Complex numbers · Grade 12

In Grade 12 you learn: the Argand diagram and modulus–argument form.

Complex numbers extend the real numbers with i, where i² = −1, so that every quadratic equation has solutions. Written as a + bi, they are points on an Argand diagram. In modulus–argument form they turn multiplication into rotation and scaling. That is why engineers and physicists rely on them.

Key facts

• i^2 = −1. Add and multiply like algebra, replacing i^2 by −1.
• The conjugate of z = a + bi is z* = a − bi, and z × z* = a² + b². To divide, multiply top and bottom by the conjugate of the bottom.
• Modulus: |z| = √(a^2 + b^2). The argument is the angle from the positive real axis, with −180° < θ ≤ 180°. Sketch it to get the quadrant right.
• Modulus–argument form: z = r(cos θ + i sin θ). Multiplying multiplies the moduli and adds the arguments.
• De Moivre’s theorem: (r(cos θ + i sin θ))^n = r^n(cos nθ + i sin nθ).
• A polynomial with real coefficients has its non-real roots in conjugate pairs.

Worked examples

Other years

• Complex numbers · Grade 11