Prologue

It once fell to my lot to teach Mathematics to first-year students at an Australian university. It is often said that you truly learn only when you start to teach. And so it was in my case.

As I dug deeper into what numbers are and how many sets of numbers there are or could be, I realized that there was a vast chasm of ignorance in my knowledge. The terrain that lay beyond was not yet mapped out in my mind.

The more I attempted to know, the more forbidding was the safety net around the scaffolding of abstract algebra that politely said, “Workers Only. Keep Out”. To cross that net and venture to understand what lies beyond it is the burden of this blog.

Our journey starts with the enigmatic equation \(x^2 + 1 = 0\). It wends its way around concepts like sets, fields, vector spaces, arithmetic closure, algebraic closure, polynomial roots, winding numbers, algebraic structure, and the work of two mathematical greats: Gauss and Galois.

There are many brief excursions into side alleys before we resume our main journey. They explain as much abstract algebra, or topology, or some other mathematics, as needed, to help the reader mentally retain the developing image of mathematics without letting it dissolve into the foam of incomprehension or dust of despair.

The centrepiece is the Fundamental Theorem of Algebra, which is usually glossed over with a passing reference, and often without even a plausibility argument. Because of the theorem’s importance, I strike a balance here between being too rigorous and too lax, so that there is sufficient tension to sustain student interest, and sufficient detail to ensure that the proof is persuasive.

This blog is written for high school and first year students of mathematics whose primary interest is in the application of what they learn. Many of the pre-requisites have been covered in my other blogs, and links to them are peppered all over this blog. But, in order to make the reading self-contained and focused, I have condensed and repeated much of that material here so that this story is gripping. Questions are asked, and later answered, to highlight the evolution of mathematics as a logical science.

Prior reading

In my blogs The Wonder That Is Pi, and The Two Most Important Numbers: Zero and One, I have introduced what I call the number menagerie. Different types of numbers were inducted into the art of reckoning or arithmetic, which later flowered into the science of quantity and symmetry called mathematics.¹

I also strongly recommend the unfamiliar reader to also read my blog Expressions, Equations, and Formulae to get a firmer footing on what we will discuss subsequently.

The equation that started it all: \(x^2 + 1 = 0\)

The equation \[ \begin{aligned} x^2 + 1 &= 0\\ x^2 &= -1 \end{aligned} \qquad{(1)}\] triggered the great change which led from the real numbers to the complex numbers.

And why did the change happen? Because Equation 1 asserts that the square of a number can be negative. But, there is no real number whose square is negative. Hence, a new type of number having this property needed to be invented. It was granted an identifying tag \(i\) to show that it was imaginary, not real! Simply put, \[ i^2 = -1. \qquad{(2)}\]

We may now proceed to factorize Equation 1 so: \[ \begin{aligned} x^2 + 1 &= 0\\ x^2 - (-1) &= 0\\ x^2 - (i^2) &= 0\\ x^2 - i^2 &= 0\\ (x - i)(x + i) &= 0\\ x &= \pm i. \end{aligned} \qquad{(3)}\]

But where does \(\pm i\) live? Let us catch our breath first, with some definitions, and then answer this question.

Sets, Fields, Vector Spaces, and Closure

We now take our first detour to define some basic terms:

1. A set is a collection of objects obeying some criterion. For our purposes, sets are collections of numbers.

2. A binary operation acts on two members in a set to generate a result. If the result is always in the set, we say that the set is closed under that operation.

3. There are four standard arithmetic operations, namely, \(+, -. \times, \div\).

4. A set that is closed under these operations—where order does not matter for addition or multiplication, and division by zero is excluded—is called a field.² A number system capable of arithmetic is a field.

5. A vector space consists of two sets: a set of vectors and a scalar field—like the real numbers—used to multiply or scale those vectors.

The rational numbers, \(\mathbb{Q}\), and real numbers \(\mathbb{R}\) are fields. The natural numbers, \(\mathbb{N}\), and integers, \(\mathbb{Z}\) are not fields. A field is a behavioural template and need not consist only of numbers; it could be a collection of other mathematical objects that obey the rules of a field.

Vectors arise naturally in Physics. They are usually drawn on paper with two perpendicular coordinate axes. Vectors are shown as line segments with arrowheads: their lengths denote magnitude, and arrows direction. Vectors reside in vector spaces.

Every field can be regarded as a vector space over itself, but not every vector space is a field, since vector spaces do not generally have multiplication defined between vectors.

The familiar three-dimensional space we inhabit is denoted as \(\mathbb{R}^3 = \mathbb{R} \times \mathbb{R} \times \mathbb{R}\), which is a Cartesian triple product space, where every point in space is given a unique triplet of three real numbers, denoting its position from some chosen origin. This triplet behaves as a vector.

The irrationals

The “holes” in the number line left by the integers were filled by the rational numbers. The remaining holes, like the number \(\sqrt{2}\) were supposedly to be filled by the irrationals. The first on my list of questions was how are the irrationals defined? How many are there? Is there any way of defining them other than as in the binary not rational classification?

Dedekind’s schnitt or cut

These questions also troubled the mathematicians of the mid-1800s. How does one define an irrational number without recourse to geometry? If the whole numbers are all we confidently have—numbers we can touch and relate to—how can we define something like \(\sqrt{2}\) without a geometric detour as the diagonal of a unit square?

