Modulus, Argument and Polar Representation

Modulus, Argument and Polar Representation

This lesson covers the following key concepts:

• Modulus (absolute value) of complex number
• Argument (amplitude) of complex number
• Principal value of argument: -π < arg(z) ≤ π
• Argand diagram and geometric representation
• Polar form: z = r(cos θ + i sin θ)
• Euler's form: z = re^(iθ)
• Properties of modulus and argument

Important Formulas

• $$|z| = \sqrt{a^2 + b^2}$$
• $$\arg(z) = \tan^{-1}\left(\frac{b}{a}\right) \text{ (with quadrant check)}$$
• $$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$
• $$|z_1 z_2| = |z_1| |z_2|, \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$$
• $$\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$$
• $$\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$$
• $$|z_1 + z_2| \leq |z_1| + |z_2| \text{ (triangle inequality)}$$