AHL 1.12 (HL)—Complex numbers

Write a complex number as z = a + bi, where i² = −1. The real part a gives the horizontal coordinate and the imaginary part b gives the vertical coordinate, so z can be plotted as (a,b) on an Argand diagram.

Add and subtract complex numbers component by component. For multiplication, expand normally and replace i² with −1: (2 + i)(3 − i) = 7 + i. The conjugate of a + bi is a − bi, and z z̄ = a² + b² is real.

i is not a variable whose square is positive; i² is defined as −1. Keep the real and imaginary parts separate when interpreting a result geometrically.

For division, multiply numerator and denominator by the denominator's conjugate: $(3+2i)/(1-i)=(1+5i)/2$. Powers follow repeated multiplication or technology in Cartesian form. A real quadratic with negative discriminant has a conjugate pair of roots; for $x^2-4x+13=0$, $x=[4\pm\sqrt{-36}]/2=2\pm3i$, plotted symmetrically about the real axis on an Argand diagram.