Complex Number to Polar Form Calculator

The polar form expresses a complex number z using its distance from the origin (modulus r) and the angle it makes with the positive real axis (argument θ). It is written as z = r(cos θ + i sin θ), as opposed to the standard rectangular form z = a + bi.

Given z = a + bi, calculate the modulus r = √(a² + b²) and the argument θ = atan2(b, a). Then write the polar form as z = r(cos θ + i sin θ). The atan2 function correctly handles all quadrants, unlike the simple arctan(b/a).

The exponential form is z = r·e^(iθ), derived from Euler's formula e^(iθ) = cos θ + i sin θ. It is equivalent to the trigonometric polar form and is often more compact and convenient for multiplication and division of complex numbers.

The imaginary unit i has a = 0 and b = 1. Its modulus is r = √(0² + 1²) = 1, and its argument is θ = π/2 (or 90°). So the polar form of i is 1·(cos(π/2) + i sin(π/2)), or simply e^(iπ/2).

The complex number 0 has modulus r = 0 and its argument θ is undefined (or conventionally taken as 0). The polar form is 0·(cos 0 + i sin 0) = 0. Because r = 0, the angle carries no geometric meaning.

The argument θ lies in the range (−π, π]. For a complex number in the first quadrant (a > 0, b > 0), θ is positive. In the second quadrant (a < 0, b > 0), θ is between π/2 and π. In the third quadrant (a < 0, b < 0), θ is between −π and −π/2, and in the fourth quadrant (a > 0, b < 0), θ is negative. Using atan2(b, a) accounts for all cases automatically.

Multiplying two complex numbers in polar form is straightforward: multiply their moduli and add their arguments. If z₁ = r₁·e^(iθ₁) and z₂ = r₂·e^(iθ₂), then z₁·z₂ = r₁r₂·e^(i(θ₁+θ₂)). This is much simpler than expanding (a + bi)(c + di) in rectangular form.

The modulus (also called the magnitude or absolute value) of a complex number z = a + bi is r = √(a² + b²). It represents the Euclidean distance from the origin to the point (a, b) in the complex plane and is always a non-negative real number.