Complex Numbers and Properties

1. The Field of Complex Numbers $\mathbb{C}$

The set of complex numbers, denoted by $\mathbb{C}$, extends the real number system $\mathbb{R}$ to allow for the solution of equations such as $x^2 + 1 = 0$.

• The Imaginary Unit: Defined as $i$, where $i^2 = -1$.
• Standard (Cartesian) Form: A complex number $z$ is written as:
  $$z = a + bi$$
  where $a, b \in \mathbb{R}$.
• Real Part: $\text{Re}(z) = a$
• Imaginary Part: $\text{Im}(z) = b$ (Note: the imaginary part is the real coefficient $b$, not $bi$).
• Purely Real: If $\text{Im}(z) = 0$, then $z$ is a real number.
• Purely Imaginary: If $\text{Re}(z) = 0$ and $b \neq 0$, then $z$ is purely imaginary.

Equality of Complex Numbers

Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ are equal if and only if their real parts are equal and their imaginary parts are equal:
$$\text{Re}(z_1) = \text{Re}(z_2) \text{ and } \text{Im}(z_1) = \text{Im}(z_2)$$

COMMON MISTAKE: When solving equations like $(x+yi)^2 = -18i$, students often forget to equate the real part to 0 and the imaginary part to $-18$. Always separate the equation into two distinct real equations.

2. Arithmetic of Complex Numbers

Addition and Subtraction

Performed by combining like terms (real with real, imaginary with imaginary).
$$ (a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i $$

Multiplication

Performed using the distributive law (FOIL), remembering that $i^2 = -1$.
$$ (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i $$

Powers of $i$

The powers of $i$ follow a cyclic pattern of length 4:
* $i^1 = i$
* $i^2 = -1$
* $i^3 = -i$
* $i^4 = 1$

STUDY HINT: To evaluate $i^n$, find the remainder when $n$ is divided by 4. For example, $i^{102} = i^2 = -1$ because \$102 \div 4$ has a remainder of 2.

3. The Complex Conjugate and Division

The Conjugate

The complex conjugate of $z = a + bi$ is denoted by $\bar{z}$ and is defined as:
$$\bar{z} = a - bi$$
Properties of the Conjugate:
1. $\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$
2. $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$
3. $z + \bar{z} = 2\text{Re}(z)$ (always real)
4. $z - \bar{z} = 2i\text{Im}(z)$ (always purely imaginary)
5. $z\bar{z} = a^2 + b^2 = |z|^2$ (always real and non-negative)

Division

To divide complex numbers in Cartesian form, multiply the numerator and the denominator by the conjugate of the denominator to “realise” the denominator.
$$\frac{z_1}{z_2} = \frac{z_1}{z_2} \times \frac{\bar{z}_2}{\bar{z}_2} = \frac{z_1 \bar{z}_2}{|z_2|^2}$$

KEY TAKEAWAY: Multiplying a complex number by its conjugate $z\bar{z}$ results in a purely real number equal to the square of its distance from the origin. This is the standard method for simplifying complex fractions.

4. The Argand Diagram

The Argand diagram is a geometric representation of $\mathbb{C}$ where the horizontal axis is the Real axis and the vertical axis is the Imaginary axis.

• A complex number $z = a + bi$ is represented by the point $(a, b)$ or by a position vector from the origin to $(a, b)$.
• Addition: Geometrically follows the parallelogram law of vector addition.
• Conjugation: $\bar{z}$ is the reflection of $z$ in the Real axis.
• Negation: $-z$ is the reflection of $z$ through the origin (rotation of $\pi$).

5. Modulus and Argument

Modulus

The modulus of $z = a + bi$, denoted by $|z|$, is the distance from the origin to the point $z$ in the Argand diagram.
$$|z| = \sqrt{a^2 + b^2}$$
Properties of Modulus:
* $|z_1 z_2| = |z_1||z_2|$
* $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$
* $|z^n| = |z|^n$
* $|z_1 + z_2| \le |z_1| + |z_2|$ (The Triangle Inequality)

Argument

The argument of $z$, denoted $\arg(z)$, is the angle $\theta$ the line segment $Oz$ makes with the positive Real axis, measured anticlockwise.
* Principal Argument ($\text{Arg } z$): The unique value of the argument such that $-\pi < \theta \le \pi$.
* For $z = a + bi$:
$$\tan \theta = \frac{b}{a}$$
Care must be taken to identify the correct quadrant based on the signs of $a$ and $b$.

EXAM TIP: Always sketch the position of $z$ on an Argand diagram before calculating the argument to ensure you select the correct quadrant. For example, if $z = -1 - i$, $\tan \theta = 1$, but the angle is in the 3rd quadrant, so $\text{Arg } z = -\frac{3\pi}{4}$.

6. Polar (Modulus-Argument) Form

A complex number can be expressed in terms of its modulus $r$ and argument $\theta$:
$$z = r(\cos \theta + i \sin \theta) = r \operatorname{cis} \theta$$
where $r = |z|$ and $\theta = \arg(z)$.

Arithmetic in Polar Form

Let $z_1 = r_1 \operatorname{cis} \theta_1$ and $z_2 = r_2 \operatorname{cis} \theta_2$:
* Multiplication: Multiply moduli and add arguments.
$$z_1 z_2 = r_1 r_2 \operatorname{cis}(\theta_1 + \theta_2)$$
* Division: Divide moduli and subtract arguments.
$$\frac{z_1}{z_2} = \frac{r_1}{r_2} \operatorname{cis}(\theta_1 - \theta_2)$$

De Moivre’s Theorem

For any integer $n$:
$$(r \operatorname{cis} \theta)^n = r^n \operatorname{cis}(n\theta)$$

VCAA FOCUS: VCAA often tests the properties of $\operatorname{cis} \theta$. Remember that $\overline{\operatorname{cis} \theta} = \operatorname{cis}(-\theta) = \frac{1}{\operatorname{cis} \theta}$.

7. Subsets of the Complex Plane (Loci)

Complex numbers can define paths or regions in the Argand diagram:

REMEMBER: For equations like $|z - (1+i)| = 2$, the centre is at $(1, 1)$. Always factor out the negative sign inside the modulus to clearly see the “fixed point” (the centre).