The concept of imaginary numbers can seem like an especially esoteric or detached-from-reality notion, perhaps dreamt up in the overactive imaginations of mathematicians. The absurdity of the concept doesn’t improve when one learns that the origin of imaginary numbers comes from trying to take the square root of a negative number.
Positive numbers have a pleasing solidity to them, in that they correspond to countable “things.” Five apples, ten people, and even fractional amounts, such as 2.2 pounds of rice. The number zero becomes less concrete, referring merely to the absence of something — such as, zero cows in the barn. Although most people are more comfortable with negative numbers than imaginary numbers, when we encounter a number like negative five, we have already departed from a simple association of the number with tangible things.
The ancients rejected negative numbers as being without meaning because they could see no way physically to interpret a number that is less than nothing….As late as the sixteenth century we find mathematicians referring to the negative roots of an equation as fictitious or absurd or false.1
Nonetheless, we learn fairly early on in school how to do simple arithmetic with negative numbers. In particular, we learn that the square of any (real) number is positive. So, how could a negative number even have a square root? Surely, the label “imaginary” is appropriately applied to the result of any such endeavor.
Why Bring Up Imaginary Numbers?
I raise the issue simply because imaginary numbers have shown themselves to facilitate, despite their moniker, mathematics without which we would be seriously impeded in our ability to describe and quantify reality. To understand their nature, it will help to introduce complex numbers. If the square root of -1 is designated with the letter i, then a complex number, z, might be written as z = a+bi, where a and b are real numbers. So, a complex number has two parts, a real part, a, and an imaginary part, b.
We are accustomed to real numbers being laid out on a “number line,” typically with the positive numbers to the right of zero and the negative numbers to the left of zero. In this sense, real numbers are one-dimensional, with a number corresponding to every point on the line. Analogously, it can be useful to consider complex numbers as two-dimensional quantities, with a complex number corresponding to every point on a plane.
This is exactly how complex numbers are visualized — as points on a so-called complex plane. The familiar number line makes up the x-axis, or real axis, so any point on this line has its imaginary part b=0. Points in the plane above or below the real axis have a positive or negative imaginary part, respectively. Since complex numbers correspond to the points in a plane and the entire real number axis is just a single line in that plane, it makes sense that learning to work with complex numbers should lead to expanded functionality in mathematics.
And Such Is the Case!
What kind of practical benefit has come to us through the use of complex numbers?
When people first considered taking square roots of negative numbers, they felt very uneasy about the problem….They certainly would not have believed that the new numbers could be of any practical use. Yet complex numbers are of great importance in a variety of applied fields; the electrical engineer would, to say the least, be severely handicapped without them.2
Solving mathematical problems involving fluid flow, oscillatory motion, and quantum mechanics are all facilitated through theorems and procedures for handling functions of complex variables. At this point, someone might object to all this by asserting that the physical world corresponds to real things, so how could invoking imaginary numbers avoid a departure from reality? Remember, however, that the form of a complex number, z = a+bi, contains two real numbers, a and b, and it’s these real numbers that end up corresponding to actual properties of real-world phenomena.
An electrical engineer can analyze an alternating current circuit by assuming a complex-number form of the electrical current. The actual current corresponds to just the imaginary part, but the complex resistance, known as the impedance, has physical meaning in both its real (ohmic resistance) and imaginary parts (capacitive and inductive impedance).
The term, imaginary, for the part of a complex number that lies off the real number axis may contribute to our sense that mathematicians are dabbling in something out of a fairy tale. René Descartes, in 1637, is credited with being the first to assign this label to results involving the square root of a negative number.
Before Descartes’ introduction of this term, the square roots of negative numbers were called sophisticated or subtle.3
I think these earlier labels would’ve actually been more appropriate, considering the enormous benefit complex number theory has given us in mathematical descriptions and calculations of many aspects of the physical world. Exploring the possibilities found in the realm of complex numbers (perhaps not too dissimilar, after all, to an adventure in fairy land) we reach a particularly magical point known as Cauchy’s integral formula. This ultimately useful mathematical result comes from doing calculus with complex numbers.
