IB Mathematics AI – Topic 1

Standard Form:

\[ z = a + bi \]

where:

• a: Real part, denoted \(\text{Re}(z)\)
• b: Imaginary part, denoted \(\text{Im}(z)\)
• i: Imaginary unit where \(i^2 = -1\), so \(i = \sqrt{-1}\)

Important Powers of i:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• Pattern repeats every 4 powers

Complex Conjugate:

If \(z = a + bi\), then the complex conjugate is \(z^* = a - bi\)

\[ z \cdot z^* = (a+bi)(a-bi) = a^2 + b^2 = |z|^2 \]

⚠️ Common Pitfalls & Tips:

• Remember \(i^2 = -1\), not \(i = -1\)
• The imaginary part \(b\) is just the coefficient, not \(bi\)
• Conjugate changes sign of imaginary part only
• For division, multiply top and bottom by conjugate

Polar Form:

\[ z = r(\cos\theta + i\sin\theta) = r\text{cis}\,\theta \]

Modulus (r):

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

Always positive; represents distance from origin

Argument (θ):

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

Angle measured counterclockwise from positive real axis

Principal argument: \(-\pi < \theta \leq \pi\) (or \(-180° < \theta \leq 180°\))

Converting FROM Cartesian TO Polar:

1. Calculate \(r = \sqrt{a^2 + b^2}\)
2. Find \(\theta = \tan^{-1}(b/a)\)
3. Adjust \(\theta\) based on quadrant:
   • Quadrant I (a > 0, b > 0): \(\theta\) is positive
   • Quadrant II (a < 0, b > 0): \(\theta = \pi - |\tan^{-1}(b/a)|\)
   • Quadrant III (a < 0, b < 0): \(\theta = -\pi + |\tan^{-1}(b/a)|\)
   • Quadrant IV (a > 0, b < 0): \(\theta\) is negative

Converting FROM Polar TO Cartesian:

\[ a = r\cos\theta, \quad b = r\sin\theta \]

⚠️ Common Pitfalls & Tips:

• CRITICAL: Always check which quadrant the complex number is in
• Calculator gives \(\tan^{-1}\) in limited range; adjust for correct quadrant
• Use GDC to convert forms (check mode settings)
• Argument of zero is undefined

Solution:

Step 1: Find modulus r

\[ r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4 \]

Step 2: Find argument θ

Since \(a = -2\) (negative) and \(b = 2\sqrt{3}\) (positive), point is in Quadrant II

\[ \tan\theta = \frac{2\sqrt{3}}{-2} = -\sqrt{3} \]

Reference angle: \(\tan^{-1}(\sqrt{3}) = \frac{\pi}{3}\)

In Quadrant II: \(\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\)

Step 3: Write in polar form

\[ z = 4\left(\cos\frac{2\pi}{3} + i\sin\frac{2\pi}{3}\right) = 4\text{cis}\,\frac{2\pi}{3} \]

Answer: \(z = 4\text{cis}\,\frac{2\pi}{3}\) or \(4\text{cis}\,120°\)

Euler's Formula:

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

Exponential (Euler) Form:

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

where r = modulus, θ = argument

Euler's Identity:

When \(\theta = \pi\):

\[ e^{i\pi} + 1 = 0 \]

Connects five fundamental constants: e, i, π, 1, and 0

Conversion Between Forms:

• Cartesian to Euler: \(a + bi = re^{i\theta}\) where \(r = \sqrt{a^2+b^2}\), \(\theta = \arg(z)\)
• Euler to Cartesian: \(re^{i\theta} = r\cos\theta + ri\sin\theta\)
• Polar to Euler: \(r\text{cis}\,\theta = re^{i\theta}\)

⚠️ Common Pitfalls & Tips:

• Euler form makes multiplication and division much easier
• The exponent must have \(i\) (imaginary unit)
• Can write \(re^{i\theta}\) or \(e^{i\theta} \cdot r\)
• Use for powers and roots (De Moivre's Theorem)

In Cartesian Form:

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 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 top and bottom by conjugate of denominator

In Polar/Euler Form:

Multiplication:

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

Multiply moduli, add arguments

Division:

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

Divide moduli, subtract arguments

Powers (De Moivre's Theorem):

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

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

⚠️ Common Pitfalls & Tips:

• For multiplication/division, polar/Euler form is much easier
• For addition/subtraction, Cartesian form is easier
• Don't forget \(i^2 = -1\) when expanding
• Always simplify conjugate products: \((c+di)(c-di) = c^2+d^2\)

Solution:

(a) Multiply in Euler form:

Multiply moduli: \(2 \times 3 = 6\)

