Complex Numbers and Applications

The study of resolving polynomials have long since been prominent in the ages of old Father Time. The Babylonians (about 400 BC) are often credited as the first solvers of the quadratic equation, and we are very familiar with the ways of resolving the roots through various methods that we have been educated with. In spite of this, the ancients have failed valiantly at an attempt to find solutions of polynomials of higher degrees, such as cubics and quartics. Although we now know that there are no general equation for solving polynomials with a degree bigger than 5 thanks to the Abel-Ruffini Theorem and Galois theory, general equations does exist for cubics and quartics, as proven and demonstrated by Gerolamo Cardano in 1545. These innovations were only made possible by the study of complex numbers, which are still regarded as one of the most important human inventions to date.

We may first start by considering the following equation $x^2+1=0$ . By rearranging the equation, we obtain the following answer for $x$ ( $x = \pm \sqrt{-1}$ ). However, such an answer is impossible in the real world as there exists no such number which forms a negative number when multiplied by itself. However, we might coin a term ‘imaginary’ as the solution, namely , where . This solution to the equation was originally disputed by many mathematicians in the 17th century, with René Descartes coining it ‘imaginary’ as a derogatory term and regarded it as useless and fictitious. However, the term soon gained significant popularity and wide acceptance following the work of Leonhard Euler and Carl Friedrich Gauss. In fact, it was Rafael Bombelli, an Italian mathematician who fully tackled this problem and invented the notation of $i$ and $-i$ which we still use today.

The first recorded appearance of complex numbers was around 60 AD, when the Greek mathematician and engineer Hero of Alexandria presented a calculation involving the square root of a negative number. However, it was Cardano who started popularising the concept of such a number through the infamous Cardano’s formula for cubic equations.

The formula is as follows:

$\forall a \in \mathbb R_{\ne 0},\text{ } \forall b,c,d \in \mathbb{R}$ The equation $ax^3+bx^2+cx+d=0$ will have solution :

Monstrous, I know.

This is Cardano’s formula for solving cubics. Take the equation $x^3 = 15x + 4$ as an example. The formula equates to $x=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}$ .

However, as we might observe, there are negative numbers inside the square root. However, if we calculate it using $i$ and convert it to a polar coordinate (this will be explained below), we obtain the value of $x=4$ and $-2\pm\sqrt3$ respectively after a hefty calculation utilising de Moivre’s formula. The reason why one equation is able to return 3 results is out of scope of this paper, however it can be calculated through Galois theory, which states that solutions of a polynomial must lie equidistance apart in a unit circle, and this fact can be manipulated to locate all solutions of an equation with one equation only. This act of complex numbers returning real solutions much fascinated mathematicians, and thus the pursue of complex roots of equations commenced.

Complex roots are also crucial to completely factorising a polynomial. Take the polynomial $x^2-6x+25=0$ as an example. By using the quadratic equation, we obtain the value of $x = \frac{6\pm\sqrt{36-4(25)}}{2}\Longrightarrow{x=3\pm\sqrt{-16}}\Longrightarrow{x=3\pm4i}$ .

The function can now be completely factorised as $(x-3+4i)(x-3-4i)$ . Such factorisation methods are much exploited in financial calculators, where interest calculations are needed to output a solution quickly. By factorising equations, less computational power is required, and this reduces the amount of time needed. It is also important in calculus, where through factorising more complex integration problems can be solved quickly along with the help of employing partial fractions. It is also worth noting that for any polynomial with all real coefficients, complex roots must appear in conjugates. This can speed up the factorisation of polynomials greatly – if one root is known, the other is also obtained.

Unit circles utilising complex numbers

Complex numbers also appears in the famous Euler’s equation $e^{\pi{i}}+1=0$ . This relates to the study of unit circles, which can be used to convert Cartesian coordinates to polar coordinates and vice versa. Euler’s more generalised formula, $e^{ix}=\cos(x)+i\sin(x)$ (plug $\pi$ into the formula to obtain the equation), when combined with de Moivre’s formula, is incredibly powerful at simplifying and locating roots of equations, especially of imaginary roots where approximation methods such as Newton-Raphson iteration may not be suitable. Polar coordinates are also incredibly useful in daily life, as applied in navigation systems as the study of circular and orbital motion.