Almost Insurmountable Difficulty
Anyone who has taken a calculus course learns the process of integration for finding the area under the curve described by a mathematical function. One also learns, however, that integrating certain functions poses almost insurmountable difficulty. Cauchy’s integral formula, however, melts the difficulty of doing a wide class of integrals by taking advantage of the two-dimensional number space of complex numbers.
When I’ve taught complex variable theory as part of an advanced mathematics course for physics majors, introducing this topic to students feels something like giving them a secret power to do math. The author of the book already referenced on imaginary numbers describes his first encounter with Cauchy’s formula while studying at Stanford University:
For me, complex function theory was a revelation bordering on a mystical experience….With Cauchy’s theory of complex integration one could almost without effort, calculate the values of a seemingly endless number of incredibly odd, strange, and downright wonderfully mysterious-looking definite real integrals….Such calculations were to me, then, seemingly possible only if one had the powers of a sorcerer.4
One example of a practical application of Cauchy’s formula for complex variables is in solving an integral from the equation of motion for a planet with an elliptical orbit, which leads to the famous result of Kepler’s Third Law of Planetary Motion. A completely different use of complex numbers arises in the research field of computational nanoelectronics. Analyzing electron transmission through nanoscale structures routinely includes considering information revealed by the imaginary part of the complex variable associated with the energy of the electrons. The following statement is from an article on currents circulating around a benzene ring molecular structure.
The positions of the transmission zeros and poles in the complex energy plane, and their possible interference or even complete cancellation of each other, are shown to correlate with the amplitude and resonance structure of the circular transmission resonances.5
What Does All This Have to Do with Intelligent Design?
I think that the discovery of the almost unbelievably practical applications of complex number theory — derived from the bold intellectual extension of real numbers into the two-dimensional realm — is so “wonderfully mysterious” that it seems more consistent with a buried treasure we were intended to find than a prodigious bit of unexpected luck encountered by accidental beings in a happenstance universe.
For centuries, mathematicians discounted the “imaginary” solutions of equations involving the square roots of negative numbers. But when these nonsensical outcomes of mathematical manipulations were tentatively embraced and explored, they revealed a hidden usefulness without which our quantitative formulations of reality would be severely curtailed. If the extension of our familiar one-dimensional number line to a two-dimensional complex number plane proved fruitful, who’s to say that a further extension of numbers to encompass a complete three-dimensional volume might not disclose further unimaginable treasures?6
- Paul J. Nahin, An Imaginary Tale: The Story of √(-1), (Princeton University Press, 1998), 5-6.
- Mary L. Boas, Mathematical Methods in the Physical Sciences, 3rd ed. (Hoboken, NJ: John Wiley & Sons, Inc., 2006), 47.
- Boas, Mathematical Methods in the Physical Sciences, (2006), 6.
- Nahin, An Imaginary Tale, (1998), 188.
- Eric R. Hedin, Arkady M. Satanin, and Yong S. Joe, “Circular transmission resonances and magnetic field effects in a ring of quantum dots connected to external leads in the meta-configuration,” Jnl. of Computational Electronics, June 2019, Volume 18, Issue 2, pp 648–659. DOI 10.1007/s10825-018-01291-2.
- A speculation on what a 3-dimensional number might correspond to is suggested by the mathematical impossibility of the square root of a negative number, which led to the two-dimensional plane of complex numbers. Another mathematical “impossibility” is the problematic case of division by zero, giving infinity. Perhaps the three-dimensional space above and below the complex number plane is populated by quantities (call them complete numbers) that correspond to complex numbers divided by zero. This hint is not without basis, as complex numbers that lead to infinities in real quantities (such as the transmission amplitude) have turned out to have practical significance (signifying the energy width of the transmission resonance). Furthermore, Cauchy’s formula, referred to above, actually exploits the existence of points in the complex plane that lead to infinites.