NSW Syllabuses

Mathematics Extension 2 Stage 6 - Year 12 Extension 2 - Complex Numbers MEX-N2 Using Complex Numbers

Outcomes

A student:

• MEX12-1

understands and uses different representations of numbers and functions to model, prove results and find solutions to problems in a variety of contexts

• MEX12-4

uses the relationship between algebraic and geometric representations of complex numbers and complex number techniques to prove results, model and solve problems

• MEX12-7

applies various mathematical techniques and concepts to model and solve structured, unstructured and multi-step problems

• MEX12-8

communicates and justifies abstract ideas and relationships using appropriate language, notation and logical argument

Subtopic Focus

The principal focus of this subtopic is to develop and apply complex number knowledge to situations involving trigonometric identities, powers and vector representations in a complex number plane.

Students develop appreciation of the interconnectedness of complex numbers across various topics of mathematics and their applications in real life.

Content

• N2.1: Solving equations with complex numbers
• Students:
• prove De Moivre’s theorem for integral powers using proof by induction (ACMSM083)
• solve power equations (ACMSM082)
• use De Moivre's theorem to derive trigonometric identities such as $$\cos 3 \theta = 4 \cos^{3} \theta - 3 \cos \theta$$
• solve quadratic equations of the form $$ax^2 + bx+c = 0$$, where $$a$$, $$b$$, $$c$$ are complex numbers (ACMSM072, ACMSM074, ACMSM075)
• define and use conjugate roots for polynomials with real coefficients, and show that roots of polynomials with real coefficients occur as conjugate pairs (ACMSM090)
• solve problems involving real polynomials with conjugate roots
• N2.2: Geometrical implications of complex numbers
• Students:
• given the points representing $$z_1$$ and $$z_2$$, find the position of the point representing $$z_1+z_2$$ and $$z_1-z_2$$
• describe the vector representing $$z_1+z_2$$ or $$z_1-z_2$$ as corresponding to the relevant diagonal of a parallelogram with vectors representing $$z_1$$ and $$z_2$$ as adjacent sides
• examine and use the geometric interpretation of multiplying complex numbers, including rotation and scaling in the Argand plane (ACMSM085)
• explain why multiplying by $$i$$ gives a $$\frac{\pi}{2}$$ rotation anticlockwise about $$(0,0)$$, and multiplying by $$-1$$ gives a rotation of $$\pi$$ in the Argand plane
• recognise the geometrical relationship between the points representing a complex number $$z=a+ib$$, and the points representing $$\bar z$$, $$cz$$ (where $$c$$ is real) and $$iz$$ (ACMSM073)
• determine and examine the $$n$$th roots of unity and their location on the unit circle (ACMSM087)
• determine and examine the $$n$$th roots of complex numbers and their location in the Argand plane (ACMSM088)
• solve problems using $$n$$th roots of complex numbers
• identify subsets of the Argand plane determined by relations, for example, $$\left|z-3i\right|\le 4$$, $$\frac{\pi}{4} \le Arg(z) \le \frac{3\pi}{4}$$, $$Re(z) > Im(z)$$ and $$\left|z-1\right| = 2\left|z-i\right|$$ (ACMSM086)