In more detail
In a + bi, a is the real part and b the imaginary part. You add complex numbers part by part and multiply them like brackets, replacing i² by −1. The conjugate of a + bi is a − bi, and a number times its conjugate is the real number a² + b². The size is |a + bi| = √(a² + b²), so |3 + 4i| = 5. In polar form a number is a length and an angle; multiplying two numbers multiplies their lengths and adds their angles, which is why complex numbers describe rotations so neatly.
Why it matters
Complex numbers give every quadratic an answer (see Quadratic formula) and describe oscillations: electrical engineers write AC impedance as a complex number, and signal processing such as the Fourier transform is built on them. Software needs to be told: in Python math.sqrt(-1) raises a ValueError, while cmath.sqrt(-1) returns 1j (Python writes i as j).
Example
(3 + 4i) + (1 − 2i) = 4 + 2i
(3 + 4i) × (1 − 2i) = 3 − 6i + 4i − 8i² = 11 − 2i
(3 + 4i) / (1 − 2i) = −1 + 2i
|3 + 4i| = 5 angle of 3 + 4i = 53.13°
Division multiplies top and bottom by the conjugate of the denominator: (3 + 4i)(1 + 2i) / (1² + 2²) = (−5 + 10i) / 5 = −1 + 2i.