2.1. Complex Numbers

Definition 2.1 (Imaginary unit \(\i\))

The imaginary unit \(\i\) is defined as that \(\i^2=-1\).

For example, if we have the following equation

\[z^2=-4,\]

the roots will be

\[z=\pm\sqrt{-4}=\pm\sqrt{4\i^2}=\pm 2 \i.\]

Definition 2.2 (Complex number and Complex conjugate)

A complex number is a number in the form of

(2.1)\[\begin{equation} z=x+\i y,\qquad x \in \R, y\in \mathbb{R}, \end{equation}\]

where \(x\) is the real component of the complex number and \(\i y\) is its imaginary component. We often call \(y\) the imaginary component for simplicity. The modulus of the complex number is calculated as

(2.2)\[\begin{equation} |z|=\sqrt{x^2+y^2}. \end{equation}\]

Its complex conjugate is defined as

(2.3)\[\begin{equation} \bar{z}=x-\i y. \end{equation}\]

For example \(z_1= 2+ 3\i\) and \(z_2=2-3\i\) are two complex numbers, and they are each other’s complex conjugate.

Definition 2.3 (Complex Arithmetic)

The arithmetic operations on two complex numbers given as \(z_1=x_1+\i y_1\) and \(z_2=x_2+\i y_2\) are defined as

(2.4)\[\begin{align} \text{``$+$''}:\quad ~ & ~ z_1+z_2=(x_1+\i y_1)+(x_2+\i y_2)=(x_1+x_2)+\i (y_1+y_2), \\ \text{``$-$''}:\quad ~ & ~ z_1-z_2=(x_1+\i y_1)-(x_2+\i y_2)=(x_1-x_2)+\i (y_1-y_2), \\ \text{``$\times$''}:\quad ~ & ~ z_1\cdot z_2=(x_1+ \i y_1)\cdot(x_2+\i y_2)=(x_1 x_2 - y_1 y_2)+\i (x_1 y_2+x_2 y_1), \\ \text{``$\div$''}:\quad ~ & ~ \frac{z_1}{z_2}=\frac{(x_1+\i y_1)}{(x_2+\i y_2)}=\frac{(x_1+\i y_1)(x_2-\i y_2)}{(x_2+\i y_2)(x_2-\i y_2)}=\frac{(x_1+\i y_1)(x_2-\i y_2)}{x_2^2 + y_2^2}. \end{align}\]

Example 2.1

Applying four arithmetic operations on \(1+2\i\) and \(2-5\i\), calculate

• \((1+2\i) + (2-5\i)\)

• \((1+2\i) - (2-5\i)\)

• \((1+2\i) \times (2-5\i)\)

• \(\dfrac{1+2\i}{2-5\i}\)

Solution

\[\begin{align*} & (1+2\i) + (2-5\i)= 3-3\i \\ & (1+2\i) - (2-5\i)= -1+7\i \\ & (1+2\i) \times (2-5\i)= 2-5\i +4\i -10\i^2=12-\i \\ & \frac{1+2\i}{2-5\i}=\frac{(1+2\i)(2+5\i)}{(2-5\i)(2+5\i)}=\frac{-8+9\i}{29} \end{align*}\]

Definition 2.4 (Complex number set)

We denote the set of all the complex numbers as

(2.5)\[\begin{equation} \mathbb{C}=\{z|z=x+\i y, ~x\in \R, y\in \mathbb{R}\}. \end{equation}\]

For the real number set \(\mathbb{R}\), we can use the number line to represent it. For the complex number set \(\mathbb{C}\), we can use the complex plane to represent it.

Fig. 2.1 The complex plane \(\mathbb{C}\).

For \(z=x+\i y\), we can also write it in the second form (polar form) as

\[ z = r\left(\cos \theta + \i \sin \theta\right) \]

or the third form (exponential form)

\[ z = r e^{\i \theta}, \]

where \(\theta\) is called the argument of \(z\), defined by the real axis and the straight line connecting the origin and \(z\) in the complex plane, and \(r=|z|\).

Remark 2.1 (Three forms of a complex number)

• Standard form: \(z=x+\i y\)

• Polar form: \(z=r(\cos\theta+\i\sin\theta)\)

• Exponential form: \(z=r e^{\i \theta}\)

where \(r=|z|=\sqrt{x^2+y^2}\), \(\tan \theta = \dfrac{y}{x}\).

If we use the exponential form to represent complex numbers, then the multiplication and division of two complex numbers can be expressed in simpler ways. Let \(z_1= r_1 e^{\i \theta_1}\) and \(z_2 = r_2 e^{\i \theta_2}\), then

\[\begin{split} \begin{align} & z_1 \cdot z_2 = r_1 e^{\i\theta_1} \cdot r_2 e^{\i\theta_2}=r_1 r_2 e^{\i(\theta_1+\theta_2)}\\ & \frac{z_1}{z_2} = \frac{r_1 e^{\i\theta_1}}{r_2 e^{\i\theta_2}}=\frac{r_1}{r_2}e^{\i(\theta_1-\theta_2)} \end{align} \end{split}\]