Complex numbers in Cartesian form

i is the number whose square is −1: Some equations, like x² = −1, have no real solution — no real number squares to a negative.

So mathematicians invented a new number i with the single rule i² = −1.

A complex number is then written z = a + bi, where a is the real part and b is the imaginary part (both are ordinary real numbers).

This is exactly how engineers describe alternating currents and how physicists handle waves — i is a genuine tool, not a trick.

Add and subtract: just collect like terms: Treat i like a letter (an x): add the real parts together and the imaginary parts together. Nothing new to memorise.

Multiply: expand, then replace i² with −1: Multiply out the brackets exactly like (a + b)(c + d). The only new step is that whenever you get i², swap it for −1.

The conjugate flips the sign of the imaginary part: The conjugate of z = a + bi is z* = a − bi (some books write z̄).

Its magic property: z × z\* is always a real number, because

(a + bi)(a − bi) = a² − (bi)² = a² + b² — the i's cancel.

That is exactly what lets us divide complex numbers.

To divide: multiply top and bottom by the conjugate of the bottom: We can't divide by 'i' directly, so we clear the i from the denominator — just like rationalising a surd. Multiply the fraction by (conjugate of the bottom)/(conjugate of the bottom). The bottom becomes a real number and you split into real + imaginary parts.

Picture a complex number on the Argand diagram: An Argand diagram plots z = a + bi as the point (a, b) — real part across, imaginary part up.

The modulus |z| is then simply the distance from the origin to that point, found by Pythagoras:

|z| = √(a² + b²). It measures the 'size' of the complex number.