Complex numbers (HL)

Learning objectives

Define i with i² = −1, write complex numbers in Cartesian form z = a + bi, and add, subtract, multiply and divide them, using the conjugate to simplify quotients

Find the real part, imaginary part, conjugate, modulus and argument of a complex number, and represent complex numbers and operations on them as points or vectors in the complex (Argand) plane

Convert between Cartesian, modulus–argument (polar) and Euler forms, and multiply and divide complex numbers in polar or Euler form, interpreting the results geometrically as rotations and enlargements

Apply De Moivre's theorem, including its extension to rational exponents, to calculate powers of complex numbers and to simplify expressions involving them

Find the nth roots of a complex number, including the roots of unity, and describe how they are arranged in the complex plane

Use the fact that non-real roots of polynomial equations with real coefficients occur in conjugate pairs to solve quadratic and higher-degree polynomial equations