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Topic 2: Complex arithmetic and algebra

Complex arithmetic using polar form

4 hours

• Use the modulus |z| of a complex number z and the principal argument Arg(z) of a non-zero complex number z.
• Understand the difference between the argument, arg(z), and the principal argument, Arg(z), of a non-zero complex number z.
• Express a complex number in Cartesian form z = a + bi and polar form z = r(cos θ + i sin θ) or z = r cis θ.
• Convert between Cartesian form and polar form. 1 interactive
• Understand and use multiplication and division of complex numbers in polar form and the geometric interpretation of these. 1 interactive
• Sketch and use complex numbers in polar form as polar coordinates.

Subsets of the complex plane (the Argand plane)

3 hours

• Identify and sketch subsets of the complex plane determined by straight lines and circles, e.g. |z − 3i| < 4, π/4 ≤ Arg(z) ≤ 3π/4, Re(z) > Im(z) and |z − 1| = 2|z − i|. 1 interactive

Roots of real quadratic equations

4 hours

• Determine complex conjugate solutions of real quadratic equations with real coefficients using factorisation, completing the square and the quadratic formula, with and without technology.
• Determine and use linear factors of quadratic polynomials with real coefficients that involve the complex conjugate root theorem, e.g. determine the coefficients of a real quadratic equation given one complex root.