ZIMSEC A Level · Pure Mathematics · Form 6
Basics
i² = −1. A complex number z = x + iy has real part x and imaginary part y. The conjugate z* = x − iy.
- Add and subtract parts separately; multiply by expanding; divide by multiplying top and bottom by the conjugate of the denominator.
Worked example: Express (3 + 2i)/(1 − i) in the form x + iy.
Multiply by (1 + i)/(1 + i): (3 + 3i + 2i + 2i²)/(1 + 1) = (1 + 5i)/2 = ½ + (5/2)i
The Argand diagram and modulus–argument form
- |z| = √(x² + y²); arg z = θ, measured from the positive real axis (−π < θ ≤ π).
- z = r(cos θ + i sin θ) = re^(iθ).
- |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂.
Roots of polynomials
Non-real roots of real polynomials occur in conjugate pairs.
Worked example: Solve z² − 4z + 13 = 0.
z = [4 ± √(16 − 52)]/2 = [4 ± √(−36)]/2 = 2 ± 3i
Loci
|z − a| = r is a circle centre a, radius r. |z − a| = |z − b| is the perpendicular bisector of a and b. arg(z − a) = θ is a half-line from a.
Exam tip: Draw the Argand diagram to find the correct quadrant for the argument.
Check yourself
- Find |3 − 4i| and arg(1 + i).
- Simplify (2 + i)(3 − 2i).
- Describe the locus |z − 2i| = 3.
Show answers
- 5; π/4
- 6 − 4i + 3i − 2i² = 8 − i
- A circle with centre (0, 2) and radius 3.
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