From the natural
to the imaginary

Michele Diego

It is in quantum mechanics that complex numbers find their greatest prominence, entering directly and explicitly into Schrödinger’s equation, which governs the behaviour of every quantum system.

American physicist Richard Feynman, one of the leaders of 20th-century physics, used to say that science is just imagination… in a straitjacket. Equipped with his words, we cannot be surprised by the historical evolution of mathematics that has marked great imaginative breakthroughs and creative endeavours; these have, however, taken place over centuries, giving our minds time to process these revolutionary thoughts. Great discoveries in mathematics are usually the result of demonstrating or inventing something that previously seemed impossible, unthinkable or absurd. Yet the seriousness of this discipline implies that the time between one revolution and the next must be ripe.
We, therefore, follow this historical course of mathematical thought at an accelerated rate during school; it is a bit like those animal foetuses that, during their weeks of gestation, seem to trace their species’ evolutionary history. From a young age, in fact, we have an intuition of the existence of the so called natural numbers (0, 1, 2, 3, …). All of us, from our earliest years, can distinguish between having nothing, having one thing and having many. Given their structure, we can imagine these natural numbers as equidistant points on a half line that starts at zero and continues to infinity. The gap between two subsequent points is fixed and is equal to 1, thus for every number “n” follows “n+1”. There are no other natural numbers between “n” and “n+1”.

We then find out about the existence of integer numbers (…, -3, -2, -1, 0, 1, 2, 3, …), which in addition to natural numbers, include also negative numbers extending the half line to below zero, towards a negative infinity. Hence the half line has actually become a real straight line extending from minus infinity to plus infinity. The separation between numbers, however, remains the same. Every number “n” always has a number “n+1” as its successor.
In any case, the existence of negative integer numbers alone can give us some trouble: for example, what does minus an apple mean? Our mind, nevertheless, manages to overcome this obstacle, and it does not seem that strange that negative speed represents an object moving backwards, or that a negative acceleration means slowing down, or that our bank account is in deficit.
The following step is the introduction of rational numbers (3/4, 1/2, -8/3, …): if I divide a cake into sixteen parts, each slice is equal to 1/16 of the entire cake; rational numbers allow us to express this concept. If we visualise the straight line containing our numbers again, rational numbers fill in the gaps between integer numbers. The interesting thing is that, as we have seen, integer numbers lie on the line as equal-spaced points. This means that between the number “n” and the number “n+1” there is no integer number, while between two rational numbers there are always infinitely many others. For example, between 1/4 and 1/2, there are infinitely many other rational numbers (e.g. 1/3, 2/5 or 4/9 to name a few). In other words, unlike natural and integer numbers, having given one rational number, there is no next one.
The last step we take, in successfully completing a standard school curriculum, is the introduction of the so called real numbers. Our straight line of numbers still has “holes” in it. This may seem strange if we think about what we said earlier: if we take any two points on the straight line on which the numbers lie, there are an infinite number of rational numbers between those two points. It would appear that we have completed the entire straight line, that there can be no empty spaces left, and yet numbers like √2 or π cannot be written in the form of rational numbers. This was a problem even for Ancient Greeks, since Pythagoras’ theorem applied to a triangle with sides of one metre in length, requires a length of √2 metres for the hypotenuse. Real numbers, therefore, also fill in those holes left by rational numbers, thereby finally completing the straight line of numbers.
But those who continue studying mathematics soon realise that things do not cease there. There are, in fact, other numbers that we apparently do not have to deal with in everyday life – so much so that ordinary calculators are not designed to use them – but which actually allow us to explain a multitude of natural phenomena that surround us. They are called complex numbers, and were invented as an attempt to solve previously unsolvable equations. In short, just as the introduction of the real numbers made it possible to calculate numbers that were previously unknown (such as √2), complex numbers further expand the boundaries drawn by real numbers and essentially allow us to calculate √-1, which was impossible to do with real numbers.
But since we have seen that the line on which the numbers are found is entirely covered, without holes of any kind, by real numbers, where do complex numbers find their place? They are not on a single line, like the numbers discussed above, but on a plane that is defined by two straight lines. Each complex number is in fact defined by a pair of numbers, one of which is called “real” and the other “imaginary”. The number that enables √-1 to be solved is called the “imaginary unit” and is denoted by the letter “i”.

Apart from abstract mathematical formalism, complex numbers have a wide range of applications in the mathematical understanding of the world. In electromagnetism, they are used to describe the spread of electromagnetic waves (essential for the study and exploitation of light in its various forms: radio, Wi-Fi, telecommunications, remote controls, mobile phones, etc.), as well as for the understanding and control of alternating current circuits (at the basis of any technology we connect to current).
In the study of mechanical dynamics, complex numbers are used to recreate both wave mechanics and those non-ideal phenomena related to the existence of dissipative forces that prevent perpetual motion; they are therefore fundamental for a realistic and ideal description of the world. However, it is in quantum mechanics that complex numbers find their greatest prominence, entering directly and explicitly into Schrödinger’s equation, which governs the behaviour of every quantum system. In this article we do not have time to explain the details of the equation, but we can at least say that from it we obtain information about the time trend of the quantum system in relation to the energy it possesses. At the very first term of the equation the already mentioned imaginary unit i appears, showing explicitly that the equation is solved in the complex field.
From natural to complex numbers, the evolution of mathematics has enabled mankind to understand and describe phenomena that are increasingly distant from the human mind’s intuition. This path has seen a mutual exchange between mathematics and physics, between what is discovered in a purely abstract way by a purely mathematical mind and what is constructed on the basis of the need to describe a physical phenomenon. When the world is ripe enough, a new branch of mathematics will be discovered, opening the door to a new understanding of the world.

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