3.9.1—Complex numbers

Write z=a+bi, where a,b are real and i²=−1. The conjugate is a−bi, modulus is |z|=√(a²+b²), and z z̄=|z|².

Collect real and imaginary parts separately, rationalise denominators with the conjugate, and use modulus as a distance from the origin.

(2+3i)(2−3i)=13, so the modulus of 2+3i is √13.

i is not a variable with arbitrary square root; i² is fixed as −1 and real/imaginary parts must match independently.