1.1 Complex Numbers and Their Properties

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Basic notions

In Ruler’s notation, the imaginary unit number is

(1)   \begin{equation*} i^2 = -1\end{equation*}

Also, a complex number is an expression of the form

(2)   \begin{equation*} z = x + iy\end{equation*}

Here, we use {\rm Re}(z) represent the real part x of z; and {\rm Im}(z) represent the imaginary part y of z.

If x = 0, we say that z is pure imaginary; on the contrary, if y = 0, we say that z is real.

We often denote an element of the complex number as z \in \mathbb{C} and the real number is denoted by x \in \mathbb{R}.

Complex plane

We represent Equation.(2) in a two-dimensional coordinate system and we call it Complex plane. See below.

The complex plane (z plane)

In the above image, the horizontal axis represents the real number, and the vertical axis represents the imaginary number.

From the above image we can see that x = r\cos \theta, y = r \sin \theta, and \tan \theta = \frac{y}{x}.

Obviously, \theta = \alpha \pi + \arctan(\frac{y}{x}) + 2\pi n, where n is any integer and the quadrant has to be specified, so as to be able to determine whether \alpha =0 or 1.

So, z can be written in the polar form

(3)   \begin{equation*} z = r \cos \theta + i r \sin \theta = r(\cos \theta + i \sin \theta)\end{equation*}

Also, the radius r is

r = \sqrt{x^{2} + y^{2}} \equiv |z|.

  1. |z| is the length of the vector associated with z and is often referred to as the modulus of z.
  2. The angle \theta is called the argument of z. It is denoted by \underline{{\rm arg} z}

Note:

  1. \underline{\theta \equiv {\rm arg} z}
  2. If {\rm z \neq 0}, there is a \theta in the interval \theta_0 \leq \theta < \theta_0 + 2\pi where \theta_0 is an arbitrary number; others differ by integer multiples of 2\pi
    For example if z = 1+ i, then |z| = r = \sqrt{2}, {\quad} \theta = \frac{\pi}{4} + 2n\pi, {\quad} n = 0, \pm 1, \pm 2, ...

Polar Exponential

The polar exponential is defined by:

\cos \theta + i \sin \theta = e^{i\theta}

From Equation.(3) we know that z can be written in

(4)   \begin{equation*} z = re^{i\theta}\end{equation*}

From the properties of standard properties of elementary calculus, we have the following forms:

    \[e^{i 2\pi } = 1     \qquad     e^{i \pi } = -1 \]

    \[e^{\frac{i \pi }{2}} = i \qquad e^{\frac{i 3 \pi }{2}} = -i\]

As shown in the following:

Complex conjugate

If z \in \mathbb{C}, then the complex conjugate \overline{z} = x-iy = re^{-i\theta} = r(\cos \theta - i \sin \theta) = r[\cos{(-\theta)} + i \sin{(-\theta)}]

So, two complex numbers are equal if and only if their real and imaginary parts are respectively equal. Besides, the complex conjugate \overline{z} of z is found in the Gaussian complex plane by reflecting z with respect to the real axis.

Note that

    \[\begin{aligned}z \overline{z} &= (x+ iy)(x -iy) \\                       &=x^2 + y^2 \\                       &=|z|^2\end{aligned}\]

This is useful for the division of two complex numbers:

    \[\begin{aligned}\frac{z_1}{z_2} &= \frac{z_1 \overline{z_2}}{z_2 \overline{z_2}} = \frac{(x_1 +i y_1) (x_2- iy_2)}{(x_2 +i y_2) (x_2- iy_2)} \\                          &= \frac{z_1 \overline{z_2}}{|z_2|^2} = \frac{(x_1 x_2 + y_1 y_2) + i(x_2 y_1 -x_1 y_2)}{x_{2}^2 + y_{2}^2} \\                          &= \frac{x_1 x_2 +y_1 y_2}{x_{2}^2 + y_{2}^2} + \frac{i(x_2 y_1 - x_1 y_2)}{x_{2}^2 +y_{2}^2}\end{aligned}\]



Theorem 1.1 For any z_1, z_2 \in \mathbb{C}, we have |z_1 + z_2| \leq |z_1| + |z_2|, and |z_1 - z_2| \geq | |z_1| -|z_2| |.