Add arguments: \(\frac{\pi}{4} + \frac{\pi}{3} = \frac{3\pi + 4\pi}{12} = \frac{7\pi}{12}\)

\[ z_1 \times z_2 = 6e^{i7\pi/12} \]

Answer: \(6e^{i7\pi/12}\)

(b) Power using De Moivre:

\[ z_1^4 = (2e^{i\pi/4})^4 = 2^4e^{i\cdot 4 \cdot \pi/4} = 16e^{i\pi} \]

Convert to Cartesian:

Using \(e^{i\pi} = \cos\pi + i\sin\pi = -1 + 0i = -1\)

\[ z_1^4 = 16 \times (-1) = -16 \]

Answer: -16 (or -16 + 0i)

Components:

• Horizontal axis: Real axis (Re)
• Vertical axis: Imaginary axis (Im)
• Point (a, b): Represents \(z = a + bi\)
• Distance from origin: Modulus \(|z| = r\)
• Angle from positive real axis: Argument \(\arg(z) = \theta\)

Geometric Interpretations:

• Modulus: \(|z|\) is the distance from origin to point z
• Argument: \(\arg(z)\) is angle counterclockwise from positive real axis
• Addition: Vector addition (parallelogram rule)
• Conjugate: Reflection across real axis
• \(|z_1 - z_2|\): Distance between two points

Loci in Complex Plane:

• \(|z| = r\): Circle centered at origin, radius r
• \(|z - z_0| = r\): Circle centered at \(z_0\), radius r
• \(\arg(z) = \theta\): Ray from origin at angle \(\theta\)
• \(\text{Re}(z) = k\): Vertical line at x = k
• \(\text{Im}(z) = k\): Horizontal line at y = k

⚠️ Common Pitfalls & Tips:

• Imaginary axis is vertical, not horizontal
• Plot (real part, imaginary part), not (modulus, argument)
• Conjugate reflects across real axis (horizontal)
• Use Pythagoras for distances: \(|z_1 - z_2|\)

Expressing Sine and Cosine:

From Euler's formula \(e^{i\theta} = \cos\theta + i\sin\theta\) and \(e^{-i\theta} = \cos\theta - i\sin\theta\)

Cosine Formula:

\[ \cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2} \]

Sine Formula:

\[ \sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} \]

Applications:

• Signal processing: Representing waves and oscillations
• Electrical engineering: AC circuits analysis
• Quantum mechanics: Wave functions
• Solving trigonometric equations: Using complex exponentials
• Fourier analysis: Decomposing functions into sine/cosine components

Simplifying Trigonometric Expressions:

Complex exponentials can simplify products and powers of trig functions

Example: \(\cos^n\theta\) can be found using \(\left(\frac{e^{i\theta} + e^{-i\theta}}{2}\right)^n\)

⚠️ Common Pitfalls & Tips:

• Don't confuse \(e^{i\theta}\) with \(e^{\theta}\) (no imaginary unit)
• Sine formula has \(2i\) in denominator, not just 2
• These formulas are in IB formula booklet
• Useful for proving trigonometric identities

Solution:

Start with Euler's formula:

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

Square both sides:

\[ e^{i2\theta} = (e^{i\theta})^2 = (\cos\theta + i\sin\theta)^2 \]

Expand the right side:

\[ = \cos^2\theta + 2i\cos\theta\sin\theta + i^2\sin^2\theta \]

\[ = \cos^2\theta + 2i\cos\theta\sin\theta - \sin^2\theta \]

\[ = (\cos^2\theta - \sin^2\theta) + i(2\cos\theta\sin\theta) \]

But also by Euler's formula:

\[ e^{i2\theta} = \cos(2\theta) + i\sin(2\theta) \]

Equate real parts:

\[ \cos(2\theta) = \cos^2\theta - \sin^2\theta \quad \checkmark \]

Bonus - equate imaginary parts:

\[ \sin(2\theta) = 2\cos\theta\sin\theta \]

Three Forms

• Cartesian: \(a + bi\)
• Polar: \(r\text{cis}\,\theta\)
• Euler: \(re^{i\theta}\)

Key Values

• \(r = \sqrt{a^2+b^2}\)
• \(\theta = \tan^{-1}(b/a)\)
• \(e^{i\pi} = -1\)

Operations (Euler)

• Multiply: \(r_1r_2e^{i(\theta_1+\theta_2)}\)
• Divide: \(\frac{r_1}{r_2}e^{i(\theta_1-\theta_2)}\)
• Power: \(r^ne^{in\theta}\)

Trig Connections

• \(\cos\theta = \frac{e^{i\theta}+e^{-i\theta}}{2}\)
• \(\sin\theta = \frac{e^{i\theta}-e^{-i\theta}}{2i}\)