Use this short revision lesson to review the polar form of a complex number and the relationship between magnitude, argument, and powers.

Polar form

A complex number z = x + iy can be written as z = r(cos θ + i sin θ), where r = √(x² + y²) and θ = arg(z).

De Moivre's theorem

For an integer n, [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). This is useful for powers and roots.

Check your understanding

1. Find the modulus of 3 + 4i.
2. Write −1 + i in polar form using a principal argument.
3. State the condition on r when finding the nth roots of a complex number.

Worked check: The modulus of 3 + 4i is √(3² + 4²) = 5.