Complex numbers
i = √−1. A complex number is z = a + bi with a (real) and b (imaginary) parts.
The complex numbers extend the reals by adding i where i² = −1. Every complex z has a conjugate z* = a − bi (sign of imaginary part flipped).
Worked examples
z = 3 + 4i; z* = 3 − 4i.
Re(z) = 3, Im(z) = 4.
Argand diagram: plot real on x-axis, imaginary on y-axis.
Frequently asked questions
Why was i invented?
To solve quadratic equations like x² + 1 = 0 with no real roots. Necessary for the fundamental theorem of algebra.
z + z*?
= 2 Re(z); always real. zz* = a² + b² = |z|² (modulus squared).