## Section4.3Cyclic Groups of Complex Numbers

The *complex numbers* are defined as

where \(i^2 = -1\text{.}\) If \(z = a + bi\text{,}\) then \(a\) is the *real part* of \(z\) and \(b\) is the *imaginary part* of \(z\text{.}\)

To add two complex numbers \(z=a+bi\) and \(w= c+di\text{,}\) we just add the corresponding real and imaginary parts:

\begin{equation*} z + w=(a + bi ) + (c + di) = (a + c) + (b + d)i. \end{equation*}Remembering that \(i^2 = -1\text{,}\) we multiply complex numbers just like polynomials. The product of \(z\) and \(w\) is

\begin{equation*} (a + bi )(c + di) = ac + bdi^2 + adi + bci = (ac -bd) +(ad + bc)i. \end{equation*}Every nonzero complex number \(z = a +bi\) has a multiplicative inverse; that is, there exists a \(z^{-1} \in {\mathbb C}^\ast\) such that \(z z^{-1} = z^{-1} z = 1\text{.}\) If \(z = a + bi\text{,}\) then

\begin{equation*} z^{-1} = \frac{a-bi}{ a^2 + b^2 }. \end{equation*}The *complex conjugate* of a complex number \(z = a + bi\) is defined to be \(\overline{z} = a- bi\text{.}\) The *absolute value* or *modulus* of \(z = a + bi\) is \(|z| = \sqrt{a^2 + b^2}\text{.}\)

There are several ways of graphically representing complex numbers. We can represent a complex number \(z = a +bi\) as an ordered pair on the \(xy\) plane where \(a\) is the \(x\) (or real) coordinate and \(b\) is the \(y\) (or imaginary) coordinate. This is called the *rectangular* or *Cartesian* representation. The rectangular representations of \(z_1 = 2 + 3i\text{,}\) \(z_2 = 1 - 2i\text{,}\) and \(z_3 = - 3 + 2i\) are depicted in Figure Figure 4.3.2.

Nonzero complex numbers can also be represented using *polar coordinates*. To specify any nonzero point on the plane, it suffices to give an angle \(\theta\) from the positive \(x\) axis in the counterclockwise direction and a distance \(r\) from the origin, as in Figure Figure 4.3.3. We can see that

Hence,

\begin{equation*} r = |z| = \sqrt{a^2 + b^2} \end{equation*}and

\begin{align*} a & = r \cos \theta\\ b & = r \sin \theta. \end{align*}We sometimes abbreviate \(r( \cos \theta + i \sin \theta)\) as \(r \cis \theta\text{.}\) To assure that the representation of \(z\) is well-defined, we also require that \(0^{\circ} \leq \theta \lt 360^{\circ}\text{.}\) If the measurement is in radians, then \(0 \leq \theta \lt2 \pi\text{.}\)

The polar representation of a complex number makes it easy to find products and powers of complex numbers. The proof of the following proposition is straightforward and is left as an exercise.

##### Proposition4.3.5

Let \(z = r \cis \theta\) and \(w = s \cis \phi\) be two nonzero complex numbers. Then

\begin{equation*} zw = r s \cis( \theta + \phi). \end{equation*}##### Theorem4.3.7DeMoivre

Let \(z = r \cis \theta\) be a nonzero complex number. Then

\begin{equation*} [r \cis \theta ]^n = r^n \cis( n \theta) \end{equation*}for \(n = 1, 2, \ldots\text{.}\)

The multiplicative group of the complex numbers, \({\mathbb C}^*\text{,}\) possesses some interesting subgroups. Whereas \({\mathbb Q}^*\) and \({\mathbb R}^*\) have no interesting subgroups of finite order, \({\mathbb C}^*\) has many. We first consider the *circle group*,

The following proposition is a direct result of Proposition Proposition 4.3.5.

##### Proposition4.3.9

The circle group is a subgroup of \({\mathbb C}^*\text{.}\)

Although the circle group has infinite order, it has many interesting finite subgroups. Suppose that \(H = \{ 1, -1, i, -i \}\text{.}\) Then \(H\) is a subgroup of the circle group. Also, \(1\text{,}\) \(-1\text{,}\) \(i\text{,}\) and \(-i\) are exactly those complex numbers that satisfy the equation \(z^4 = 1\text{.}\) The complex numbers satisfying the equation \(z^n=1\) are called the *\(n\)th roots of unity*.

##### Theorem4.3.10

If \(z^n = 1\text{,}\) then the \(n\)th roots of unity are

\begin{equation*} z = \cis\left( \frac{2 k \pi}{n } \right), \end{equation*}where \(k = 0, 1, \ldots, n-1\text{.}\) Furthermore, the \(n\)th roots of unity form a cyclic subgroup of \({\mathbb T}\) of order \(n\)

A generator for the group of the \(n\)th roots of unity is called a *primitive \(n\)th root of unity*.