I have given one solution to this conundrum in my blog How Are Numbers Built?. But the continued fraction expansions of numbers like \(\sqrt{2}\) and \(e\) and \(\pi\) are not generalizable. And without a way for the general irrational number to be defined, how do we know how many irrationals there are, and what their values are?

Richard Dedekind pondered this question and came up with a solution that was partly geometric and partly set-theoretic.³ The interested reader may consult Dedekind’s original work in translation for the details [1].

Where will the number menagerie end?

Once the real line, composed of the rationals and irrationals, was deemed complete, there was still Equation 1 to proclaim the inadequacy of the reals.

The thought that ran in my head was, “If the complex numbers well and truly account for the roots of polynomials like \(x^2 + 1\), could there be some other type of polynomial that would require an even larger repertoire of numbers in order for its roots to be solved?”

If white sheep are purely real numbers, and black sheep are complex numbers with non-zero imaginary parts, could there be brown sheep, representing another type of number entirely, that might be needed to solve even more demanding polynomial equations? Would sheep of even more colours be needed to meet future challenges? This was the disquieting thought that ran in my mind as I prepared to teach mathematics to surveyors and engineers.

Whither goes \(i\)?

We have the rational and irrational numbers filling the real number line “without any gaps”. There is no room for another type of number on the real line.

Because the numbers \(\pm i\) cannot be accommodated within it, we need to extend \(\mathbb{R}\) to place it. But the imaginary numbers are immiscible with the real numbers. They do not dissolve one into another like salt in water. They are more like grains of sand in water—distinguishable parts of one whole. A means must be found to accommodate the imaginary as well as the real numbers, while each retains its separateness and its properties.

A two-dimensional vector has components in two orthgonal or perpendicular directions, aligned with the \(x\) and \(y\) axes. Vectors are denoted as \(a\mathbf{i} + b\mathbf{j}\) where \(a\) and \(b\) are real numbers and \(\mathbf{i}\) and \(\mathbf{j}\) the unit vectors in the \(x\) and \(y\) directions respectively. The vector is a single entity although its orthogonal components are twin.

Could we not likewise, christen a complex number as a sum of two numbers, one real and the other imaginary, functioning as a single unit?

A complex number may be written as \(a(1) + b(i) = a + bi\).⁴ Clearly, the real number \(1\) and the imaginary unit \(i\) function like the orthogonal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\), spanning the two-dimensional plane. So, we may identify the set of complex numbers, \(\mathbb{C}\), with a two-dimensional vector space. And pictorially, that is exactly what an Argand diagram is.

The field of complex numbers ℂ

We are now at the threshold of the expansion of the real numbers, \(\mathbb{R}\), into the complex numbers, \(\mathbb{C}\). Both \(\mathbb{R}\) and \(\mathbb{C}\) are fields. The addition of the imaginary unit \(i\) might be seen as an ad hoc and opportunistic expansion of the real number field merely to accommodate solutions to \(x^2 + 1 = 0\). But there is a deeper and wider logic to this expansion.

The addition of \(i\) has extended the solution space of polynomials beyond \(\mathbb{R}\). And this extension was necessitated by the quadratic equation, Equation 1. Adding \(i\) to \(\mathbb{R}\) creates an algebraic extension of \(\mathbb{R}\). This means the entire complex plane, \(\mathbb{C}\), is simply an algebraic extension of \(\mathbb{R}\). What does this mean?

The real numbers can be built from a single number, namely \(1\), multiplied by some factor. So, \[ \begin{aligned} 5 &= 5 \times 1\\ -6.3 &= -6.3 \times 1 \end{aligned} \] The number \(1\) is the basic building block for the real numbers—like a LEGO^® brick from which objects may be built. Accordingly, the number \(1\) qualifies as a basis vector for the one-dimensional vector space of the real numbers \(\mathbb{R}\). The scalar multipliers are also supplied by \(\mathbb{R}\) and include any real number.

The mathematical notation describing how \(\mathbb{R}\) is extended to \(\mathbb{C}\) goes like this: \[ [\mathbb{C}:\mathbb{R}] = 2 \qquad{(4)}\] What does this mean? The complex numbers \(\mathbb{C}\) are built from the scalar multipliers drawn from the real numbers \(\mathbb{R}\). Exactly two bricks or basis vectors, \(1\) and \(i\), are needed to do this. The dimension of \(\mathbb{C}\) is therefore 2.

To reiterate, the complex numbers in \(\mathbb{C}\) may be built as a linear combination of the two numbers, \(1\) and \(i\). The dimension of \(\mathbb{C}\) is two, and its scalar multipliers are drawn from \(\mathbb{R}\).

Can we multiply two complex numbers? Yes we can, as detailed in my blogs A Tetrad of Captivating Problems, and Demystifying Fractional Powers.

It is usual to specify a vector space over a (scalar) field where the scalar multipliers are supplied by the field and the vectors by the vector space. We may say that \(\mathbb{C}\) is a vector space over the field \(\mathbb{R}\). Equally, we may say that \(\mathbb{C}\) is a vector space over the field \(\mathbb{C}\). Therefore, \(\mathbb{C}\) is both a field and a vector space.

