The origin of complex numbers dates back to the sixteenth century, were a number of Italian mathematicians played a significant role leading up to their discovery. The first of whom was Gerolamo Cardano (1501-1576), an Italian mathematician, physician and philosopher who published a book called Ars Magna (The Great Art) in 1545. This book contained algebraic solutions to both cubic and quartic equations along with the first recorded occurrence of complex numbers, although Cardano did not know this at the time. Cardano acknowledges in Ars Magna that it was another mathematician, Niccolo Fontana (also known as Tartaglia) who had given him the solution to solve cubics of the form x^3+ax=b (with a,b >0) and his student Lodovico Ferrari who found a way of solving quartic equations, though it depended heavily on Fontana’s solution. Cardano had spent years extending Fontana’s formula to work with other types of cubic equations, it was through this work that he discovered some solutions contained square roots of negative numbers [2]. Although he would actively try to avoid working with negative numbers because they were not fully understood, there were still a number of cases that he proceeded with anyway.
It was in chapter 37 that Cardano proposed a problem, he said;
“If someone says to you, divide 10 into two parts, one of which multiplied into the other shall produce 40, it is evident that this question is impossible. Nevertheless, we shall solve it in this fashion.” ([1], par. 6)
He found the solutions to be 5+√(-15) and 5-√(-15). Though Cardano’s work would have been described as geometric algebra and at this time negative numbers were still treated with some suspicion. This is due to it being very difficult to conceive of any physical reality corresponding to them. Cardano called the solutions “sophistic”, as he saw no meaning in them and regarded them as useless.
The next step leading to the discovery of complex numbers was made by another Italian mathematician, Rafael Bombelli (1526-1572) almost 30 years after Ars Magna was published. He worked with the formula that Cardano had published in his book, known as the Cardano-Tartaglia formula. It was one solution to the depressed cubic equation x^3+ax+b=0, given as;
x=∛(b/2+√(b^2/4+a^3/27)) +∛(b/2-√(b^2/4+a^3/27))
An explanation of how the Cardano-Tartaglia formula is derived can be found in ([8], p.12-13.)
Bombelli considered cases when b^2/4+a^3/27 ∛(2+√(-121)) =a+√(-b)
from which he deduces,
∛(2-√(-121)) =a-√(-b)
and obtains, after algebraic manipulations, a=2 and b=1. Thus
x=a+√(-b)+a-√(-b)=2a=4
as desired. This gave Bombelli the correct solution and convinced him there was some validity in his ideas about complex expressions written in the form a+√(-b).
Bombelli then went on to lay the groundwork for complex numbers as he developed rules of multiplication and addition. He also introduced some early notation, he used ptm (plus than
It was Leonhard Euler (1707-1783) in 1777 who first introduced the notation i=√(-1), which retained the basic property, i^2=-1. The symbol i is sometimes called the imaginary unit because it was still not fully understood at the time. ([6], pp.17). With him, the notation a+bi originated for complex numbers which is still being used today. Euler worked with complex numbers extensively and in 1748, he created a mathematical formula known as Euler’s formula, which established the fundamental relationship between trigonometric functions and the complex exponential function. It states for any real number x;
e^ix=cos(x)+isin(x)
It is ubiquitous in mathematics, physics and engineering and was remarked as being “our jewel” and “the most remarkable formula in mathematics” by physicist Richard Feynman [7]. This helped establish the cogency of complex numbers in the eyes of other mathematicians at this time.
Then in 1797, Caspar Wessel (1745-1818), a Norwegian-Danish mathematician released a fundamental paper on complex numbers and the complex plane. Though it had been released in Danish and was rarely read outside of Denmark, so it went unnoticed for nearly a century until a French translation of the paper was released in 1897. In his paper he had treated complex numbers as vectors and correctly observed that to accommodate complex numbers, one would have to abandon the two directional real line. It was here that Wessel introduced the complex plane.
The complex plane is a geometric representation of the complex numbers established by the real axis and the orthogonal axis, known as the imaginary axis, as seen in figure 1.1. Wessel also derived most properties of complex numbers with the addition of complex numbers being very similar to the addition of vectors and also multiplication in the trigonometric form.
Figure 1.1. Diagram showing the complex plane [10].
Johann Carl Friedrich Gauss (1777-1855), a German mathematician with great influence within the mathematical community gave a tremendous boost to the acceptance of complex numbers through his work in 1831. Initially Gauss had actually doubted the usefulness of complex numbers as he had been quoted saying, “the square root of negative one is illusive”. Then a few years later he overcame his doubts with the application of complex numbers to number theory. He was also responsible for truly popularizing the practice of using the complex plane as a graphical representation of complex numbers and the use of the notation a+bi.
Finally in 1833, William Rowan Hamilton (1805-1865) introduced a more abstract formalism for the complex numbers with the notation (x,y) as he was looking for ways to extend complex numbers to higher spatial dimensions. He failed to find a useful three dimensional system but in working with four dimensions he created quaternions ([8], pp.12-13).
It took almost 200 years for complex numbers to be understood and widely accepted within the mathematical community. Their evolution has continued but it was the work of these mathematicians that pioneered their existence and made valuable contributions to the understanding of such numbers. Presently complex numbers have an array of modern applications including; electromagnetism, quantum mechanics and fluid dynamics ([3], section 9.3-7).