Complex numbers

Need to know

What you need to know

• The modulus is always non-negative, as it represents a distance.
• The argument must be calculated carefully. Always sketch the point on an Argand diagram to ensure you find the angle in the correct quadrant.
• The **principal argument** is the unique value of $\theta$ such that $-\pi < \theta \le \pi$. This is the standard range required in Cambridge exams unless specified otherwise.

Explanation

Numbers in 2D

1. A complex number z = x + iy is plotted as the point (x, y) in the Argand diagram, with a real (x) and imaginary (y) axis.
2. The modulus |z| is the distance from the origin, calculated as √(x² + y²). The argument, arg z, is the angle from the positive real axis.
3. The polar form uses the modulus (r) and argument (θ) to write z = r(cos θ + i sin θ), which simplifies to the exponential form z = re^(iθ).
4. De Moivre’s theorem provides a powerful shortcut for powers: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ. This means to raise a complex number to a power, you raise the modulus to that power and multiply the argument.