A worked example: 3+4i to polar form
The complex number 3+4i converts to polar form as magnitude 5, angle 53.13010235° — the classic 3-4-5 right triangle, with the angle measured counterclockwise from the positive real axis.
The formula: r = √(a² + b²), θ = arctan(b ÷ a)
Treating the real part as an x-coordinate and the imaginary part as a y-coordinate, the magnitude is just the Pythagorean distance from the origin, and the angle is the same arctangent calculation used to find any angle from x and y coordinates.
Converting back: polar to rectangular
Going the other direction, real part = r × cos(θ) and imaginary part = r × sin(θ). Taking the magnitude 5 and angle 53.13010235° from the example above and converting back gives exactly 3 + 4i again — an exact round trip with no rounding drift.
Why polar form is useful at all
Rectangular form (a + bi) is natural for addition and subtraction, but polar form makes multiplication and rotation far more intuitive — multiplying two complex numbers in polar form just multiplies their magnitudes and adds their angles, a much simpler operation than the rectangular-form expansion.
Where the angle convention comes from
The angle is measured counterclockwise from the positive real axis, in degrees — the same convention used for angles in standard 2D coordinate geometry, which is exactly why the same arctangent formula applies directly.