Polynomials

In a previous blog, I have classified polynomials in the section entitled “the …nomial” family. Take a look at it if interested. For those needing a comprehensive refresher on polynomials, I highly recommend the student-friendly, free online textbook from OpenStax [2].

The polynomial originally arose from the study of algebra in which the solution of equations was the primary goal. Any arithmetic equation, however complicated it might look, can be finally reduced to a polynomial on the left side and zero on the right [3].⁵

When calculus evolved, polynomials were identified as continuous and differentiable functions with certain maxima-minima properties. Still later, it has been suggested that polynomials may be viewed as “… a clothesline on which we hang up a sequence of numbers for display.” [4]. If you are interested in polynomials and their place in the history of mathematics, Merzbach and Boyer is a good reference [5].

A real-valued function \(f\) assigns to each real input value \(x\) some real output value \(y\) according to some rule. This may be written as: \[ \begin{aligned} f&: \mathbb{R} \to \mathbb{R}\\ f(x) &= y \end{aligned} \qquad{(5)}\]

A polynomial is a continuous function [2] that has the specific form: \[ f(x) = a_nx^n + a_{n - 1}x^{n - 1} + \dots + a_{1}x + a_0. \qquad{(6)}\]

If all the \(a_{i}\)’s are real numbers and \(n\) is not zero, we call this a real-valued polynomial of degree \(n\).

Plotting a real-valued polynomial

For \(f: \mathbb{R} \to \mathbb{R}\), the input variable, \(x \in \mathbb{R}\) is a real number. And so is the output variable \(f(x) = y \in \mathbb{R}\).

The familiar Cartesian plane, \(\mathbb{R}^2\), is actually two real number lines at right angles to each other. This setup is referred to as the Cartesian product of \(\mathbb{R}\) with itself, denoted by \(\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}\). It leads to ordered pairs of real numbers \((x, y)\), which relate the output \(y\) to the input \(x\).

Let \(f\) be a real-valued polynomial. The roots of the polynomial are the values of the input that lead to zero output. If \(\alpha\) is a root, then, \(f(\alpha) = 0\). Pictorially, a zero or root of \(f\) is the value of the real variable \(x\) when the graph of the polynomial crosses the \(x\)-axis. Roots are located at the zero-crossings on the graph.

Graph of \(f(x) = x^2 + 1\)

Consider the famous polynomial \(f(x) = x^2 + 1\) which has led us here. If we plot \(y = f(x) = x^2 + 1\) on \(\mathbb{R}^2\), we get Figure 1. The curve is a conic section called a parabola, which never crosses the \(x\)-axis. This means that the equation has no real solutions.

Before we look at the complex-valued polynomial, we need to look at a theorem that constrains continuous functions to behave in a certain way.

Meaning of “root” of a real polynomial

When we plot a real polynomial \[ \begin{aligned} f &: \mathbb{R} \to \mathbb{R}\\ y &= f(x) \end{aligned} \] on a graph paper with \(x\) and \(y\) axes, it is important to bear in mind that the root(s) is/are the value(s) of \(x\) at which \(y = 0\).

You might say, “Wait a second! Is it not where the curve cuts the \(x\)-axis?” And you would be correct. But bear in mind always that the \(x\)-axis is also the line \(y = 0\).

Why is this important? Because it forces attention onto the output \(y\) rather than the input \(x\). The root(s) are those input values for which the output is zero. That is the true meaning of zero crossings in the context of finding the roots of real polynomials. The zeros being crossed are those of \(y\), not \(x\).⁶

The Intermediate Value Theorem (IVT)

When a real polynomial is plotted on co-ordinate axes, the input ranges from the very negative to the very positive. We carefully count the zero-crossings and determine the roots. Multiple roots touch—but do not cross—the \(x\)-axis, as shown in Figure 7 for the value \((3, 0)\) on the \(x\)-axis.

Before plotting the same polynomial with two different domains, I want to make a small detour into something called the Intermediate Value Theorem.

Simply stated, “If you are an ant crawling on a table, you cannot cross a boundary without touching it.” This is what the Intermediate Value Theorem (IVT) states. In mathematical terms, if a function \(f: \mathbb{R} \to \mathbb{R}\) is continuous within an interval \([a, b]\), then for any value \(u\) between \(f(a)\) and \(f(b)\), there exists some \(c \in (a, b)\) for which \(f(c) = u\).

Consequently, if \(f(a)\) and \(f(b)\) are opposite in sign, there must exist some \(c \in (a, b)\) for which \(f(c) = 0\). This is called the Theorem of (Bernard) Bolzano.

In other words, a continuous function, like a polynomial, that changes sign within a closed interval must assume the value zero at some point inside that interval. A corollary of this theorem is that all real odd-degree polynomials must have at least one real root.

Redefining the domain

We have seen that if we re-define the function \(f\) to have a complex domain, solutions do exist for \(f(x) = x^2 + 1 = 0\).

Let us now define \(g:\mathbb{C} \to \mathbb{C}\) with \(g(z) = z^2 + 1\). We have deliberately used a different function name, \(g\), and variable, \(z\), to avoid ambiguity with the real version and accord with convention regarding complex variables.⁷ It must emphasized that the polynomial is the same, only its domain has been re-defined.

