The common definition of an imaginary number, or complex number, is the square root of a negative number. The concept of i is straightforward enough: since i is the square root of -1, then it logically follows that i squared must be equal to -1. Although the square root of a negative number is often considered undefined, the brilliance of complex numbers lies in the assumption that this square root is nonetheless meaningful. Since almost all of the basic properties of algebra or real numbers apply to complex numbers, they are merely an extension of the real number system. In a way, complex numbers are just as “real” as real numbers.
Complex numbers have had a rather intricate history in their naming. Originally termed imaginary numbers by Rene Descartes, the term complex number became more popular following Carl Friedrich Gauss’s work in the subject. Gauss was the first to represent a complex number as a point in a plane, where the real part would be located on the x-axis and the imaginary part would be located on the y-axis. Other great mathematicians, including Euler, Cauchy, and Riemann, would also contribute to the discovery of the properties and applications of this new number system.
Complex numbers were rejected by most of the mathematical community upon their discovery, as the general view was that these numbers lacked meaning and were therefore preposterous. However, Rafael Bombelli thwarted these claims (perhaps more by chance than intent, as he was not fully convinced of this new concept) when he showed a way to reach the solution of a depressed cubic equation by using complex numbers. A depressed cubic equation is simply a cubic equation without an x^2 term. For example, the depressed cubic equation that Bombelli solved through the use of complex numbers was x^3 – 15x - 4 = 0. Using the Ferro-Tartaglia formula, which gives a solution to a depressed cubic equation like the quadratic formula gives the solutions to a quadratic equation, Bombelli found that x was equal to a term containing the square root of -121. Previous mathematicians had ignored this result because it involved the square root of a negative number and considered this particular depressed cubic equation to be an exception to the general formula.
Bombelli wondered if perhaps this solution for x did have significance despite containing a seemingly absurd negative square root. Bombelli manipulated the term using accepted rules of algebra. By treating the square root of -1 as a separate identity, he was able to simplify the result for x from the cubed root of (2 + the square root of -121) + the cubed root of (2 – the square root of -121) into (2 + the square root of -1) + (2 – the square root of -1). This term is obviously equal to 4, which shocked the mathematical world because 4 was undoubtedly a solution to the depressed cubic equation, yet the solution had been derived through an unconventional method. Regardless of whether complex numbers had any tangible meaning, it could no longer be argued that they were not beneficial.
Following Bombelli’s achievements, the art of complex analysis has been refined and perfected. A complex number is traditionally written as z = x + iy, but it can also be written in polar form, where z = r * e^(i * theta), and r represents the distance between the point and the origin on the complex plane. This stems from Euler’s famous identity, which states that e^(i * theta) = cos(theta) + i * sin(theta).
Euler succeeded in revealing a connection between the exponential function and trigonometric functions in the complex domain, a connection that would have otherwise been unnoticed for centuries had mathematicians dealt solely with real numbers.
Once complex numbers are clearly defined, complex functions can also become significant. Using Gauss’s model that represents a complex number as a point in a plane, a complex function can be thought of as a function that takes a region on one complex plane as the domain and maps it onto the corresponding region on another complex plane.
Since complex functions exist, then it naturally follows that one can take derivatives and integrals of these functions. A complex function is said to be analytic if its derivative exists not only at a point, but also at all the points within a disk of radius r around that point. Essentially, a function is analytic if its derivative can be found at a point and at nearby points.
Similarly, the integral of a complex function is also defined. Since a complex number is a point on a plane rather than a point on a number line (as is the case for real numbers), a complex integral is the line integral over a contour in the complex plane, rather than simply over a segment of the number line. One of the highlights of complex integration is the Cauchy-Goursat theorem, which claims that if a complex function is analytic in a simply connected domain, and there is also a simply closed contour inside the domain, then the integral of the function over this contour is zero. The terms “simply connected” and “simply closed” are merely ways of saying that the domain and the contour both enclose a region and the boundary of that region does not overlap with itself. For example, a basic circle in the complex plane would constitute a simply closed contour.
The usefulness of this theorem lies in its extension, which states that if there are two contours both located in a domain, one contour is interior to the other, and the domain contains the entire region between the two contours, then the complex integrals of a function over these two contours are the same. It is obviously very advantageous to use this theorem in problem solving, because complex integrals over very complicated and irregularly shaped contours can, for example, be simplified into complex integrals over ordinary circles.
Using similar techniques, a branch of complex analysis called residue theory can be used to solve integrals of real functions over real domains that, without the aid of complex analysis, would be impossible to solve. Essentially, the real integral can be transformed into a solvable complex integral, which yields the same result that the real integral would have produced if there was a method to solve it, making the Cauchy-Goursat theorem extremely beneficial to mathematicians.
Additionally, complex analysis has many practical applications in the real world. Solutions to many physics and engineering problems can be found with the help of complex variables. For example, complex analysis is used to analyze fluid flow (flow of gases or liquids). A technique used regularly to design airplane wings, conformal mapping, allows a complicated problem such as the flow of air around an airplane wing to be converted into a much simpler problem, such the flow of air around a circle, which can be solved much more easily than the former. Complex analysis is also used in elliptic curve cryptography. An elliptic curve is a quadratic equation that has complex variables for x and y instead of the traditional real variables. In elliptic curve cryptography, conjugate points on the curve are used to encode and decode a message. This method is commonly used to secure websites and confidential information from public view. Complex analysis is used in electromagnetism to solve for the electric and magnetic fields in electrical systems as well. As a result, it it useful in designing microwave cavities that power microwave ovens, and it can also be utilized to design antennas for radios, televisions, and cell phones.
The complex number was once scoffed at by mathematicians and continues to remain an elusive concept for many. Yet it has thoroughly proved its value by becoming both essential in problem solving and intertwined with numerous technological advances.
Bibliography
Churchill, Ruel, James Brown, and Roger Verhey. Complex Variables and Applications. New York: McGraw-Hill Inc., 1976.
Mathews, John and Russell Howell. Complex Analysis for Mathematics and Engineering. Sudbury, MA: Jones and Bartlett Publishers, 2001.
Trim, Donald. Complex Analysis and Its Applications. Boston: PWS Publishing Company, 1996.