Complex analysis is a predominant topic in the study of complex numbers, and links topics seemingly unrelated such as number theory and analytic combinatorics together. It heavily employs complex numbers to create modelling functions of real-life phenomenons such as periodic motions (water, light waves, etc.) and particle movement. Then, through complex analysis, we can solve the equation and predict the movement of the wave or the particle. These give insight into studies of fluid mechanics, particularly the Navier-Stokes equation, one of the seven unsolved Millennial Problems. These models are of enormous use in applied maths and physics, as they give insight into how to pump oil in oil rigs, how earthquakes shake buildings and when tides might arrive, and Contour integration, a direct cause of complex analysis, is also heavily featured in the study of inertial navigation and is the fundamental backbone of the Nyquist stability criterion, which is a graphical technique for determining the stability of a dynamical system. This is massively employed in electronics, where it is crucial to understand whether a system is stable or not.

Complex analysis is also the foundation of another Millennial Problem, the Riemann hypothesis. This is a problem concerning the non-trivial zeros of the Euler-Riemann zeta function, which is defined as follows :

$\zeta(z)=\sum^\infty_{n=1}\frac1{n^z}$

For example, if we input 2 into the Riemann zeta function, we would obtain $\zeta(2)=\frac{\pi^2}6$ . This is the solution of the famous Basel problem. However, the function is only defined for all $z > 1$ , as the harmonic series does not converge when summed to infinity. In spite of this, complex analysis can extend the domain of the function towards negative infinity, and we can obtain the generalised Riemann function instead. This allows for the Riemann hypothesis, which states that all non-trivial zeros (trivial zeros are defined as the negative even integers which return zero when inputted) have complex value $\frac12+it, \text{ }t \in \mathbb{R}$ .

This is closely linked to number theory and the twin prime theorem, where it is stated in the Heath-Brown theorem if the conjecture was proven true, there must be infinitely many twin primes due to the non-existence of Siegel zeros¹.

The study of complex numbers are also fundamental to understanding motions of particles and occurs quite naturally in the study of quantum physics. The wave function, usually denoted by 𝚿, assigns a complex value to each point $x$ at each time $t$ . Then, through utilising the Born rule, named after physicist Max Born, we are able to take square roots of these complex numbers and obtain the probability density function of the particle. The well-known 1-dimensional Schrödinger equation also relies on complex numbers as shown below :

$-\frac{\hbar^2}{2m} \frac{\partial^2 \Psi}{\partial x^2} + V(x, t) \Psi = i\hbar \frac{\partial \Psi}{\partial t}$

The equation relates the collection of probability amplitudes that pertain to one moment of time to the collection of probability amplitudes that pertain to another. This allows for a calculation of a pathway the particle in question might take, which grants better understanding of particles overall. It also led to the famous Heisenberg uncertainty principle, and helped physicists understand how things work at a subatomic scale, completely departing from classical mechanics of which we are so used to.

Fourier Transform (FT) is another field of which complex numbers are required heavily. FTs uses integration to find the centre, or rather an average mean of all points in a complex plane, and is able to separate a superposition of frequencies into a graph which shows obvious peaks when the numbers match one of the original frequencies.

$\hat{g}(f)=\int^{t_2}_{t_1}g(t)e^{-2\pi{ift}}dt$

The Fourier Transform integral

FFT (Fast Fourier Transform) is an extremely powerful tool built on Discrete Fourier Transforms (DFT) and much used in forensics and computer programs to identify individual frequencies of a certain recording, but is much faster than the original FT integral, due to the easy adaptability of the function to create an inverse of itself. In sound editing, the program can automatically recognise frequencies of notes played and return the notes played. This allows for editing tones by increasing or decreasing frequencies, and notes can be easily added or removed by applying a spike at the corresponding frequency and applying the Inverse Fourier Transform (IFT) to the function.

JPEG image compression also makes use of FFTs to compress images. It works by distorting some of the highest frequencies in the image which the human eye is less sensitive to.

