How are Complex Numbers used in Science and Engineering?
In science applications, complex numbers are used to combine two variables into the same equation to make the calculations easier to solve, while keeping the variables separated. In circuit analysis, voltage and current are both functions of time and complex numbers can represent the voltage and current in sinusoidal oscillation. Complex numbers convert a difficult feedback problem involving differential equations into a much more straightforward problem in geometry, within the Argand plane. The Argand plane is richer in its mathematics than the Cartesian plane (with both axes real) because it includes the operations between points as complex numbers. For example we can multiply two complex numbers in a meaningful way, and we can take the conjugate of a complex number. These extra properties allow using geometrical solutions to algebraic problems.
Instead of calculating voltage and current separately using their trigonometric identity, we can combine them into the same equation to simplify the calculation process. Mechanical vibration is similar to simple harmonic motion where all acceleration, velocity and displacement are function of time. Similar to the application in circuit analysis, we can combine those variables into the same equation using complex numbers. This is much more significant in complex mechanical systems where it is impossible to perform calculation by hand.
It is important to know the applications of complex numbers because it helps student find the value in learning this topic.
Here are a few Youtubes to illustrate the complex numbers part in those applications.
- matrices
- electronics with brief explanation (Christensen, n.d.)
- mechanical vibrations
Some students may find those examples interesting and want to research further, which we should encourage. However, it is not important for students to fully understand the complex numbers in those applications because they are highly specialised and the students are lacking the prior knowledge required to appreciate them.
The matrix video lays out the linkage to another step in our sequence of mathematical structures, but they are for another year, and now is not the moment to make another great leap of invention! (in a better world, perhaps a syllabus less obsessed with 17th and 18th century mechanics, we might have less calculus and more modern mathematics in mathematics advanced, maybe even a little something in stage 5)
The topic complex numbers is one of the few mathematical topics that require students to focus on the maths itself. The NSW syllabus also emphasises the connections the various forms of complex numbers, instead of dialing in to the applications in science. Before students are able to understand and appreciate the applications, students must completely understand the concept of complex numbers and the background knowledge of the application. It is like the sport rock climbing. It is unwise to bring a beginning climber outside to the cliff because it is extremely dangerous. A climber should be skillful at indoor climbing before heading outside. In order to climb the cliff, the climber also needs to be knowledgeable about the rock structure, the effect of the environment, the effect of weather condition and everything essential to the climb.
Introduction to Complex Number in Physics/Engineering (Crumley, 2004) provides a brief summary of some uses, as does Using Complex Numbers in Circuit Analysis and Review of the Algebra of Complex Numbers (Johnson, 2014).