The Growing Use of Complex Numbers in Mathematics

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The Growing Use of Complex Numbers in Mathematics

Overview

Complex numbers are those numbers containing a term that is the square root of negative one. Initially viewed as impossible to solve, complex numbers were eventually shown to have deep significance and profound importance to our understanding of physics, particularly those parts of physics involving electricity and magnetism. Complex numbers were finally recognized as a legitimate branch of mathematics in the last years of the eighteenth century, and they have been regarded as indispensable to some fields since the early nineteenth century.

Background

In 1850 b.c. an unknown Egyptian mathematician wrote on a papyrus the formula for calculating the volume of a truncated cone, called a frustum [V = ¹/₃*H*(a² + ab + b²)]. Interestingly, the formula is identical to the one used today, although deriving and solving this formula is usually thought to require mathematics far more advanced than existed at that time. In later years, about a.d. 100, Hero of Alexandria (fl. a.d. 62) investigated this equation further. At some point in his calculations, he appears to have purposely overlooked the fact that, in an example he wrote down, he had to calculate a value for the square root of a negative number. In fact, his manuscript shows that the negative number was written as a positive number, making extraction of the square root possible by mathematics as it existed at that time. Hero made virtually no errors in his other work; it is nearly certain that he purposely chose not to solve this problem, possibly because negative numbers had no meaning to him, let alone the square root of such numbers. So Hero missed the historic opportunity to "discover" the square root of a negative number.

Negative numbers reared their heads repeatedly in subsequent centuries, continuing to bedevil mathematicians and students because their very existence seemed counter-intuitive. One could count stones, for example, but how could one count negative stones? It made no sense to have a number representing quantities that were less than nothing. Carrying this absurdity further, how could one then perform mathematical manipulations with such a number? Or, more precisely, how could one possibly propose to extract the square root of a negative number? These questions perplexed mathematicians for over 1,000 years after Hero's first-century manuscript. For the most part, mathematicians were simply not interested, so they tried to find some way to either avoid or ignore working with negative numbers.

One example of this is the work of the brilliant Italian mathematician Scipione del Ferro (1465-1526). Until del Ferro, it was widely thought that cubic equations (that is, equations in which one of the terms is a cube, or a variable raised to the third power) were impossible to solve. Del Ferro was able to show that a certain class of cubic equations could be solved, the "depressed cubics" in which one of the terms (the second degree, or squared term) was missing. Depressed cubics could be solved, but to do so threatened to generate both real and imaginary roots. The real roots were the ones corresponding to numbers that del Ferro understood and that, to him, had physical significance. However, in his solutions to problems of depressed cubics, del Ferro simply disregarded all except the real roots.

The next advances in solving cubic equations was made by Italian mathematician Girolamo Cardano (1501-1576), who developed a correct, but roundabout, method of extracting the roots of cubic equations. He didn't understand why his method worked; this was explained by Rafael Bombelli (1526-1572), but neither man yet understood the meaning or the reason for the numerous negative numbers and their square roots that showed up in these calculations. A century later, Gottfried Leibniz (1646-1716) arrived at some of the same conclusions as Bombelli and, not knowing of Bombelli's priority, thought he had come up with the thoughts first.

The first person to make a mathematically sophisticated effort to understand the significance of the imaginary roots was the Englishman John Wallis (1616-1703). Unfortunately, while Wallis earned a fine reputation in mathematics, his contributions towards a deeper understanding of what the square root of -1 actually meant were not terribly significant. However, Wallis did point out that negative numbers can have a physical meaning—they are simply the numbers that lie to the left of zero (in the negative direction) on a line of numbers (often called the number line). This explanation, while seemingly trivial today, was a great leap forward at the time because, previously, this explanation was simply not considered. This also suggested that, since negative numbers could now be visualized, perhaps their square roots could be, too.

During this time, too, some examples of problems requiring imaginary roots began to make themselves known. For example, if one person moving at a constant velocity is chasing a ball rolling downhill, it is not difficult to set up an equation to find out when the person will catch the ball. However, if the person is moving too slowly to do so, this equation will have an imaginary root; that is, a root expressed in terms of √(-1). In a plot, such an equation might look like a parabola that doesn't quite reach the x-axis. Since the roots of an equation are those values that make the equation equal zero (i.e., where the plot intersects the x-axis), such an equation cannot have real roots and can only be solved using imaginary numbers.

The term "imaginary" to describe numbers that are the square roots of negative numbers comes from the great mathematician Leonhard Euler (1707-1783), who wrote "All such expressions as √(-1)...are consequently impossible or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible."

Another term often heard in this branch of mathematics is complex numbers. A complex number is a number that contains both a real number and an imaginary one. The real part of the complex number might be zero, leaving just the imaginary part, or it can be any real number. Complex numbers are written to show both the "real" and "imaginary" parts. An example of this would be the complex number 2+3i, in which 2 is the real part and 3i is the imaginary part of the complex number.

The person who finally helped make sense of this number was a Norwegian surveyor named Caspar Wessel (1745-1818). In a paper presented to the Royal Danish Academy of Sciences in 1797, Wessel proposed plotting complex numbers on a two-dimensional graph with the real part of the number along the x-axis and the complex part along the y-axis. This turned out to be a practical solution to the vexing question of trying to visualize something that was thought to have no physical meaning. By doing this, Wessel also led the way to developing techniques to actually use complex numbers, most typically in various branches of physics.

Impact

Wessel's paper generated little excitement outside of Denmark because Danish scientific journals received little attention. Nearly 100 years later, in 1895, Wessel's paper was rediscovered, and he received the credit he was due for this momentous interpretation of the meaning of imaginary and complex numbers and the manner in which they could be represented graphically. In the interim, others made the same discoveries, putting complex numbers to good use.

It turns out that complex numbers are a particularly useful means of mathematically handling operations such as multiplying vectors and angles. Because of this, they are ideally suited for describing the physics of alternating electrical current and AC circuitry, in which the voltage varies cyclically between positive and negative values, following a sine wave pattern. In fact, many physical phenomena behave in a similar manner. This means that, over time, the values for, say, electrical current voltage and direction cycle through the same values repeatedly, and, plotted in what is called polar coordinates, this forms a circle that repeats endlessly. At any instant in time, then, the AC current sine wave will have a specific voltage, whether positive or negative. These coordinate—voltage and time—can be described using complex numbers just as easily as by using conventional numbers. Drawing a line from the origin to a point representing the voltage at a particular time gives a vector, a line that represents a quantity with both magnitude (say, 25 volts) and a direction (say, 30 degrees up from the x-axis in the positive direction).

Of course, we can go through this exercise with real numbers, too, so the utility of complex numbers seems elusive. And, for simply describing quantities in this manner, there is not a tremendous amount of utility to complex numbers. However, when it comes time to add or multiply vectors, they become enormously useful, and complex numbers greatly simplify vector mathematics. This simplification is not just noticed in electrical circuits, either. Many physical phenomena are cyclic in nature, from the rotation of the Earth to the oscillation of pendulums to the orbits traced by satellites to the beating of our hearts. All of these phenomena are described easily and elegantly using complex analysis, giving greater insight to physicists and engineers into the workings of our world and helping them to design more useful machines.

P. ANDREW KARAM