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.