Through this redefinition, we have gained a solution where there was none previously. Categorically, \(g(z) = z^2 + 1 = 0\) implies \(z = \pm i\).

Complex polynomials

A complex polynomial \(P: \mathbb{C} \to \mathbb{C}\). It takes a complex value, \(z\), as input and maps it to another complex value, \(P(z)\), as output. The complex zero, is defined as \((0, 0)\): its real and imaginary parts are both equal to zero. For convenience, we write it simply as \(0\).

When a complex polynomial is plotted on the complex plane, its zeros are defined as those values of its input that map to the complex origin \((0, 0)\), on the complex plane. The roots occur when the graph of the complex polynomial passes through the point \((0, 0)\). Symbolically, if \(z_{0}\) is a root of \(P\), \(P(z_{0}) = 0\).

Graph of the complex polynomial \(y = x^2 + 1\)

Let us now define \(P: \mathbb{C} \to \mathbb{C}\) where \(P(z) = z^2 + 1\) where I have made two concessions to convention by replacing \(x\) by \(z\) and \(f\) by \(P\).

As noted in the previous section, the input as well as the output of a complex polynomial is a two-dimensional vector. So, how do we plot a complex polynomial on a two-dimensional plane? Clearly, something must give. But what is it that gives?

We trade in our ability to see graphical picture of the entire function at once, as in Figure 1. We must view it in instalments. Let us see how to accomplish this.

How do plots of complex polynomials look like?

We know that for every \(r\), the plot of \(P(z)\) is a closed curve. And how many such plots are there? One for every value of \(r\). But that means an uncountable infinity of plots!

This is the tradeoff: we cannot see all of \(P(z)\) in one graph. We have to slice through the picture to isolate and display selected values of \(r\) to understand the behaviour of the curve.

Because \(\theta\) varies from \(0\) to \(2\pi\), the closed curve for \(r\) tells us what we need about \(\theta\).

The way to plot a complex polynomial in the two-dimensional complex plane is as a series of curves that show how they vary with \(\theta\) for any given value of \(r\). We index by \(r\) and allow \(\theta\) free rein to vary.

As a complex polynomial is plotted, it traces a path on the complex plane, much as an ant makes a trail as it crawls on a table. With the \((r, \theta)\) parameterization of the complex input, we start with \(r = 0\) and let it increase without limit. The \(\theta\) value is, of course, periodic: each successive revolution adds \(2\pi\) to its value.

Properties of complex polynomials

Roots are the values of \((r, \theta)\) for which \(P(z) = 0\). As \(r\) increases, the polynomial traces larger and larger curved paths on the complex plane. But, each and every root is defined to pass through the complex zero, \((0, 0)\).

So, rather than sweeping from left to right across the \(x\)-axis, as with a real polynomial, we watch the origin on the complex plane through an aperture whose radius keeps on increasing as the modulus of the input variable becomes larger. And rather than focusing on the intersection of a graph with a line, to find zeros, we focus on when the value of \(P(z)\) equals \(0\).

Repeated and conjugate roots

When a root \(\rho\) is repeated, the curve for \(r = \rho\) becomes a cusp at the origin.

Moreover, when complex conjugate roots exist, the curve splits into two symmetrical paths that both pass through the origin.

How do we plot complex functions?

A real-valued function of a real variable fits neatly into the two-dimensional graphical representation \(\mathbb{R}^2\) of Figure 1.

Since a complex number has two parts, graphing a complex-valued function of a complex variable means that we have to contend with a total of four variables: two input and two output. But we do not have four axes!

How do we accommodate complex polynomials into the two-dimensional plane? This dilemma was faced by mathematicians in the early days of complex function plotting.

If this question has never occurred to you, kindly pause awhile to ponder it. By asking such fundamental questions, you deepen your understanding of the subject, and also attune your thinking to how it evolved.

Injecting rigour into the notation

With two complex planes and four variables, there is ample scope for confusion and ambiguity when dealing with complex polynomials. Let us settle the notation unequivocally first. We enumerate the following:

1. Let the polynomial \(P\), of degree \(n \geq 1\), with real coefficients, map one complex number to another.

2. Let the output of \(P\) be single-valued rather than multi-valued.

3. Let the input complex plane be called the \(z\)-plane with \(z = x + iy\).

4. Let the output complex plane be called the \(w\)-plane with \(w = u + iv\).

5. Let the third vertical axis, if used, be called the \(\zeta\)-axis, or zeta axis, to avoid confusion with the input variable \(z\).

6. Then, \(P(x + iy) = P(z) = w = u + iv\).

We may now write down the following mathematical statements to avoid confusion about what is being referred to at any stage of the visualization of the complex polynomial. \[ \begin{aligned} P &: \mathbb{C} \to \mathbb{C}\\ P &: z \mapsto P(z)\\ P &: z \mapsto w\\ P &: (x + iy) \mapsto (u + iv)\\ P(z) &= a_nz^n + a_{n-1}z^{n-1} + \dots + a_1z + a_0 \text{, with $n \in \mathbb{Z}$ ; $n \geq 1$ and $a_i \in \mathbb{R}$}\\ P(z) &= P(x + iy)\\ &= w\\ &= u + iv \end{aligned} \qquad{(7)}\] As you can see, it is quite a mouthful, but we will not get into strife if we keep the variables dis-entangled as we go along.

