Complex numbers

The equation x² + 1 = 0 has no real solution: no real number squared gives −1. Mathematicians simply added a number that does — i, with i² = −1 — and obtained a number system in which every quadratic equation has roots; engineers use it every day to describe alternating current and signals. In the CSCA syllabus complex numbers share the line “Vectors and complex numbers” with the lesson “Plane vectors”, and only the basics are tested: the algebraic form, the complex plane and the four operations. Know the handful of rules in this lesson, and a typical item is two lines of algebra.

The imaginary unit and the algebraic form

The complex plane, the conjugate and the modulus

Every complex number z = a + bi corresponds to the point Z(a, b) of the plane and to the vector OZ⃗ from the origin. The horizontal axis is the real axis, the vertical one the imaginary axis; every point of the imaginary axis except O is a pure imaginary number. Adding complex numbers means adding their vectors — the parallelogram rule from “Plane vectors” — so |z₁ − z₂| is the distance between the two points.

The four operations

Addition and subtraction work part by part, exactly as with vectors: z₁ + z₂ is the diagonal of the parallelogram built on OZ₁⃗ and OZ₂⃗. Multiplication is ordinary expansion of brackets plus one substitution, i² = −1. Division needs a trick, because i must not stay in a denominator: multiply the numerator and the denominator by the conjugate of the denominator, since (c + di)(c − di) = c² + d² is real.

1. 1
   Write the conjugate

   Write the conjugate of the denominator: c + di → c − di.

2. 2
   Multiply

   Multiply both the numerator and the denominator by it.

3. 3
   Denominator

   The denominator becomes c² + d², a positive real number.

4. 4
   Numerator

   Expand, replace i² by −1, and collect the real and the imaginary parts.

5. 5
   Answer the question

   Divide both parts by c² + d² and read off what is asked: a part, the conjugate, the modulus or the quadrant.

Powers of i and quadratic equations

Because i⁴ = 1, the powers of i come back to where they started every four steps, like the hand of a clock. So any power can be reduced to i⁰, i¹, i² or i³ by dividing the exponent by 4 — a favourite one-line CSCA item. Powers of 1 + i and 1 − i work the same way: square first, since (1 ± i)² = ±2i.

How CSCA asks about this

The CSCA mathematics paper has 48 questions in 60 minutes — about 75 seconds each, and a complex-number item should take less. The usual patterns:

• simplify a product or a quotient, then give the real or imaginary part, the conjugate, the modulus or the quadrant;
• a parameter: z is real, pure imaginary, or its point lies in a given quadrant;
• equations for z (z times an expression, z together with z̄) and the equality of two complex numbers;
• powers of i and sums of consecutive powers;
• the complex plane: the distance |z₁ − z₂| and simple sets such as |z − z₀| = r;
• the roots of a real quadratic with D < 0 come in a conjugate pair.