Introduction into complex numbers
In school we have been taught that √(-1) has no solutions, but what if it did. In 1545 an italian mathematician called Gerolamo Cardanoin thought why don’t we just assign √(-1) a variable and this created our imaginary unit i. Imaginary numbers are just the product of the imaginary unit i with any real number b so it can be written as bi. This is similar to the real numbers as our real unit is just 1 instead of i.
Complex numbers are the combination of a real number and an imaginary number so it can be written as a+bi. Even though it doesn’t seem intuitive that something which doesn’t exist in real life can help, we use these types of concepts all the time. The quadratic formula uses irrational numbers (numbers that cannot be expressed as fractions) to sometimes get rational roots, showing that sometimes when we use complicated ideas such as irrational numbers or complex numbers we get a less complicated solution. Even within the cubic formula there are imaginary numbers being used; further proving this point.
This point about counterintuitive concepts being very useful has been adopted in many sciences such as engineering and physics. In engineering it has allowed us to have a greater understanding of circuits and signals. Within physics it has allowed us to analyse concepts such as damping far more quickly. By manipulating their properties we can use complex numbers to make creative solutions to problems that would be very difficult or take a long time to solve if we just used real numbers.
One of the distinct properties of complex numbers is that they can act like a vector. This idea allows us to graph a complex number as a vector on an imaginary axis against a real axis called an Argand diagram. This allows us to describe problems involving rotation using complex numbers due to their vector-like properties as we can see below.
Here we have a clear example of a complex number graphed on an Argand diagram. We have defined the distance from the origin to be |z| and the angle from the real axis to be , which allows us to express complex numbers in new ways. We can write any complex number as z=|z|(cos(α)+isin(α)) which is called mod-arg form. We know this using Pythagoras as |z|cos(α) = 4 and |z|sin(α) = 6i so when we add these components together it allows us to write it in our mod-arg form.
Euler, an 18th century Swiss mathematician, noticed something really special about the mod-arg form from looking at the Maclaurin series. The Maclaurin series are simply non-polynomials written using an infinite series of polynomials, and is what our calculators use when getting approximations for our sin functions. Here are some examples of non-polynomials as an infinite series of polynomials:
Euler realised that complex numbers can be written in its exponential form, which can be derived using our Maclaurin series:
As we can clearly see there are 2 parts to all complex numbers, the imaginary part and the real part. We can use this fact to do more manipulation by thinking of the real coefficient of the complex number to be cos(α) and the imaginary coefficient to be sin(α). To make use of this idea we use the Re(z) function, which is equal to the real coefficient of the complex number z and Im(z) which means the same thing but for the imaginary coefficient of z. The benefit of these new ways of writing functions is that it allows us to use the properties of exponential numbers enabling more mathematical operations. The whole point of complex numbers is having less limitations to make things easier.
Applications of Complex numbers
In Electrical Engineering:
The importance of complex numbers in electrical engineering is to do with the idea of phase. Phase is the distance shift of sinusoids, complex numbers allow us to add two functions with different phases together relatively easily. This is an important idea as components can produce voltages with different phases so we need to get the resulting voltage to understand the circuit.
Three-phase power which delivers electrical energy to houses uses 3 alternating currents with different phases. We use three different phases to smoothen the power transfer otherwise our lights will flicker at 50 Hz, which is visible to us and would be very annoying. This is why we have introduced three-phase power. To figure out the resulting power you have to add different phases together as the total power is the product of the resulting voltage and current.
Knowledge of how to add functions with different phases is also useful for adding capacitors to our circuits as they cause impedance, which shifts the voltage, causing phase difference. Capacitors are used for signal processing due to their property of causing impedance and are used in products such as speakers.
We can add functions with different phases by using our 3 forms of complex numbers, switching between them depending on which operation we want to do. Here is an example of how this can be done:
As we can see complex numbers have allowed us to use a more intuitive way of answering this type of question. It has allowed us to use relatively simple ideas such as factorisation and power laws to create an easy method of adding functions with different phases.
In Signal Analysis:
A lot of signal analysis requires complex numbers such as the Fourier transform. The first idea that is necessary to understand the Fourier transform is that any signal can be expressed by adding different sinusoids together.
As we can see adding two different sinusoids together can make a non-sinusoidal signal. With this knowledge the goal is to know what frequencies make up the signal as we can do much more to analyze and manipulate it. This knowledge allows us to do things such as sound editing so if there is an annoying high frequency sound, we can remove it using a specially designed filter.
The Fourier Transform is a tool for understanding the sinusoids which make up the signal and it is used to understand their coefficients. With the Fourier transform, we make an assumption that a given signal g(t) is made from a sum of sinusoids at an infinite range of frequencies.
G(ω) is a frequency spectrum of the signal g(t). You can plot a frequency spectrum with amplitude on the y-axis and omega on the x-axis and see the amplitude of every sinusoid that makes up the signal g(t). We say that the signal g(t) is in the time-domain, because it varies with time, and G(ω) is in the frequency-domain because it varies with frequency. For example, if we did a Fourier transform on a simple sinusoidal with ω=2π and an amplitude of 1, the frequency spectrum G(ω) would have a single spike of magnitude 1 when the x-axis is at 2π, and it would be zero for all other x values.
