Gauss and Regular Polygons: Complex Numbers

Introduction to Complex Numbers

Complex numbers are not really complex! In fact they are reasonably simple to understand and operate upon. The concept is definitely strange on a first look, but is damn powerful and has diverse ramifications in various branches of mathematics. Now, to illustrate the point that these numbers are really simple, we are gonna define them in terms of quantities already known.

In coordinate geometry a point is referred to using its coordinates in the form $(x, y)$. Thus to every point in the plane there corresponds a unique pair of real numbers and vice versa. Complex numbers are these ordered pairs of real numbers with certain operations defined on them. The formal definitions for these operations are as follows:

A complex number is an ordered pair of real numbers denoted by $(x, y)$ with the operations  of additions and multiplication defined on them as follows:
\begin{align} (a, b) + (c, d) &= (a + c, b + d)\notag\\ (a, b) \cdot (c, d) &= (ac - bd, ad + bc)\notag \end{align} Moreover $x$ is called the real part and $y$ is called the imaginary part of the complex number $(x, y)$. When using a single letter $z$ to denote a complex number, its real part is denoted by $\Re(z)$ and the imaginary part is denoted by $\Im(z)$.

With these definitions of addition and multiplication, it is easy to check that the complex numbers of the form $(x, 0)$ behave exactly the same as the real number $x$ with regard to these operations. Thus we have \begin{align} (x, 0) + (y, 0) &= (x + y, 0)\notag\\ (x, 0) \cdot (y, 0) &= (xy, 0)\notag \end{align} Therefore the numbers of the form $(x, 0)$ are identified with the real numbers $x$ and thus the set $\mathbb{C}$ of all complex numbers properly contains the set $\mathbb{R}$ of real numbers.

It is also easy to see that for any non-zero complex number $z = (x, y)$(meaning at least one of the $x$ and $y$ is non-zero) there is complex number $$z' = \left(\frac{x}{x^{2} + y^{2}}, \frac{-y}{x^{2} + y^{2}}\right)$$ such that $zz' = z' z = (1, 0) = 1$ and this is called the reciprocal of $z$ and denoted by $1 / z$ or $z^{-1}$.

It can be easily verified by the reader that the complex numbers form a field under the operations defined on them with $(1, 0) = 1$ as the multiplicative identity and $(0, 0) = 0$ as the additive identity.

The Imaginary Unit $i$

Lets the consider the complex number $i = (0, 1)$. Clearly we have $$i^{2} = ii = (0, 1) \cdot (0, 1) = (-1, 0) = -1$$ so that the complex number $i$ acts as the square root of $-1$!. This number is called the imaginary unit and using it any complex number $z = (x, y)$ can be written in the form $$(x, y) = (x, 0) + (0, y) = (x, 0) + (0, 1)\cdot(y, 0) = x + iy$$ Henceforth the complex numbers will be written in this form (one writes $x + yi$ if $y$ is some numeric value like $2.3$).

Argand Diagram

Since complex numbers are just ordered pair of real numbers they can be used to represent points in a plane as in coordinate geometry. It turns out that this is very fruitful and the source of all the usefulness of complex numbers in mathematics. The coordinate plane is then called the complex plane or the Argand Diagram, X-axis is called the real axis and Y-axis is called imaginary axis.

In this setting if $P$ is a point in the complex plane then the complex number $z$ representing $P$ acts as the vector $\overrightarrow{OP}$ where $O$ is the origin. The rule of addition of complex numbers matches exactly the rule of addition of vectors.

Fig 1. Geometrical Interpretation of Addition of Complex Numbers

The geometrical interpretation of multiplication of complex numbers requires us to cast the complex number $z = x + iy$ in a different form. If $P$ is the point corresponding to $z$ and $OP$ makes angle $\theta$ with the real axis then $$x = r\cos\theta, \,\,\, y = r\sin\theta$$ where $r$ is the length of line segment $OP$. Consequently we have $$r^{2} = x^{2} + y^{2},\,\,\, \tan\theta = \frac{y}{x}$$ and then the complex number can be written in terms of parameters (actually called polar coordinates and the form called polar form) $r$ and $\theta$ as $$z = x + iy = r(\cos\theta + i\sin\theta)$$ where $r$ is called the modulus of $z$ and denoted by $|z|$ and $\theta$ is called the amplitude or argument of $z$ and denoted as am(z) or $\arg(z)$.

Fig 2. Polar Form of Complex Numbers

Multiplication in this form is easily seen to be $$r_{1}(\cos\theta_{1} + i\sin\theta_{1}) \cdot r_{2}(\cos\theta_{2} + i\sin\theta_{2}) = r_{1}r_{2}(\cos(\theta_{1} + \theta_{2}) + i\sin(\theta_{1} + \theta_{2}))$$ so that the moduli get multiplied and the arguments get added during multiplication. The number $z = r(\cos\theta + i\sin\theta)$ is also written in the form $$z = re^{i\theta}$$ which is just a formal notation which helps us remember the above product rule. (One does not need to think that some kind of imaginary power to a base $e$ is involved here.)

Therefore in geometrical language, multiplying a given complex number $z$ represent point $P$ by another complex number $r(\cos\theta + i\sin\theta)$ amounts to scaling the vector $\overrightarrow{OP}$ by a factor of $r$ and then rotating it in counterclockwise direction by angle $\theta$.

The Complex Number $\cos\theta + i\sin\theta$

The number $z = x + iy = \cos\theta + i\sin\theta$ represents a point on the unit circle $x^{2} + y^{2} = 1$. If $\theta$ takes all the values from $0$ to $2\pi$, $z$ traverses the full circle. Let us now locate a point $P_{0}$ on the unit circle such that the it represent the complex number $1$. Let the point represented by $\cos\theta + i\sin\theta$ be called $P_{1}$.
Fig 3. Constructing a Polygon

Similarly we represent points $P_{k}$ corresponding to the complex number $z^{k} = \cos(k\theta) + i\sin(k\theta)$ for each value of $k = 1, 2, 3, \ldots$. From the construction it is easily seen that the radial vectors $\overrightarrow{OP_{k}}$ are equally spaced out with the angle between consecutive radial vectors being $\theta$. It clearly follows that all the lengths $P_{k}P_{k + 1}$ are equal. If for some value of $k$, say $k = n$ the radial vector $\overrightarrow{OP_{k}}$ turns out to be the same as the initial radial vector $\overrightarrow{OP_{0}}$ then we have a regular polygon $P_{0}P_{1}P_{2}\ldots P_{n - 1}P_{0}$ of $n$ sides inscribed in the unit circle. Since $\overrightarrow{OP_{n}} = \overrightarrow{OP_{0}}$ we get \begin{align} &z^{n} = 1\notag\\ &\Rightarrow \cos(n\theta) + i\sin(n\theta) = 1\notag\\ &\Rightarrow \cos(n\theta) = 1,\,\,\, \sin(n\theta) = 0\notag \end{align} The simplest solution (apart from $\theta = 0$) for the above equations is $\theta = 2\pi / n$.

It now follows that construction of a regular polygon of $n$ sides is intimately connected with finding a complex number $z$ such that $z^{n} = 1$. Such numbers are called $n^{th}$ roots of unity and the next post will deal exclusively with them and the polynomial equations satisfied by them.