Plotting complex functions to visualize them

Plotting complex functions is like trying to stuff too many things into a small suitcase. Something has to give. For example, instead of taking an extra pair of shoes, you might take just the pair you are wearing and save some space. The critical question is “What gives?”.

The book by Needham entitled Visual Complex Analysis [6] will prove invaluable for visualizing complex functions. Do refer to it if you get stuck or are confused.⁸

Physicists and engineers need to plot complex functions as part of their work. It is therefore a subject, not only of academic, but also practical importance.

Because I do not find it easy to visualize complex function plots, I will be taking detours to anchor in key concepts, and explain what happens in slow motion. Please bear with me.

Conformal mapping: the \(z\) and \(w\) planes

If there is a definitive, real McCoy representation of how a complex function is plotted showing both domain and co-domain, this is it. Taking the bull by the horns, we concede that four variables need two planes.

Traditionally, the input plane is called the \(z\)-plane, and its axes are the real and imaginary components of the input variables: \(z = x + iy\).

Likewise, the output plane is called the \(w\)-plane, and the real and imaginary parts of the output variables are called \(u\) and \(v\), i.e., \(w = u + iv\).

We plot the input on the \(z\)-plane on one side and the output on the \(w\)-plane on the other. Needless to say, the labels must clearly identify which plane we are looking at. Colours, arrows, or other devices may be used to match corresponding sets of inputs and outputs. But it does require the willingness and effort to visualize.

On the left, we have the rectangular grid of the input \(z\) plane. On the right, we have the rectangular grid of the output \(w\) plane. Then imagine that we feed the entire grid of values on the \(z\) plane as inputs to the polynomial \(P(z)\), and that we plot the resulting outputs on the rectangular \(w\) plane.

When \(P\) is continuous and differentiable, this process is called conformal mapping [6]. Lines that are orthogonal on the input plane retain their orthogonality on the output plane.

For the case of \(P(z) = z^2 + 1\), we will have two graphs side-by-side. On the left, the undisturbed rectangular co-ordinate grid of the \(z\) plane. On the right the values of \(P(z)\) plotted on the pristine \(w\) plane.

Note that both planes are just that: two-dimensional planes. There is no vertical axis on either: so, no extraneous variable to confuse the issue. Figure 2 illustrates this.

There are several noteworthy points about the above two graphs:

1. Both graphs depict the complex plane. The one on the left shows the input \(z\)-plane, while the one on the right shows the output \(w\)-plane.

2. The red-and-blue lines on the rectangular grid on the left shows the input. The red-and-blue lines on the right show the output.

3. The origin at \((0, 0)\) on the \(z\)-plane has been mapped to \((1, 0)\) on the \(w\)-plane.

4. The red-and-blue rectangular grid on the left has been warped by the polynomial \(P(z)\) into the red-and-blue parabolic grid superimposed on the \(w\)-plane. This is the clearest depiction of the action of the polynomial.

5. Although both the left and right images are on the same scale, the right image appears magnified in comparison to the original grid on the \(z\)-plane. This is due to squaring in the polynomial map.

6. The intersections of the lines in the transformed red-and-blue grid on the right are at ninety degrees just as in the original on the LHS.⁹ This feature of conformal mapping makes it valuable for solving practical problems.

Though they are fascinating, we do not need conformal mapping to prove the FTA, and so will now go on to other methods for plotting complex curves.

The Modulus Surface Plot and Cassini Contours

What can we do to reduce the plotting real estate and still get a good picture of what is going on? One solution is to use the third axis—the vertical axis on the \(z\)-plane, mutually perpendicular to the other two—and use it to represent some real-valued function of the output.

This hybrid setup, with the input variables on the base, complex plane and a real-valued function of the output on the vertical axis, allows us to better visualize some of the relationships we are trying to understand.

Let us use Equation 7 to re-orient ourselves to the variables: \[ \begin{aligned} P(x + iy) &= P(z)\\ &= w\\ &= u + iv. \end{aligned} \qquad{(8)}\] The absolute value, or magnitude, or modulus of a complex number, is a non-negative, real-valued function of the complex number. In our case, the modulus of \(P(z)\) is denoted and defined as: \[ \begin{aligned} \lvert P(z) \rvert &= \lvert w \rvert\\ &= \sqrt{u^2 + v^2}. \end{aligned} \] We may then collapse the output complex number \(P(z) = w\), with its two dimensions, to its one-dimensional modulus. This latter, being real and non-negative, may be easily represented on a single axis.

Thus, we could plot the input variables \(\Re(z)\) on the \(x\)-axis, \(\Im(z)\) on the \(y\)-axis, and the modulus of the output, \(\lvert P(z) \rvert\), on the vertical axis.¹⁰ We will call this third, vertical axis the \(\zeta\) axis to avoid confusion with \(z\).¹¹ The resulting surface is called the modulus surface. It is a topographic representation of the absolute value of the output complex number mapped over the plane of the input complex number.

