map projection. Jean D'Alembert used them in hydrodynamics, while Euler, D'Alembert, and Lagrange used them in their incorrect proofs of the fundamental theorem of algebra, which states that every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. Also note, Euler was the first to give the square root of -1 the symbol i.

Johann Carl Friedrich Gauss published the first correct proof of the fundamental theorem of algebra in his doctoral thesis of 1797, but still claimed that "the true metaphysics of the square root of -1 is elusive" as late as 1825. By 1831 Gauss overcame some of his uncertainty about complex numbers and published his work on the geometric representation of complex numbers as points in the plane. In 1797, a Norwegian surveyor named Caspar Wessel and in 1806 a Swiss clerk named Jean Robert Argand had results similar to Gauss's, but these went unnoticed for the most part. William Rowan Hamilton expressed complex numbers as pairs of real numbers (a,b) in 1833.

Complex numbers continued to develop after this. Work on them led to the fundamental theorem of algebra and a branch of mathematics called complex function theory. They are aids in various branches of geometry and in solving certain Diophantine equations in number theory. They are used in quantum mechanics and electric circuitry. What many once believed impossible, ridiculous, and even fictitious, has become a reality.