One reason that this is particularly useful is that we can use it to easily design filters. Often, there are specific frequencies in a signal that we want to get rid of. However, it would be very hard to come up with a time-domain filter that cancels out these frequencies, but we can use a clever trick to get around this.
We can first design a filter in the frequency domain by drawing a frequency spectrum that has a value of 1 for the frequencies that we want to keep, and zero for the frequencies that we want removed. We can then use the Inverse Fourier transform, which allows us to turn this frequency spectrum into a normal signal that we can use as a filter.
To get rid of the frequencies that we don’t want in our signal, we combine the time-domain filter that we created using the Inverse Fourier transform with the signal. This cancels out the frequencies that we don’t want in our signal, so now if we did a Fourier transform on the filtered signal we would see that the frequencies that we designed our filter to block would have an amplitude of zero.
So in summary, the Fourier transform and the Inverse Fourier transform allow us to easily transform signals from the time-domain to the frequency-domain and back again. This is particularly helpful for designing filters, as it is easy to switch between domains so we can actually use them. Complex numbers have allowed us to understand the properties of the sinusoids that make up our arbitrary irregular signal in an intuitive equation for us to use to understand the signals more deeply.
In Oscillating Springs:
Oscillations occur everywhere in our lives as all objects have a natural frequency of oscillation. The reason why glass breaks when playing a specific frequency is a result of glass having a natural resonant frequency. This idea is close to a concept called free vibrations which happens when an object is left to oscillate at its natural frequency; this means there are no external forces affecting the object. Using a mass attached to a spring we can see free vibrations occur if we just leave it to bounce. The mass keeps going up and down reaching the same maximum distance from the midpoint, this can be expressed on a graph with constant amplitude. It’s important to note that the greater the amplitude the more energy there is in the system.
Damping occurs when resistive forces cause the energy to be transferred to different areas (like heat and sound) so the energy dissipates- a principle used by shock absorbers in cars. When damping occurs the kinetic energy of the system decreases throughout each oscillation as it is being transferred to other types of energy.
Dampeners are essential for bridges to stay stable by countering the effect called resonance, which makes them less stable. Resonance happens when an external object’s frequency matches the natural frequency of the system causing energy to be transferred. For example, this occured when people walked across the Millenium Bridge after its opening and their walking frequency matched the natural frequency of the bridge. This allowed kinetic energy to easily transfer to the bridge, causing it to sway. If the amplitude of the swaying reaches a certain point it can cause the bridge to collapse. This is why dampeners are important as they dissipate this energy making the bridge far more safe.
The energy of the system relates to the amplitude therefore dampening causes the amplitude to decrease over time as energy is transferred. This is exactly what happens in a damping oscillator function as the amplitude decreases at an exponential rate as energy dissipates from the system.
It should be clear that here we have a sinusoid function with a decreasing amplitude so we can intuitively express mathematically by just scaling the sinusoid using an exponential:
Where a expresses how quickly the oscillation dampens, b determines the time between each oscillation, c and d determine the starting points of the function. The a and b values can easily be worked out with the use of complex numbers and the quadratic equation. The link between the oscillator function and the quadratic equation is a result of second order differential equations, which I won’t be going into for this article. For convenience I provide the result straight away:
Where m represents the mass in the system, r is the resistive force which dissipates the energy and k is the stiffness of the spring which is dissipating the energy. When solving this equation using the quadratic formula we get a complex number x = a+bi; the a and b values are the same ones as in the oscillator function allowing for an easy understanding of the rate at which the energy dissipates and speed of oscillation.
Oscillating springs are just another example of how complex numbers allow for a method of solving the problem by using quadratic, instead of a computationally difficult solution. What makes this method unique to the other examples is that even though it isn’t that intuitive it is a really fast method which allows us to get information without spending too long on the problem.
Throughout this article I hope I’ve convinced you that complex numbers have their uses within the real world, as a new way of thinking about problems. The main strength of complex numbers is that they create intuitive or quick solutions to problems that would otherwise be very difficult to solve. These benefits are very powerful as intuitive ways of solving problems goes a long way in terms of teaching as it allows more people to access it. Quick solutions allow people to not spend as long on tedious problems and allows us to get the answer in only a few steps.
We have seen how complex numbers are important in electrical engineering because they allow us to solve problem adding functions with different phases efficiently, to help understand certain circuits. In signal analysis it made explaining the Fourier transform easy to understand and intuitive. In Oscillating springs it made working out the properties of the damping function a quick task. These are only three of the many real-world applications of complex numbers, showcasing it’s unique versatility in science.
Even though you will never be able to measure a complex number in real life, it’s a concept as useful as real numbers. This fact isn’t unique, negative numbers are also something we cannot measure in real life but yet we use them constantly to describe certain relationships and complex numbers are used in a similar way.
The fact that complex numbers are used in so many different areas in science proves that it is a useful topic to pursue. I have found that complex numbers play a pivotal role in understanding the world around us as this allows us to look at problems from new perspectives, even if we can’t directly see complex numbers in real life.