The modular surface of the complex polynomial \(P(z) = z^2 + 1\) is shown in Figure 3. Yet another view of the same surface is shown in Figure 4.

Observe from Figures 3,4 the following:

1. The complex plane at the base of both figures is the input complex plane or \(z\) plane.

2. To avoid confusion with the \(z\)-plane, the vertical axis perpendicular to the base plane is called the \(\zeta\)-axis.

3. The modulus surface touches the \(x\)-\(y\) plane at only two points: the roots \(\pm i\).

4. The real parabola \(y = x^2 + 1\) is plotted and appears on the modulus surface. It does not ever contact the complex plane or the line \(y = 0\). This shows the consistency between the real-valued and complex-valued versions of the same equation.

5. The closed loops at the bottom of both figures are the topographic contour maps of the modulus surfaces. In our case, they are called Cassini ovals, and are illustrated in Figure 5.

Plotting the loci of constant \(r\)

Since it is the set \(\mathbb{C}\) that yields the hard won solutions to this equation, we must plot it on the complex plane to gain further insights. The Cartesian to polar parameterization of Equation 12 holds the key to manageable graphing of complex functions without losing track of what is happening.

Bear in mind that the \(x\)-axis represents the real numbers and the \(y\)-axis the imaginary numbers.

The third solution is to plot only the output variables on the \(w\)-plane because they are the ones holding the interesting information. Let the output speak for itself. We only need one pair of variables.

But merely plotting the output is meaningless if there is no link to the input. We could keep some aspect of the input fixed and see how the output varies. This is a compromise solution.

Also, what do we keep fixed, and what do we vary in the input? I invite the reader to take a short detour and read Appendix A at the end of this blog for one way to do this.

We need to condense the input and link it somehow to the output. One way is to fix the input and plot the output. Repeating this with several different inputs will give us a series of output curves. This is called parametriztion and is a well known device for parsimonious visualization.

The oft-adopted solution is to keep the modulus or absolute value, \(\lvert z \rvert = r\), of the input complex number fixed and plot the graph of the output on the complex plane. This is what we do below.

Let the \(P(z) = z^2 + 1\) be a complex polynomial that satisfies Equation 1. We already know that \[ \begin{aligned} P(z) &= z^2 + 1 \\ &= (z - i)(z + i)\\ \end{aligned} \] But the complex number \(z\), expressed in polar form is \(z = re^{i\theta}\). Then, \[ \begin{aligned} P(z) &= (re^{i\theta})^2 + 1\\ &= r^2e^{i(2\theta)} + 1\\ &= r^2(\cos2\theta + i\sin2\theta) + 1\\ &= r^2\cos2\theta + 1 + i(r^2\sin2\theta). \end{aligned} \qquad{(9)}\] If the above steps seem unfamiliar, read my blog A Tetrad of Captivating Problems where the complex exponentials are carefully explained.

When moving from the Cartesian \((a, b)\) representation of the complex number \((a + ib)\) to the polar \((r, \theta)\) representation of the same number as \(re^{i\theta}\), we trade in signed values for magnitudes and angles.

So, the roots \((0, i)\) and \((0, -i)\) simply become the polar coordinates \((0, \frac{\pi}{2})\) and \((0, -\frac{\pi}{2})\) respectively. Observe that for both roots, the magnitude, \(r = 1\). The circle with radius one touches the origin, \((0, 0 )\).

We may now set the real and imaginary parts to \(x\) and \(y\) respectively and we have our parameterization for the complex plane: \[ \begin{aligned} x &= r^2\cos2\theta + 1; \text{ and}\\ y &= r^2\sin2\theta. \end{aligned} \qquad{(10)}\]

The parametric plot put out by gnuplot is shown in Figure 6.¹² As stated above, the observant among you will notice that we have three concentric circles, all of whose centres have been shifted to the right of the origin by one unit.

We could have deduced the following from Equation 9 even without the plot:

1. The contours of constant \(r\) values are closed concentric circles.

2. The centres of the circles is offset from the origin and is at \((1, 0)\).

3. For every increase of \(2\pi\) in \(\theta\), the curve cycles through twice around the origin.

We get the different contours by allowing \(r \geq 0\) to increase freely and trace out the paths dictated by \(\theta\) as it does so. Each value of \(r\) gives rise to a single coloured curve in Figure 6.

The most intuitive way to understand what Figure 6 demonstrates is to think of the three \(r\) values as hoops trying to touch the origin of the complex plane, \((0, 0)\). The first hoop with \(r = 0.5\) falls short and does not reach the origin. The third hoop with \(r = 1.5\) over-reaches and goes beyond the origin. The second hoop with \(r = 1.0\) precisely captures the origin as it touches the complex origin \((0, 0)\). But what does this actually mean?

The changing radius indicates the changing absolute value of \(z\) in \(P(z)\). The function falls short of the solution for \(r = 0.5\), gives the exact solution for \(r = 1.0\), and overshoots the solution for \(r = 1.5\).

Contours of constant \(r\) are closed

Let \(P(z) = P(re^{i\theta})\) be a complex polynomial. We could choose to plot it by keeping the \(r\) value fixed and letting \(\theta\) vary. We could then repeat the process by changing the fixed value of \(r\) and letting \(\theta\) vary again. The result would be a series of contours of constant \(r\).