Suppose we start with a grayscale image, which is represented as a matrix where each entry is an integer between 0 (black) and 255 (white). The encoding process works by taking the FT of the JPEG image, and the matrix will reduce greatly in size due to the non-zero entries being concentrated in the top left of the matrix. This allows for an easier compression of images as zero entries can be compressed easily.

A graph of how Discrete FTs work, however periodic summation is used here instead of integration (Credit : Wikipedia)

FTs are also utilised in large-integer multiplication algorithms and polynomial multiplication. If we wanted to multiply two large integers A and B of size N, we could first transform them into their polynomial coefficient representation on base $x$ . We store the resulting coefficients into vectors a and b, respectively. According to the convolution theorem, if c is the convolution of two input vectors a and b, $c=a\cdot b$ , then the DFT of c is equal to the pairwise multiplication of the DFT transform of each input vector, $\text{DFT}(c)=\text{DFT}(a)\times \text{DFT}(b)$ . This result is applied in deep-learning algorithms, where quick multiplication of large integers are required.

Additionally, the analysis of electrical circuits makes use of complex numbers to describe sinusoidal current and voltages, and the mathematics of complex numbers can be applied to AC currents and voltages. To simplify the process of adding together voltages of AC currents for example, we might resort to describing it in terms of a complex function. Since complex numbers can be represented in terms of sine and cosine functions, by only taking the real value of the function, we are able to obtain the cosine output easily and simplify calculations as a result. For example, sinusodial currents must be converted to polar form in order to carry out additions of circuits. An addition voltages in phasor form, for example the addition of $V_1=120\text{ }\angle30^{\circ}$ and $V_2=160\text{ }\angle50^{\circ}$ , one must convert it to rectangular form, namely $V_1=60\sqrt{3}+60j$ and $V_2=103+122.6j$ .² Addition can hence be carried out and a result can hence be obtained by converting the result back to polar form, namely $V = 216\text{ }\angle41.4^{\circ}$ . This can be further extended to the calculation of electrical reactance, a measure of resistance of a change of current, where a complex value of voltage and current is supplied in a calculation of AC circuits.

This can be further extended to the calculation of electrical reactance, a measure of resistance of a change of current, where a complex value of voltage and current is supplied in a calculation of AC circuits.

$-jX_c = -j{\frac{1}{\omega{C}}}=-j\frac{1}{2\pi{fC}}$

The formula for Electrical Reactance

In conclusion, the journey of complex numbers from their humble origins to their profound impact on the modern world is a testament to the power of human ingenuity and the timeless pursuit of understanding the intricacies of mathematics. The invention of complex numbers by mathematicians like Cardano, Bombelli, and Euler marked a pivotal moment in the history of mathematics, as it expanded our understanding of numbers beyond the constraints of real quantities. Through their evolution, complex numbers have transcended their theoretical origins to become indispensable tools in various scientific and engineering disciplines.

Today, complex numbers find applications in fields as diverse as electrical engineering, quantum mechanics, fluid dynamics, and signal processing. They provide the mathematical framework for solving problems that were once deemed unsolvable and enable us to describe phenomena that were previously elusive. From designing efficient electrical circuits to simulating quantum systems, complex numbers have become an essential part of our technological advancements.

In essence, the invention and application of complex numbers have opened doors to new realms of understanding and problem-solving, shaping the course of human knowledge and innovation. As we continue to explore the vast potential of these enigmatic numbers, we can only anticipate that they will play an increasingly significant role in unraveling the mysteries of the universe and in the advancement of human civilization. Complex numbers, born from the minds of visionary mathematicians, continue to inspire us to push the boundaries of what is possible and to embrace the beauty of mathematical elegance in the modern world.

Siegel zeros are values which are very close to 1 and return 0 when inputted into the Riemann zeta function, but are neither trivial or non-trivial zeros, in other words a counter example to the Riemann hypothesis. However, a recent breakthrough by Yitang Zhang is very close to disproving the existence of such numbers. ↩︎
In physics, $j$ is often used instead of $i$ , as $i$ may indicate current in electrical components and cause confusion. ↩︎

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