Complex Numbers

Complex Numbers - Formulas & Properties

IB Mathematics Analysis & Approaches (SL & HL)

🔷 The Imaginary Unit

Definition:

\[i = \sqrt{-1} \quad \text{or} \quad i^2 = -1\]

Powers of i (Cyclic Pattern):

\(i^1 = i\)

\(i^2 = -1\)

\(i^3 = -i\)

\(i^4 = 1\)

The pattern repeats every 4 powers: \(i^5 = i\), \(i^6 = -1\), etc.

📐 Cartesian (Rectangular) Form

General Form:

\[z = a + bi\]

where \(a\) is the real part and \(b\) is the imaginary part

Key Notation:

\[\text{Re}(z) = a \quad \text{(Real part)}\]

\[\text{Im}(z) = b \quad \text{(Imaginary part)}\]

Equality of Complex Numbers:

\(a + bi = c + di\) if and only if \(a = c\) and \(b = d\)

🔢 Operations with Complex Numbers

Addition:

\[(a + bi) + (c + di) = (a + c) + (b + d)i\]

Subtraction:

\[(a + bi) - (c + di) = (a - c) + (b - d)i\]

Multiplication:

\[(a + bi)(c + di) = (ac - bd) + (ad + bc)i\]

Use FOIL method and remember \(i^2 = -1\)

Division:

\[\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\]

Multiply numerator and denominator by the conjugate of the denominator

🔄 Complex Conjugate

Definition:

The conjugate of a complex number is obtained by changing the sign of the imaginary part.

\[\text{If } z = a + bi, \text{ then } \overline{z} = a - bi\]

Important Properties:

\[z \cdot \overline{z} = a^2 + b^2 = |z|^2\]

\[\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}\]

\[\overline{z_1 \cdot z_2} = \overline{z_1} \cdot \overline{z_2}\]

\[\overline{\left(\overline{z}\right)} = z\]

📏 Modulus and Argument

Modulus (Absolute Value):

The modulus represents the distance from the origin to the point in the complex plane.

\[|z| = |a + bi| = \sqrt{a^2 + b^2}\]

Argument (Angle):

The argument is the angle \(\theta\) made with the positive real axis (measured counterclockwise).

\[\arg(z) = \theta = \tan^{-1}\left(\frac{b}{a}\right)\]

Note: Adjust \(\theta\) based on which quadrant the complex number is in

Modulus Properties:

\[|z_1 \cdot z_2| = |z_1| \cdot |z_2|\]

\[\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\]

\[|z| = |\overline{z}|\]

🎯 Polar (Modulus-Argument) Form

General Form:

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

where \(r = |z|\) is the modulus and \(\theta = \arg(z)\) is the argument

Shorthand Notation:

\[z = r \text{ cis } \theta\]

where cis stands for "cosine + i sine"

Converting from Cartesian to Polar:

\[r = \sqrt{a^2 + b^2}\]

\[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]

⚡ Euler's (Exponential) Form

Euler's Formula:

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

Exponential Form of Complex Number:

\[z = re^{i\theta}\]

where \(r = |z|\) and \(\theta = \arg(z)\)

Special Cases:

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

\[e^{i\pi} + 1 = 0 \quad \text{(Euler's Identity)}\]

✖️ Operations in Polar Form

Multiplication:

\[z_1 \cdot z_2 = r_1r_2 \text{ cis}(\theta_1 + \theta_2)\]

Multiply moduli, add arguments

Division:

\[\frac{z_1}{z_2} = \frac{r_1}{r_2} \text{ cis}(\theta_1 - \theta_2)\]

Divide moduli, subtract arguments

In Exponential Form:

\[z_1 \cdot z_2 = r_1e^{i\theta_1} \cdot r_2e^{i\theta_2} = r_1r_2e^{i(\theta_1 + \theta_2)}\]

\[\frac{z_1}{z_2} = \frac{r_1e^{i\theta_1}}{r_2e^{i\theta_2}} = \frac{r_1}{r_2}e^{i(\theta_1 - \theta_2)}\]

🌟 De Moivre's Theorem

General Formula:

For any complex number in polar form and any real number \(n\):

\[[r(\cos\theta + i\sin\theta)]^n = r^n(\cos n\theta + i\sin n\theta)\]

Alternative Notations:

\[(r \text{ cis } \theta)^n = r^n \text{ cis } (n\theta)\]

\[(re^{i\theta})^n = r^ne^{in\theta}\]

Usage:

• Raising complex numbers to integer powers
• Finding roots of complex numbers
• Simplifying trigonometric expressions
• Works for all real values of \(n\) (including negative and fractional)

√ Roots of Complex Numbers

Finding nth Roots:

To find the \(n\)th roots of \(z = r \text{ cis } \theta\), use:

\[z_k = r^{1/n} \text{ cis}\left(\frac{\theta + 360k}{n}\right)\]

where \(k = 0, 1, 2, \ldots, n-1\) gives all \(n\) distinct roots

In Radians:

\[z_k = r^{1/n} \text{ cis}\left(\frac{\theta + 2\pi k}{n}\right)\]

The \(n\) roots are equally spaced around a circle of radius \(r^{1/n}\)

📊 Complex Plane (Argand Diagram)

Definition:

A complex number \(z = a + bi\) is represented as a point \((a, b)\) in the complex plane.

Horizontal axis (Real axis): Represents the real part
Vertical axis (Imaginary axis): Represents the imaginary part
Distance from origin: Equals the modulus \(|z|\)
Angle with real axis: Equals the argument \(\arg(z)\)

Geometric Interpretation:

Distance between two complex numbers:

\[|z_1 - z_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}\]

🔑 Important Identities & Properties

Sum and Product Properties:

\[z + \overline{z} = 2\text{Re}(z) = 2a\]

\[z - \overline{z} = 2i\text{Im}(z) = 2bi\]

Triangle Inequality:

\[|z_1 + z_2| \leq |z_1| + |z_2|\]

Reciprocal:

\[\frac{1}{z} = \frac{\overline{z}}{|z|^2} = \frac{1}{r}e^{-i\theta}\]

💡 Key Tip: Use Cartesian form for addition and subtraction. Use Polar/Exponential form for multiplication, division, powers, and roots. Always make sure your calculator is in complex mode when working with complex numbers.