How does such a plot behave? Since the trigonometric functions are periodic, we expect the polar plot to repeat itself after one revolution. Consider adding one revolution of \(2\pi\) to any arbitrary argument \(\theta\): \[ \begin{aligned} P(re^{i(\theta + 2\pi)}) &= P(re^{i\theta}e^{i2\pi}); \text{ since $e^{i2\pi} = 1$,}\\ &= P(re^{i\theta}(1))\\ &= P(re^{i\theta}) \end{aligned} \]

Why only three curves?

Why only three curves? Why not more? That is a very astute question. The three curves we have chosen to display are actually snapshots out of an infinity of curves. This is the tradeoff I spoke about before. Imagine a stack of rings of different radii stacked one on top of another, like cross-sections of a cylinder, an inverted cone, or a parabola. That is what the polynomial surface actually looks like. We have chosen to display three of these rings.

But which is it? Cylinder, cone, or parabola? From Figure 1 we would guess that it should be a parabola.

Algebraic Closure

Algebraic closure means that the solution to a polynomial lies within the field over which it is defined. The complex field exhibits algebraic closure and this is the kernel of the FTA.

The definitive discovery of Gauss

Once the real number line had been completed with a definition of the irrationals, the next examination of “vacancies” belonged in the realm of the complex numbers. Fortunately, Carl Gauss banished the thought of numbers “above” the complex with his Fundamental Theorem of Algebra, the subject of this blog. All we have are white and black sheep; we do not have sheep of other colours.

Gauss and his proof

The Fundamental Theorem of Algebra (FTA) states that every non-constant polynomial of a single variable, with complex coefficients, must have at least one complex root. Here complex root also includes real roots.

This statement is also equivalent to saying that every non-constant polynomial, \(P(z)\), of degree \(n\) with complex coefficients, has exactly \(n\) complex roots, where repeated roots are counted as separate roots.

Gauss presented a proof in 1799 that relied on a geometric intuition that we will look at carefully in this blog. But because they could find gaps in the logic where Gauss had made a leap of the imagination, not all mathematicians agreed with his proof.

The branch of mathematics called [topology]](https://en.wikipedia.org/wiki/Topology) that deals with spatial properties had not been invented at the time Gauss presented his proof. It was only after Alexander Ostrowski filled the gap in 1920, that the mathematical community accepted the first proof of Gauss on the FTA.

This theorem allows mathematicians to heave a sigh of relief that no more unusual types of numbers are lying in wait to pounce upon them at an opportune moment in the future. It also puts paid to my concern about brown sheep, and sheep of other colours.

But how is this grand reassurance earned? A good start would be to extend the IVT to two dimensions and see a visual demonstration of the theorem.

I would now like to pause and take stock of what Gauss did when he adapted the IVT to a complex polynomial. These are the steps that he took through the then uncharted mathematical territory to tame the roots of complex polynomials.

Include all the mathematicians who have contributed to this.

Winding numbers

Gauss used the properties of winding numbers in his proof. The winding number “of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve’s number of turns” [7]. In our case, the point in question is the complex origin.

We already know that all complex-valued polynomials form closed curves on the complex plane when they are parametized by the absolute value¹³ \(r\) of the complex number, and its argument, \(\theta\).

The following properties of the winding number are relevant to our discussion:

1. The complex origin, \((0, 0)\) is the point around which the windings are counted for the FTA.

2. The winding number is an integer that is equal to or greater than zero.

3. The winding number of the curve for any value \(r\) equals the number of roots it encloses.

4. The winding number is cumulative at any point on the plot, i.e., it does not decrease with increasing \(r\).

5. For a polynomial of degree \(n\), the winding number stabilizes at \(n\) after all \(n\) roots have been enclosed, and remains so thereafter.

The winding number rings a note of finality to the quest for the roots of a complex polynomial of degree \(n\). After \(n\) roots have been found, no others exist to be found. And this is a vital finding in the quest to tame complex polynomials.

Polynomials are continuous functions. If you graph a continuous function on paper, as the independent variable increases in value, you can trace the associated curve without lifting the pencil off the paper. This intuitive explanation of continuity is sufficient for our purpose in this blog.

Because \(P(z)\) is a continuous polynomial, it must cross the origin for some \(r\). If small values under-reach it and large values over-reach it, for some value in between the polynomial must evaluate to zero. This capturing the solution on the complex plane is the equivalent of the IVT for the real line.

The technique we have used is the same as what the epsilon-delta definition does to trap a limit within a box,¹⁴ or the squeeze theorem in a similar context.

A second polynomial

For a second example of plotting a polynomial with real coefficients on the complex plane, we free to construct our own polynomial. To cover the spectrum of possibilities, we choose the polynomial to have one real root, two repeated real roots, and one pair of complex conjugate roots.¹⁵

Let us define our polynomial to be \[ \begin{aligned} P(z) &= (z - 1)(z - 3)^2(z - 2i)(z + 2i)\\ &= z^5 - 7z^4 + 19z^3 - 37z^2 + 60z - 36. \end{aligned} \]

We can repeat what we did with the previous example. We may plot the polynomial on the two-dimensional real plane \(\mathbb{R}^2\).

