Core Pure 1: Complex Numbers

This lesson introduces imaginary numbers, where i =√-1. You’ll learn to simplify square roots of negatives, add and multiply complex numbers, and solve quadratics with complex solutions. Watch for sign errors and remember i^2 = -1.

Teacher’s tip: Powers of i repeat every four—use that to simplify quickly.

This lesson covers complex conjugates and operations with complex numbers. You’ll learn to add, multiply, and rationalise expressions using a + bi and its conjugate a – bi. Common errors include sign mistakes and not applying conjugates correctly when simplifying fractions.

Teacher’s tip: Multiply by the conjugate to remove i from the denominator.

This lesson focuses on the roots of quadratic equations with real coefficients that have complex solutions. You’ll learn to solve and factorise such equations and understand how complex roots always occur in conjugate pairs.

Teacher’s tip: If one complex root is given, its conjugate is the other—use both to form or factorise the equation.

This lesson explores solving cubic and quartic equations with complex roots. You’ll learn how to use known roots—especially complex ones and their conjugates—to factor and find the remaining roots of real polynomials.

Teacher’s tip: If a complex number is a root, its conjugate is too—use this to simplify the problem and factor efficiently.