(I) Introduction

Definition

A complex number is a number written in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is a formal symbol satisfying \( i^2 = -1 \).

  • \( a \) is the real part of \( z \), denoted by \(\text{Re } z\).
  • \( b \) is the imaginary part of \( z \), denoted by \(\text{Im } z\).

A real number \( a \) is considered as a special type of complex number by identifying \( a \) with \( a + 0i \).

Conjugate and Modulus:

Some Properties

  1. \( \overline{z} = z \) if and only if \( z \) is a real number.
  2. \( \overline{w + z} = \overline{w} + \overline{z} \)
  3. \( \overline{wz} = \overline{w}\overline{z} \)
    (In particular, \( \overline{rz} = r\overline{z} \) if \( r \) is real)
  4. \( z\overline{z} = |z|^2 \ge 0 \)
  5. \( |wz| = |w||z| \)
  6. \( |w + z| \le |w| + |z| \) (Triangle Inequality)
  7. If \( z \ne 0 \), then \( |z| > 0 \) and \( z \) has a multiplicative inverse denoted by \( \frac{1}{z} \) or \( z^{-1} \) and given by \( \frac{1}{z} = \frac{\overline{z}}{|z|^2} \).
Example 1

Let \( w = 3 + 4i \) and \( z = 5 - 2i \). Compute \( z\overline{z} \), \( |z| \), and \( \frac{w}{z} \).

Solution:

\[ z\overline{z} = (5 - 2i)(5 + 2i) = 5^2 + (-2)^2 = 25 + 4 = 29 \]
\[ |z| = \sqrt{z\overline{z}} = \sqrt{29} \]
\[ \frac{w}{z} = \frac{3 + 4i}{5 - 2i} = \frac{(3 + 4i)(5 + 2i)}{(5 - 2i)(5 + 2i)} = \frac{15 + 6i + 20i + 8i^2}{29} = \frac{15 + 26i - 8}{29} = \frac{7 + 26i}{29} = \frac{7}{29} + \frac{26}{29}i \]

(II) Geometric Interpretation and Polar System

\( z = a + bi \) corresponds to a point \( (a, b) \) in the plane \( \mathbb{R}^2 \).

To give a graphical representation of complex multiplication, we use polar coordinates in \( \mathbb{R}^2 \).

Given a nonzero complex number \( z = a + bi \):

Let \( \varphi \) be the angle between the positive real axis and the point \( (a, b) \). This is the principal value of the argument of \( z \), where \( -\pi < \varphi \le \pi \).

A geometric diagram of a complex number in polar coordinates. The horizontal axis is labeled 'Real axis' and the vertical axis is labeled 'Imaginary axis'. A blue arrow originates from the origin (0,0) and ends at a point in the first quadrant labeled z = a + bi. Dashed lines project from this point down to the real axis at a distance labeled 'a', and horizontally to the imaginary axis at a distance labeled 'b'. The length of the blue vector is labeled r = |z|. A red arc near the origin measures the angle phi from the positive real axis to the vector, illustrating Euler's formula representation.
Figure: Geometric Interpretation of a Complex Number in Polar Coordinates
\[ a = |z|\cos\varphi \] \[ b = |z|\sin\varphi \] \[ |z| = \sqrt{a^2 + b^2} \] \[ \tan\varphi = \frac{b}{a}, \quad -\pi < \varphi \le \pi \]

We can write \( z \) as:

\[ z = a + bi = |z|(\cos\varphi + i\sin\varphi) \]

Euler's Formula

\[ e^{i\varphi} = \cos\varphi + i\sin\varphi \]

This allows us to write complex numbers in exponential form:

\[ z = |z|e^{i\varphi} \]

Let \( z = |z|e^{i\varphi} \) and \( w = |w|e^{i\theta} \). The rules for multiplication, division, and powers are:

\[ wz = |w||z|e^{i(\varphi + \theta)} = |w||z|(\cos(\varphi + \theta) + i\sin(\varphi + \theta)) \] \[ \frac{z}{w} = \frac{|z|}{|w|}e^{i(\varphi - \theta)} = \frac{|z|}{|w|}(\cos(\varphi - \theta) + i\sin(\varphi - \theta)) \] \[ z^k = |z|^k e^{i(k\varphi)} = |z|^k(\cos(k\varphi) + i\sin(k\varphi)) \] \[ \sqrt[k]{z} = \sqrt[k]{|z|}e^{i(\frac{\varphi}{k})} = \sqrt[k]{|z|}(\cos(\frac{\varphi}{k}) + i\sin(\frac{\varphi}{k})) \]
Example 2

Find all real and complex roots of the equation \( x^5 = 32 \).

Solution:

We can write \( 32 \) in polar form as \( 32 = 32e^{2\pi k i} \), where \( k \) is any integer.

\[ x = \sqrt[5]{32}e^{\frac{2\pi k i}{5}} = 2e^{\frac{2\pi k i}{5}} \]

We restrict the angle \( \frac{2\pi k}{5} \) to the principal range \( (-\pi, \pi] \). This gives the integer values \( k = -2, -1, 0, 1, 2 \).

The 5 roots are:

  • \( k = -2 \implies x = 2e^{-\frac{4\pi}{5}i} \)
  • \( k = -1 \implies x = 2e^{-\frac{2\pi}{5}i} \)
  • \( k = 0 \implies x = 2e^{0} = 2 \)
  • \( k = 1 \implies x = 2e^{\frac{2\pi}{5}i} \)
  • \( k = 2 \implies x = 2e^{\frac{4\pi}{5}i} \)

(III) Theorem on Conjugate Pair Zeros of a Polynomial

If a polynomial \( f(x) \) of degree \( n \) has real coefficients and if \( z = a + bi \) with \( b \ne 0 \) is a complex zero of \( f(x) \), then the conjugate \( \overline{z} = a - bi \) is also a zero of \( f(x) \).

Example 3

Find all real and complex eigenvalues of the matrix:

\[ A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -5 & 5 \end{bmatrix} \]

Solution: Find the characteristic polynomial \(\det(A - \lambda I) = 0\):

\[ \begin{aligned} \det(A - \lambda I) &= \begin{vmatrix} 1-\lambda & 0 & 0 \\ 0 & 3-\lambda & 2 \\ 0 & -5 & 5-\lambda \end{vmatrix} \\ &= (1-\lambda)\begin{vmatrix} 3-\lambda & 2 \\ -5 & 5-\lambda \end{vmatrix} \\ &= (1-\lambda)[(3-\lambda)(5-\lambda) - (-10)] \\ &= (1-\lambda)(\lambda^2 - 8\lambda + 15 + 10) \\ &= (1-\lambda)(\lambda^2 - 8\lambda + 25) = 0 \end{aligned} \]

From \( 1 - \lambda = 0 \), we get \( \lambda = 1 \).

From \( \lambda^2 - 8\lambda + 25 = 0 \), use the quadratic formula:

\[ \lambda = \frac{8 \pm \sqrt{64 - 100}}{2} = \frac{8 \pm \sqrt{-36}}{2} = \frac{8 \pm 6i}{2} = 4 \pm 3i \]

The eigenvalues are \( \lambda = 1, 4 + 3i, 4 - 3i \).

Notice that nonreal complex eigenvalues always occur in conjugate pairs.