Plotting P(z) on the complex plane

The complex polynomial with real coefficients, \[ P(z) = z^5 - 7z^4 + 19z^3 - 37z^2 + 60z - 36 \] may be plotted on the two-dimensional complex plane as before. Only this time—because we constructed this polynomial from its roots—we have:

1. one real root at \(z = (1, 0)\);
2. one repeated real root at \(z = (3, 0)\); and
3. one pair of complex conjugate roots at \(z = (0, \pm 2i)\).

First, we must derive the polar form of \(P(z)\), by substituting \(z = re^{i\theta}\) into \(P(z)\): \[ P(re^{i\theta}) = r^5 e^{5i\theta} - 7r^4 e^{4i\theta} + 19r^3 e^{3i\theta} - 37r^2 e^{2i\theta} + 60r e^{i\theta} - 36 \] We then expand this using \(e^{in\theta} = \cos(n\theta) + i\sin(n\theta)\) and separate the real and imaginary parts: \[ \begin{aligned} x &= r^5\cos(5\theta) - 7r^4\cos(4\theta) + 19r^3\cos(3\theta) - 37r^2\cos(2\theta) + 60r\cos(\theta) - 36 \\ y &= r^5\sin(5\theta) - 7r^4\sin(4\theta) + 19r^3\sin(3\theta) - 37r^2\sin(2\theta) + 60r\sin(\theta) \end{aligned} \qquad{(11)}\]

Behaviour as r varies

How does this polynomial behave at very small or very large values of \(r\)? For \(0 < r \ll 1\), the terms involving \(z\) or its higher powers will vanish and only the constant, \(36\) will remain. For very large values of \(r\), we expect the term involving \(z^5\) to dominate. The order of magnitude will be \(r^5\).

The plot of \(P(z)\) around the roots, and for very small values, and very large values, are orders of magnitude apart, and difficult to capture on the same plot. So, we have to show them in separate plots for meaningful comparison.

The plot for small values of \(r\), near the first root, is shown in Figure 8.

These curves may also be called phase portraits and they are used to convey a static picture of a dynamic process, as often happens in mathematics and physics. Imagine a crawling ant tracing out the curve as \(\theta\) varies for the complex numbers with polar coordinates \((r, \theta)\). These curves are also called algebraic curves.

Wilf: polynomial: 2D windings around the origin equal to intersection of x-axis

Compare two plots of the same polynomial. One conventional and the other of winding numbers.

Conclusion: at least one complex root.

Along the way, the idea of one number being larger or smaller than another was lost.¹⁶

Galois’ Bridge from Polynomials to Groups

Work out (x-alpha)(x - beta) and see where it leads to.

Equate alpha and beta to coefficients. Sum and product formulae drop out.

Distinction between arithmetic and algebraic closure

My 4 questions.

Sand and water: real and imaginary

💡 Implementation Notes for your Draft:

1. Zero External Dependencies: The code relies entirely on native browser canvas elements and vanilla JavaScript APIs. It requires no external framework downloads, keeping your parsed HTML page lightning fast.
2. Mathematical Layout Match: The trigonometry within the canvas rendering pipeline maps explicitly to the algebraic formulation of your polar representation paragraphs: \(x = r^2\cos(2\theta) + 1\) and \(y = r^2\sin(2\theta)\) where \(r=1\).
3. Standalone Encapsulation: The JavaScript runs wrapped in an Immediately Invoked Function Expression (IIFE). This guarantees that none of the animation variables pollute or interfere with any other global script scopes running elsewhere on your web page template.

Good luck with updating your draft post! Let me know if you want to add higher degree terms (like mapping a winding number of \(n\)) or need help checking the Pandoc markdown layout syntax for the final compile sequence tomorrow.

Tie-into Riemann surfaces

This is one solution that works well in certain situations. See the section called logarithms of complex numbers in a previous blog of mine.

Appendix A: The Cartesian-Polar equivalences

We could use polar coordinates, specifically applied to complex numbers, to find a way out of the complex to complex plotting logjam. The two equivalent forms of representing a complex number are the Cartesian and polar forms. As noted in a previous blog \[ \begin{aligned} z &= a + ib\\ &= re^{i\theta}\\ &= r(\cos\theta + i\sin\theta)\\ \theta &= \arg(z) = \arctan\left({\frac{b}{a}}\right)\\ r &= \lvert z \rvert = \sqrt{a^2 + b^2} \end{aligned} \qquad{(12)}\] I will refer collectively to these equations as Equation 12.

A square requires two points and a side to specify itself unambiguously. A circle requires only a centre and a radius. The circle is therefore preferable to a square when it comes to parsimony in specification. That is one reason why the \((r, \theta)\) parametrization comes to the rescue when we are in a fix on the Cartesian plane.

“Galois theory is a study of the symmetries both of roots and of their field extensions.” Nathan Carter, Visual Group Theory, Chapter 10.

Scripts for graphical output

The graphs in this blog were generated programmatically. Click on the icon of the graph of interest to you to download the script that generated it.

Acknowledgements

Feedback

Please email me your comments and corrections.

A PDF version of this article is available for download here:

References

•  Galois (1)
•  polynomial (1)
•  roots (1)
•  topology (1)