Complex Numbers in Cartesian and Polar Form

Writing $z = x + yi$ fixes a complex number by its horizontal and vertical coordinates. But a point of the plane can equally be fixed by how far it sits from the origin and in which direction. That second description — the polar form — makes multiplication, division and powers almost effortless.

Concept Modulus and argument

z = x + yi = |z|\,(\cos\theta + i\sin\theta)

Modulus

|z| = \sqrt{x^2 + y^2}

— the distance from the origin.

Argument

\theta = \tan^{-1}\!\left(\dfrac{y}{x}\right)

— the direction.

Going back the other way:

x = |z|\cos\theta

y = |z|\sin\theta

The modulus comes straight from Pythagoras applied to the right triangle formed by the two coordinates.

Note Why polar form is worth the trouble

Operation	Moduli	Arguments
Multiplication	multiply	add
Division	divide	subtract
Power n	raise to n	multiply by n

In Cartesian form each of these needs an expansion; in polar form each is one arithmetic step on the modulus and one on the angle.

Example Converting

z = 3 + 4i

to polar form

|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

\theta = \tan^{-1}\!\left(\dfrac{4}{3}\right) \approx 53.1°

Both parts are positive, so the point is in the first quadrant and no adjustment to the angle is needed.

⟹ z = 5(cos 53.1° + i sin 53.1°)

Example Multiplying in polar form

Multiply $z_1 = 2(\cos 30° + i\sin 30°)$ by $z_2 = 3(\cos 45° + i\sin 45°)$ .

Multiply the moduli:

2 \times 3 = 6

Add the arguments:

30° + 45° = 75°

⟹ z₁z₂ = 6(cos 75° + i sin 75°)

Multiplication by a complex number is therefore a stretch by its modulus together with a rotation by its argument. Multiplying by $i$ , whose modulus is 1 and argument 90°, is a pure quarter-turn.

Summary

Every complex number has a Cartesian form x + yi and a polar form |z|(cos θ + i sin θ).
The modulus is |z| = √(x² + y²) and the argument is θ = tan⁻¹(y/x).
Multiplying multiplies moduli and adds arguments; dividing divides moduli and subtracts arguments.
Geometrically, multiplying by a complex number stretches by its modulus and rotates by its argument.