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“If explaining minds seems harder than explaining songs, we should remember that sometimes enlarging problems makes them simpler!” – Marvin Minsky, speaking about Carl Friedrich Gauss

 
Mare Complexionis, The Sea of Complexity

So how to approach Complex Numbers? Even the name is somewhat foreboding, at least they are not called Completely Unfathomable Numbers, so there is perhaps hope of discerning their meaning. Not for either the first time or the last, I am going to employ a rather imprecise, but hopefully illuminating device for trying to make these numeric entities more accessible. I’m going to rely upon the game of Battleships. While there have been various iterations of this game (and there is no doubt an app for that as well), I will focus on the original which consisted of nothing more than two labelled grids where the player and opponent would each pencil the position of their various boats of war, which might occupy anything from two to five squares ^[1].

I’m going to focus on just one of the grids and just one battleship, occupying three squares ^[2]. The setup would be rather like this:

Those who remember playing battleships may recall that the row and column labels might have been positioned slightly differently, but hopefully the general idea is clear. Each square on the grid can be identified by a combination of a number and a letter. So you can see that my destroyer extends from E3 to G3. I’m going to make a further tweak by labelling these 3E and 3G, the meaning remains the same of course. My opponent will attempt to “hit” my ship by calling out a square identified in the same way. I’ll then tell them whether this is a “hit” or a “miss”; so 5B is a “miss” but 3F is a “hit”. So far so simple. I’m now going to complicate matters by dispensing with the letter labels and instead using two sets of numbers as per:

The same as before I can say that my destroyer runs between 35 and 37. However, given that these look a bit like thirty-five and thirty-seven, perhaps I can make things clearer using some punctuation, so (3,5) and (3,7). That’s somewhat better, but I still need to remember whether the column label or the row label comes first. Perhaps it is easier to have some way to recognise this so that there is no ambiguity. As the row labels go up the page, I’m going to label them with a little “i“, this is because “i fyny” is the Welsh for “up” (if at this point the reader is beginning to wonder about my sanity, please bear with me a little longer). We now get:

This is much better and it is clear that (3,5i) is not the same as (5,3i). But now it’s getting a bit irritating typing all these brackets, perhaps there is a better way. How about “3 and 5i“, which we could shorten to “3 & 5i” or maybe instead “3 + 5i“. I’m going to stick with this last formulation and state definitively that my destroyer is located between 3 + 5i and 3 + 7i on our grid. And that is the essence of Complex Numbers ^[3].

If instead of using the discrete squares from Battleships, we make each axis a Real Number line (allowing for non-integer values, like √2 + 3.75i) and further extend these in all directions, then our initial picture of the Complex Numbers is close to complete. The central part of our modified Battleships grid would then look like the following:

Of course this is rather reminiscent of the two dimensional space in which we were using matrix multiplication to flip and spin shapes in the last chapter. This is not a coincidence!

Any complex number can be uniquely identified by the combination of a value on the horizontal axis, say a, and a value on the vertical axis, say bi. Then this number can be formulated as a + bi. If we further stipulate that each of a and b can take on any Real Number value (see Chapter 4 for an introduction to Real Numbers), our definition of the Complex Numbers is pretty accurate and pretty robust.

The set of all Complex Numbers is usually denoted by ℂ, a C with a bar running through it. Extending our previous nesting of different number families, we can now also say that:

ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ ⊂ ℂ

 
Fleet Manoeuvres

So, via the use of our Battleships analogy, we have defined a new extension to what we normally view as a number ^[4]. An obvious question is “what can we do with these new numbers?”. Well for a start we can combine them, the simplest way of doing this is addition and subtraction. If we go back to our destroyer at the beginning, if we want this to steam two units up the grid we could add to its starting position as follows (the brackets are just to help denote the original location):

(3 + 5i) + 2i = 3 + 7i, a new complex number

Equally if we wanted it to shimmy 3 squares rightwards, we could also add to its starting position:

(3 + 5i) + 3 = 6 + 5i

It’s worth noting that if we add (or subtract) the element tagged i, it has no impact on the element which is not tagged and vice versa, the two things are independent. This allows us to make a general definition of addition of complex numbers:

(a + bi) + (c +di) = (a + c) + (b + d)i

Subtraction follows naturally from this. What about multiplication? Here we can borrow a property of numbers in general, one that I can introduce by talking about mental arithmetic as in the box below:

Supposing I want to multiply 33 by 14 and I have memorised neither my 14 times nor my 33 times tables (and have no calculating device / software to hand), how should I proceed? As with many areas of human endeavour, a natural approach is to break the question down into simpler components, work those out and then reassemble things. In this case, I might well start by splitting the 14 in to two parts, namely 10 and 4. 10 times 33 is easy, just add a 0 to get 330. 4 times 33 is not entirely trivial, but it’s not a lot of work to get to an answer of 132. If we add together our two numbers we get 330 + 132 = 462, which is indeed the right answer. The property of numbers we have relied upon is as follows: 14 × 33 = (10 + 4) × 33 = (10 × 33) + (4 × 33) We can generalise the above to consider decomposing both the multiplicand and multiplier to note that: (a + b) × (c + d) = (a × (c + d)) + (b × (c + d)) = (a × c) + (a × d) + (b × c) + (b × d) At this point you may be able to deduce why Mathematicians normally elide the × sign and instead write: (a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd I’ll adopt this notation going forward as well. We refer to the above property by saying that multiplication is distributive with respect to addition, but the technical term is not needed for our purposes.

We can employ this rule we have discovered to look at multiplication of complex numbers as follows:

(a + bi)(c + di) = ac + adi + bci + bdi²

But now a new question arises, how are we meant to interpret something like i²?
 
 
‘Fessing Up

It’s at this point I really have to come clean. Despite my Welsh antecedents, it is not entirely true that the i I have been using to date is related to the direction up. Instead it’s a rather special number that arises essentially from the pigheadedness of Mathematicians. This group of people reasoned as follows: I can come up with an answer to the question:

What number when multiplied by itself gives me an answer of 4?

or in Mathematical language:

x² = 4, solve for x

Here x = 2 is a ready solution ^[5].

As covered in Chapter 4, we can also consider the question:

What number when multiplied by itself gives me an answer of 2?

or in Mathematical language:

x² = 2, solve for x

As we showed before, the answer here is √2, a result which led us towards a whole new class of numbers.

Given both of these results, then why can’t we also answer the question:

What number when multiplied by itself gives me an answer of minus 4?

or

x² = -4, solve for x

A less determined breed of person might settle for saying that there is no answer of course. Any positive number squared is a positive number and any negative number squared is also a positive number (-2² = 4, the second answer to this equation provided in note ^[5] below). However Mathematicians are nothing if not determined and so collectively decided to imagine that such an answer existed (which is why René Descartes coined the phrase Imaginary Numbers to describe numbers of this type). They did this by starting with the simpler question:

x² = -1, solve for x

and defined the value of x satisfying this equation as something which should now be familiar to readers, namely i ^[6]. The newly defined i could then be used to provide an answer to the square root of -4 which will be simply the square root of plus four multiplied by i. So:

(2i)² = 2²i² = 4 × -1 = -4

In general we can say that:

√(-n) = (√n)i

This is the true provenance of i‘s role in formulating the Complex Numbers. However nothing in this section invalidates what I have introduced in the earlier, more nautically-centred comments. It is extremely common to consider the complex numbers as inhabiting the type of Ocean I have created above, though it is generally described as the Complex Plane.

We can now move forward our work on the multiplication of Complex Numbers and say:

(a + bi)(c + di) = ac + adi + bci + bdi²

which given that we now know that i² = -1 means:

(a + bi)(c + di) = ac + adi + bci – bd

Rearranging we get:

(a + bi)(c + di) = (ac – bd) + (ad + bc)i

This is quite a nice and simple finding (though perhaps the result of some slightly tortuous logic of course).

There was a physical interpretation for adding Complex Numbers, this related to moving our boat (or a point in the Complex Plan) upwards, downwards, left or right. There are similar physical interpretations of multiplying Complex Numbers which involve growing or shrinking shapes, rotating them or skewing them in odd manners. If the reader begins to detect another linkage to material covered in Chapter 6, then there is ample reason for this ^[7].
 
 
The Complex Numbers as a Group

Before moving on we of course need to assess whether or not the Complex Numbers form a Group. I’ll simply state that they do under addition (using the definition provided above) and leave checking this as an exercise for the reader. Let’s work through ℂ under multiplication:

1. Closure
    
   In the previous section we established that:

   (a + bi)(c + di) = (ac – bd) + (ad + bc)i

   A moment’s thought leads to the conclusion that this also demonstrates closure.
    

2. Identity
    
   As with multiplication on each of the Rational Numbers, ℚ, and the Real Numbers, ℝ, the idenity is 1 (and not 1 + i) because:

   (a + bi) × 1 = (a × 1) + (bi × 1) = a + bi

   So we have an identity element.
    

