Fantastic Beasts: Imaginary and Complex Numbers

Before starting our tour of charming mathematics I’d better introduce you to some of the most fantastic beasts in our world – the imaginary and complex numbers. In fact, whilst introducing them I’d like to convince you that extending our number system beyond what we can see in the natural world is a relatively common affair in mathematics, and that the imaginary and complex numbers are therefore less imaginary than some people think. If you’re already something of an expert, I’m hopeful you’ll still find the historical elements of this insightful (and please do leave a comment if you’ve noticed any inconsistencies with sources you’ve come across).

Section 1: The Trials and Tribulations of Accepting Negatives and Irrationals

Let’s set the scene by beginning the story with none other than the fabulous Pythagorean theorem. I’d like you to close your eyes and imagine yourself back in Ancient Greece (approx 500 BC) at the time when the Pythagoreans were having a blast defining, inventing and discovering new things about mathematics. Given the time period, your understanding of number is wildly different, extremely tedious to work with, and in many ways quite basic when compared to what a normal primary age student understands about number today.

They used their alphabet to represent different numbers (e.g. α = 1 and π = 80, so that πα = 81) and they put a ‘diacritical’ mark before a letter to denote it as a unit fraction (e.g. ‘πα= 1/81) – although it’s important to note that they didn’t think of fractions as numbers lying on the number line, but rather proportions of positive integers . Anyhow, I’m sure – once you’d developed fluency with this system – that it wasn’t too difficult to work with, but this notation does instil me with a sense of gratitude for what we have now. The Greeks did have a way to represent fractions with a numerator greater than one but they didn’t have the equivalent decimal form that we’re now accustomed to, nor did they have a way to represent negative numbers; not to mention the fact that a number line was centuries away from being conceptualised.

Now we all know what the Pythagoreans main contribution to mathematics was. But you may not be aware that the same connection between areas of squares on the sides of right angled triangles was also known to the Babylonians. Why then, you might ask, are the Pythagoreans credited with the theorem? Simply put, it wouldn’t be a theorem if the Greeks hadn’t mounted their flag on it by proving it to be true in every possible case. The credit in mathematics almost always goes to the person or people who lay the matter to rest and provide ever-lasting certainty in the form of proof. (Note: whilst Pythagoras himself did exist, much of what is attributed to him is likely based on the work of his followers [1].  Furthermore, “most of what we know about him is conjecture, with a great many anecdotal fables thrown in.” [2]).

As an aside, I love how you can stick any old shape on the side of a right angled triangle, and the same theorem holds as long as you stick similar shapes on the other sides. Alex Bellos represented this perfectly in his fabulous book, “Alex’s Adventures in Numberland” by plonking a Mona Lisa shape on each side of a right-angled triangle and showing that the area of the two smaller Mona Lisa’s sum up to the area of the larger one.

Anyhow, after revelling in their success of proving the theorem it must have come as quite a shock to the Greeks to realise that using it resulted in numbers that didn’t exist (at least in their notion of number). When attempting to find the length of the hypotenuse of different right-angled triangles they didn’t understand why the answers weren’t nicely defined integers or fractions. There comes a huge problem; if we know our theorem is correct, but it churns out answers that don’t make any sense to us, what do we do? Maybe we put our heads under our pillows and sing ourselves to sleep without ever thinking about these un-Godly, inexpressible numbers ever again. Indeed, it’s said that they were named “unspeakables” by the Pythagoreans. Maybe we could go one step further and throw whomever discovered them overboard [3], or maybe, instead of hiding from these terrible monsters, we could conceptualise things in a new way in the hope of understanding them.

Without digging more deeply into the history of irrationals, the problem rested on the way Greeks thought of quantities as ‘discrete objects’, made up of a finite number of parts. When you think in these terms it’s darn near impossible to accept something which can’t be split up, and can’t be expressed as a well-behaved proportion. Fortunately for us, a Greek called Zeno, and then Eudoxus after him, did a good deal to lead the revolution on this, and fought to ensure that quantities be conceptualised as continuous entities rather than as discrete packets [4].

Irrationals weren’t the only stop-gap for the Greeks; they also struggled immensely with the concept of negative numbers. Indeed, if geometry is your fundamental mathematical toolkit then what would a line of negative length even mean??? (the number zero was unknown for the a similar reason). Diophantus referred to equations with negative solutions as absurd, and to be honest I don’t blame him. With no way to conceptualise what a negative number could mean I’m sure I’d say the same thing. This same lack of conceptualisation prevented negatives from being accepted in Europe for longer than we’d like to admit.

Even as late as the early 18th century, a few mathematicians continued to oppose negatives which is quite difficult to believe given how naturally we think of them today as part of the ‘real’ number line. But then we probably take our current view of the real number line for granted. It was only in the late 16th century that Bombelli and Descartes popularised the idea of numbers lying on a line spanning infinitely in both directions. This lies in stark contrast to the Chinese viewpoint on number given that around the time of Diophantus in Ancient Greece, the Chinese found negatives relative easy to accept due to the importance placed on duality in Chinese philosophy [5].

