Certain numbers have long intrigued and fascinated mathematicians in equal measure. Imaginary numbers rank high on the list of these. Together with real or “normal” numbers, they form what is known as complex numbers, those that consists of a real part and an imaginary part. For instance, 4 + 2i is a complex number with a real part equal to 4 and an imaginary part equal to 2i where i is the imaginary number expressed as the square root of minus 1. It turns out that both real numbers and imaginary numbers are also complex numbers. For example, 17 is a complex number with a real part equal to 17 and an imaginary part equal to zero, and i is a complex number with a real part of zero.
Mathematicians have grappled with imaginary numbers as early as the 16th century. They were first named “imaginary” by the French philosopher and mathematician, Rene Descartes (1596-1650) to distinguish them from real numbers, the workhorse of mathematics. Significant progress in understanding complex numbers came in the 18th century when the French mathematician Abraham de Moivre (1667-1754) showed that complex numbers were intrinsically related to trigonometry (and thus geometry). A century later, another French mathematician, who was also a bookshop manager by the name of Jean-Robert Argand (1768-1822) developed the Argand diagram, which proved indispensable for visualizing numbers in the complex plane. Argand diagrams are like normal graphs with an x and y axis, except the axes are real and the imaginary numbers. Together, these breakthroughs paved the way for complex algebraic problems to be solved using geometry.
Like so many developments in mathematics, all of this was of purely academic interest until the modern electronic age. It turns out that complex numbers are incredibly useful in analyzing anything that comes in waves (numerically represented by sines and cosines). We find such wave representations in acoustics, antenna theory, electrical circuits, optics, seismology, hydraulics and so on. So ubiquitous are imaginary numbers in the modern world that without them, there would be no internet nor wifi to speak of.
Semiconductors and cell phones, too, rely on imaginary numbers for their operations because at the atomic level, semiconductor chips are subject to the laws of quantum physics which reduces all particles to waveforms. Complex number analysis is instrumental in understanding how the strange world of the quantum can be manipulated to deliver such modern conveniences like computers, fiber-optics, GPS, MRI imaging, to name a few.
As we speak, complex numbers are making their appearance even more forcefully in quantum computing. This is because such numbers are used to represent the state of qubits and the operations that are performed on them (a qubit is a two-state quantum system that can exist in a superposition of two states simultaneously). By manipulating the probability amplitudes of qubits using quantum gates, scientists can develop quantum algorithms to solve highly complex calculations with greater speed than classical algorithms – yet another proof that the square root of minus one is imaginary in all but name.