3. Inverses
    
   Here we can extend the definition of an inverse that we developed for the Rational Numbers, i.e. the inverse of a Complex Number is its reciprocal:

   (a + bi)^-1 = 1 / (a + bi)

   An obvious question here would be: “is 1 / (a + bi) also a Complex Number?”. Let’s try to show this by starting with:

   1 / (a + bi)

   Multiply both top and bottom by a – bi to get:

   (a – bi) / [(a – bi)(a + bi)]

   Combining the items in square brackets gives us:

   (a – bi) / (a² – (bi)²)

   Recalling that of course i² = -1 by definition, this is equal to:

   (a – bi) / (a² + b²)

   Now we have no imaginary component in the denominator and can re-write this as:

   [a / (a² + b²)] – [b / (a² + b²)]i

   Where clearly both [a / (a² + b²)] and [b / (a² + b²)] are Real Numbers (because each of a and b are Real Numbers).

   Hence our result is a Complex Number and we have inverses, remembering that (as with both ℚ and ℝ) we have to exclude 0 from the set being operated on (as the reciprocal of 0, 1 / 0, is not well-defined) ^[8]. This means that the set we are considering should properly be written as ℂ\0 rather than just ℂ, it is however pretty typical to use just ℂ in this context.
    

4. Associativity
    
   Watch out, here comes some more Associativity-related hand waving! As multiplication of Complex Numbers results in effectively multiplication and addition of Real Numbers (some tagged with is) and these operations are associative, so is multiplication of Complex Numbers.

So we have found a yet more generalised Group based on our latest extension of numbers.
 
 
amo, amas, amat…

There are many other things you can do with complex numbers beyond addition and multiplication. Amongst other things, you can raise a number (either Real or Complex) to a Complex power, so something like (2 + i)^{(3 + 4i)} rather amazingly makes sense (and indeed will be another Complex Number in the form a + bi). This also leads us to one of the most famous equations in all of Mathematics, Euler’s Identity:

e^iπ + 1 = 0

We will meet Euler’s Identity ^[9] again in Chapter 11. However for our current purposes, there is just one more concept to be introduced, that of a complex conjugate. This is something we have met already – albeit unannounced – when we were looking at the inverses of complex numbers.

Compared to what has gone before, this is pretty simple. The complex conjugate of a number a + bi is a – bi, i.e. the sign of the imaginary part is changed. Going back to our Battleships / Complex Plane analogy, this is the equivalent of finding the mirror image of our boat where the mirror runs along the horizontal axis, as shown below:

Again there are echos of what we saw in Chapter 6 here as well.

In Mathematics books the complex conjugate of a complex number is written with a bar over it, so we would say:

a + bi = a – bi

Because this can get a bit complicated typographically, an asterisk is also sometimes used to denote conjugation so:

(a + bi)* = a – bi

However, I’ll try to stick to using the bar notation in the rest of the book.

We are going to use the concept of a complex conjugate in Chapter 13, in which we will move closer towards an understanding of Special Unitary Groups.
 
 
Non-imaginary Numbers

If you remain unconvinced by what you might view as the Mathematical sleight of hand introduced in this Chapter, then let’s make the idea of Complex Numbers even more concrete and also tie them to another concept we have met recently, that of matrices.

Consider the set of matrices defined as follows:

Guess what I am going to claim about them? No, not that they are a Group (though they are under addition and would be under multiplication if we exclude the zero matrix), instead I claim that these are the Complex Numbers. How do I go about doing this, well I’m going to show that I can write a mapping between the two sets such that each member of one is mapped to precisely one of the other. I’m then going to show that other behaviour of both sets is essentially the same. Let’s first of all match up and :

From the above it seems pretty evident that as both and range over , every element of and every element of matches up.

What about addition, well let’s compare:

So that seems to work. What about multiplication?

So that works as well? What about the properties of itself, well this maps to and we have:

So in effect:

We also have the potential issue that – in order to form a Group under multiplication – our matrices above must have a non-zero determinant, whereas Complex Numbers need to be non-zero. However, a moment’s thought shows us that these two conditions are equivalent as the determinant of our generic matrix is precisely , which is only zero if both and are zero. So our potential issue evaporates.

I think that this little demonstration is now at an end. However, this is something to bear in mind if you hear a nagging voice saying that Complex Numbers are really a figment of someone’s overactive imagination.
 
 
At this point it is perhaps worth asking an obvious question. If we can extend the Real Numbers by adding our new number i standing for the square root of – 1, can we add other new numbers and repeat the trick? The answer is yes and some aspects of such an extension are covered in the box below. We will come across these numbers again in a possibly surprising way later in the book.

