Complex numbers
Complex numbers for IB Analysis and Approaches HL: Cartesian, polar and Euler forms, De Moivre, n-th roots and polynomials, with IB exam questions.
Complex numbers extend the reals with the imaginary unit , which satisfies . With them every polynomial equation has a solution, and multiplication becomes rotating and stretching in the plane. It is one of the topics most specific to Higher Level and appears in every exam session, often combined with trigonometry, induction or polynomials.
Recommended order
Start with the Cartesian form (operating as with polynomials), move on to the modulus–argument (polar) form and the Euler form (modulus and argument), apply De Moivre's theorem to powers and roots, and finish with polynomials with complex roots. At the end you have IB exam questions grouped by type.
Polynomials and the quadratic equation (topic 1), trigonometry and radians (topic 4).
Sections in this topic
Cartesian form and operations
Complex numbers in Cartesian form a+bi: the imaginary unit, the Argand diagram, addition, multiplication, division by the conjugate and equations in z and z*.
3 worked examples Section 2Modulus–argument and Euler form
Modulus and argument of a complex number, modulus-argument form r cis θ and Euler form re^{iθ}, with products, quotients, quadrants and worked IB examples.
3 worked examples Section 3De Moivre's theorem and n-th roots
De Moivre's theorem with proof, powers of complex numbers, n-th roots and roots of unity, and trigonometric identities with complex numbers (IB AA HL).
3 worked examples Section 4Polynomials with complex roots
Complex conjugate roots of a real polynomial, how to factorise it, rebuild it, and use the sum and product of roots. IB Analysis and Approaches HL.
3 worked examples End of topicIB exam questions: topic 6
IB exam questions on complex numbers: Cartesian and polar form, De Moivre's theorem, roots and polynomials (Analysis and Approaches HL), with worked solutions.
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