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).