The Sign of the Four So we defined the Complex Numbers in terms of the Real Numbers by introducing i to get: ℂ = {a + ib where a, b ∈ ℝ} Where i2 = – 1 Maybe we could add just another element like i, say j. It turns out that this doesn’t really work, but if we take a further step and introduce a third new element, k, then something wonderful happens, we have found the quaternions, first discovered by Irish Mathematician William Hamilton in 1843 [10]. The set of quaternions is denoted by ℍ in honour of Hamilton. It is defined as follows: ℍ = {a + ib + jc + kd where a, b, c, d ∈ ℝ} Where i2 = j2 = k2 = – 1 and ijk = – 1 We can use these definitions to form a table capturing how the various elements combine as follows: × i j k i – 1 k – j j – k – 1 i k j – i – 1 We can also fairly readily see that numbers in ℍ are well-behaved, you can add, subtract, multiply and divide them in ways analogous to what we have demonstrated for ℂ earlier in this chapter. One point to note however is that multiplication is not commutative, indeed in general, if g, h are two distinct generators (i.e. each a different one of i, j, k) then: gh = – hg as may be seen from the table above. The quaternions also form a Group under multiplication (as always excluding 0) and it is a Group we will meet again in Chapter 14.     Of course a natural follow-on would be to wonder whether or not we can take this process of extending the concept of number further. There is one further extension, the octonions, which unsurprisingly have eight generating elements [11] analogous to the four for the quaternions. However that is then it, there is no meaningful set of numbers [12] with 16 generating elements or indeed any more. The reason is that we lose features of the number system along the way, the quaternions are not commutative, the octonions are not associative and there is nothing much left to lose beyond this while retaining meaning as a number system.

 
 
For now, our journey into the realm of the imaginary is over. In the next Chapter we will explore the idea of Groups within Groups and how this concept relates to prime numbers.
 

Concepts Introduced in this Chapter The Complex Plane A two dimensional coordinate system, where entries on the vertical (or y) axis are labelled with an i to distinguish them. A general point on the Complex Plane is thus identified by a + bi, where i is our label and both a and b ∈ ℝ. Complex Number Any point on the Complex Plane in the form a + bi. In this a is known as the Real component and bi (or sometimes just b) as the Imaginary component. Complex Addition This consists of adding the Real components of two complex numbers and separately the Imaginary. Thus: (a + bi) + (c + di) = (a + c) + (b + d)i) i A number defined as the square root of minus one. Hence i2 = -1. Complex Multiplication Essentially normal multiplication of terms in brackets, i.e. (a + b)(c + d) = ac + ad + bc + bd Introducing the imaginary component of complex numbers we get: (a + bi)(c + di) = ac + adi + bci + bdi2 Recalling that i2 = -1, we have: (a + bi)(c + di) = (ac – bd) + (ad + bc) i Complex Conjugate The number obtained by changing the sign of just the Imaginary component of a Compelx Number. So the complex conjugate of a + bi is a – bi. This can be thought of as reflecting the Complex Number in a mirror running along the x-axis. Quaternions Any number of the form a + ib + jc + kd where a, b, c, d ∈ ℝ and where i, j, k satisfy the relations: i2 = j2 = k2 = ijk = – 1. The set of quaternions is denoted by ℍ.

Groups Discovered in this Chapter (ℂ\0,×) The Complex Numbers (excluding zero) under multiplication. A subgroup of the matrices under addition and the matrices with non-zero determinant under multiplication, that is isomorphic to the Complex Numbers. (ℍ\0,×) The Quaternions (excluding zero) under multiplication.

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Chapter 7 – Notes

[1]	Mine-sweepers and Aircraft Carriers respectively if my memory serves me well
[2]	A Destroyer?
[3]	This is actually the essence of the Gaussian Integers (the subset of ℂ such that, ), but we will leave an exploration of these to another time.
[4]	In a similar way to what we did when looking at Matrices in Chapter 5.
[5]	But we should also note that x = -2 would have worked just as well of course.
[6]	Again if i2 = 1, then also (-i)2 = 1; there are always two roots of a polynomial of degree 2 and indeed n roots of a polynomial of degree n, though sometimes these can be repeated.
[7]	The reader is left to conduct their own research in this area if it interests them.
[8]	So our Complex Ocean, at least when considered as a Group under multiplication, has a hole in its middle. Let’s hope that all the water doesn’t drain out!
[9]	Together with the related Euler’s Formula, of which Euler’s Identity is a special case.
[10]	And famously carved by him into the stone of a Dublin bridge.
[11]	These are {1, e1, e2, e3, e4, e5, e6, e7}, the rules for multiplying these generating elements together are more complicated than for quaternions and I won’t cover them here. You can see more details here.
[12]	By meaningful, I am referring to the concept of a Division Algebra.

Text: © Peter James Thomas 2016-18. Images: © Peter James Thomas 2016-17, unless stated otherwise. Published under a Creative Commons Attribution 4.0 International License.