Complex numbers seem to have an almost mystical aura surrounding them. Perhaps it’s because of the terminology used for them – for example, they have a ‘real part’ and an ‘imaginary part’, and they are ‘complex’. Surely something that is imaginary is non-existent?
Furthermore, these are self-proclaimed complex numbers we are talking about, as opposed to the simple ones that we are used to. It seems useless trying to understand complex numbers like i (the square root of -1).
It was René Descartes who coined the name ‘imaginary numbers’ in his 1637 essay La Géométrie. In his opinion, such numbers are useless and the product of fiction; the name he chose for them reflected these disdainful ideas. Before him, Italian Rafael Bombelli was the first to attempt to put complex numbers on a solid footing, in his 1572 book L’ Algebra.
Bombelli called the number i ‘plus of minus’ and the number −i ‘minus of minus’, thus alluding very early on to the fact that imaginary numbers are neither positive nor negative. Unfortunately, these new numbers were still not understood very well (if at all!) by mathematicians, prompting Descartes to call them ‘imaginary’. Even more unfortunately, the term ‘imaginary’ stood the test of time and is still what we call these numbers nowadays, even though mathematicians have made huge leaps forward in their understanding of these numbers since then.
Complex numbers seem to have an almost mystical aura surrounding them
What about the word ‘complex’? This was introduced much later by Carl Friedrich Gauss in 1831. Gauss was originally one of the doubters of imaginary numbers ‒ he famously quipped “the true metaphysics of i is hard” in 1825. However, by 1831, he had completely accepted them by interpreting them geometrically, calling them ‘numeros complexos’ in the second memoir of the Theoria residuorum biquadraticum.
The Latin word complexos means embracing, grasping, including. Hence, Gauss intended a complex number to mean a number that embraces, or includes, two numbers within it, which are its real and imaginary parts.
Perhaps a modern usage of the word complex in the English language that has roughly this sense is in the phrase ‘shopping complex’, which refers to several shops housed, or embraced, in one building. This is the sense we should evoke when there is mention of the phrase ‘complex number’; not complex as in complicated, but complex as in including more than one number within itself.
Are complex numbers useful?
It needs to be said that, like many other things, the usefulness of complex numbers depends very much on the person’s line of work. The answer to the question “Are complex numbers useful to you?”, for most people, is “no”. For some others, it is “a little bit”. And for others, it is “yes! a lot!”. It depends on what you want to do and what you’re doing.
I used to think that people who specialise in certain areas of mathematics do not require complex numbers in their research. In reality, complex numbers are becoming increasingly important in seemingly unrelated mathematical fields.
For instance, I used complex numbers in a research paper which was published in a graph theory journal which is usually devoid of such numbers. Other researchers are using graph theory to study quantum walks, which require complex numbers to be understood well.
In many areas of physics and engineering, complex numbers have also proved to be very useful to describe natural phenomena, for example, in stability analysis or to model electrical currents and impedances.
Moreover, perhaps shockingly, complex numbers simplify other areas of mathematics. This is probably the most compelling property that complex numbers possess, and the main reason why complex numbers should not be ignored, not even by people who do not use them.
Alexander Farrugia is the author of the book But What is i, Exactly? He is also a mathematics senior lecturer at the University of Malta Junior College.
Sound Bites
• In 2016, the author of this page (Alexander Farrugia), fresh from obtaining his PhD, used complex numbers in a paper that gave an expression for the difference in energy in a molecule when one chemical bond is removed or introduced. This expression had eluded the efforts of mathematicians and chemists since graph energies were introduced in 1978 by the eminent Ivan Gutman. The paper is Alexander Farrugia, Coulson-type Integral Formulae for the Energy Changes of Graph Perturbations (MATCH Communications in Mathematical and in Computer Chemistry, 77 (2017), 575-588).
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DID YOU KNOW?
• In his memoir Theoria residuorum biquadraticum, Gauss also introduced the symbol i to stand for the square root of -1, which is still the symbol we use for this number today.
• The original reason for complex numbers to be studied was to work out solutions for cubic equations, which arise in problems involving finding the dimensions of boxes (cuboids) having a certain volume.
• Nowadays, complex numbers have a vast array of applications, including electrical theory, quantum theory, signal processing, control theory, cryptography, artificial neural networks, seismic analysis and time series analysis in economics.
• Complex numbers do not have an inherent order. Unlike the real numbers, where, for example, one is bigger than zero and two is bigger than one, the number i is neither bigger than zero nor smaller than zero. (It is also not equal to zero.)
For more trivia, see: www.um.edu.mt/think.