When you sit down and think for a moment, it’s not at all obvious that numbers should be represented on an infinitely spanning line. Given how we experience number in the real world in terms of discrete, countable objects, or even measurable objects which can be approximated as discrete quantities, representing numbers on a continuous line spanning in two directions is not intuitive. The number line is an abstract representation of numbers that does not match with our observations in the real world, and this is especially the case with negative numbers. Thus, extending the number line to span in the opposite direction to account for numbers that we do not directly see in the world is an unintuitive, abstract leap, but it is a necessary one in order to progress our understanding.

Section 2: The Development of Number 

So before moving on, let’s summarise the major points of that discussion. It took a long time for irrational numbers and negative numbers to be accepted because they do not appear in our perceived world view, and we had no immediate way to conceptualise them. Let me think? – are there any others number that fall into a similar category…ah yes, the imaginary numbers. Like negative numbers they initially revealed themselves as the nonsensical solutions to equations. Historically speaking this goes back to the 16th century in which the first people to accept the existence of imaginary numbers were the likes of Cardano and Bombelli in the solutions of cubic equations, but a simpler equation to begin thinking about imaginary numbers is the quadratic below:

                   x² + 1 = 0                        (1)

There is clearly no whole number, negative number, fraction or irrational which we can substitute in to solve this equation. Having said that, if we take a bold, abstract leap and assume that there is a solution – as we did with irrationals and negatives before-hand – then all we need to do is find a way to conceptualise that solution. Low and behold – just as we extended the number line in the opposite direction to accommodate for negative numbers – the likes of Casper Wessel in the late 18th century, and then Argand and Gauss in the early 19th century, decided to conceptualise imaginary numbers by extending the number line once again.

So whilst the development of the number line was far messier than this, let’s look at a simpler, logical progression in the context of extending our number system…

Step 1: Discrete natural numbers = {1, 2, 3, 4, 5, … }

Step 2: All Positive Numbers – Re-conceptualisation of numbers to accommodate firstly fractions, and then irrationals. i.e. transitioning from a discrete view of numbers to a continuous one.

Screen Shot 2018-12-21 at 21.52.40

Step 3: First extension of the number line in the opposite direction to accommodate negative numbers; all integers (including zero).

Step 4: Second extension of the number line to accommodate imaginary numbers. This second extension allows us to conceptualise solutions to equation (1):

Screen Shot 2018-12-21 at 22.08.03

Thus, there are two imaginary solutions to the quadratic equation,

Screen Shot 2018-12-22 at 21.12.34

Each co-ordinate on this grid, which is most often referred to as the complex plane, represents a complex number with both a real part and an imaginary part. For example, 1 + 2i can be represented as a co-ordinate or vector on the complex plane, and we can then begin to understand geometrically what it might mean to add, subtract, multiply and divide complex numbers.

To provide some small insight into the complex world, if we multiply 1 + 2i by i then we obtain:

Screen Shot 2018-12-22 at 21.17.38.png

If you then look to see what multiplying by i represents on the complex plane, then -2 + i is a rotation of 90 degrees anticlockwise from 1 + 2i. It is then trivial to show that multiplying any number by i – real, imaginary or complex – represents an anticlockwise rotation of 90 degrees in the complex plane – the geometric meaning of i is thus unveiled!

The moment geometric understanding is achieved is often the moment that community acceptance is catalysed.

I’d completely forgive you if this all still seems completely, well….imaginary, and unfortunately I can’t continue this post forever to show the applications of complex numbers to electrical circuits with impedance, for example, but I hope to have shown that irrational numbers and negative numbers went through a similar stage of scepticism until familiarity and geometric understanding begins to develop an acceptance, and real-life application further validates their use.

We now have a widened perspective of numbers which allows us to progress mathematics. Just because we do not immediately perceive negative numbers, irrational numbers and imaginary numbers in our worldview, it does not mean that we cannot conceptualise them in more abstract terms, and then develop our understanding to ultimately apply them to real world phenomena. It is unfortunate that these numbers were initially given names which make them appear more odd, foreign and strange then they should [6], and that is because they were named before they were understood and applied. Ultimately, the improvement of our concept of number from discrete to continuous, alongside the extension of the number line to incorporate negative numbers, shows that a third extension in the form of the Imaginary axis and the complex plane is not particularly shocking in the history of mathematical development.

References:

[1] Wilczek, F. (2016). A Beautiful Question: Finding Nature’s Deep Design. UK: Penguin Books: UK.

[2] Muir, J. (1963). Of Men and Numbers. New York: Dodd, Mead and Company.

[3] One legend claims that this very thing happened. Hippapus, Pythagoras’ student, is attributed with the proof of the existence of irrational numbers, and he died at sea some time later. Whether his death was accidental, or whether he was killed for exposing this fact to the public, is left to the imagination…

[4] See Zeno’s Paradox of Motion if you’re interested in digging deeper into this.

[5] Hodgkin, L (2005). A History of Mathematics: From Mesopotamia to Modernity. Oxford: Oxford University Press.

[6a] McAllister, J. (2005). Mathematical Beauty and the Evolution of the Standards of Mathematical Proof. In Emmer, M. (ed.), The Visual Mind II: MIT Press.

[6b] Sinclair, N. (2011). Aesthetic Considerations in Mathematics. Journal of Humanistic Mathematics, 1(1), 2-32. Retrieved from https://pdfs.semanticscholar.org/c01f/1b7cdbe2b9a649d09311f7b3e5e1bcb88310.pdf

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