Complex Number To Polar Form Calculator

What Is the Polar Form of a Complex Number?

Every complex number z = a + bi exists in two equivalent representations: rectangular form (defined by real part a and imaginary part b) and polar form (defined by magnitude r and argument θ). The polar form is written as z = r(cos θ + i sin θ), or in compact exponential notation as z = re^iθ via Euler's formula. The complex number to polar form calculator converts between these representations using the standard formulas derived from coordinate geometry.

The Conversion Formula

Two quantities fully describe the polar form of any complex number:

• Magnitude (r): r = √(a² + b²) — the Euclidean distance from the origin to the point (a, b) in the complex (Argand) plane.
• Argument (θ): θ = atan2(b, a) — the angle the position vector makes with the positive real axis, measured counterclockwise in the range (−π, π].

Why atan2 Rather Than arctan?

The ordinary arctan(b/a) function returns values only in (−π/2, π/2), covering just two quadrants. The atan2(b, a) function examines the signs of both a and b independently to place the angle correctly in all four quadrants. For z = −1 + i, arctan(1/−1) incorrectly yields −45°, while atan2(1, −1) correctly returns 135°. This distinction is essential for unambiguous polar conversion.

Step-by-Step Derivation

Plot z = a + bi as a point on the Argand plane, with a on the horizontal axis and b on the vertical axis. The right triangle formed by a, b, and the position vector satisfies the Pythagorean theorem:

• r = √(a² + b²) (hypotenuse length by the Pythagorean theorem)
• cos θ = a/r and sin θ = b/r (trigonometric definitions of the angle)
• Therefore θ = atan2(b, a) (four-quadrant inverse tangent)

This geometric derivation is presented in detail by Lamar University's Complex Number Primer and forms a standard result in undergraduate complex analysis.

Variable Reference

• Real Part (a): Horizontal coordinate in the complex plane. For z = 3 + 4i, a = 3.
• Imaginary Part (b): Vertical coordinate; the coefficient of i. For z = 3 + 4i, b = 4.
• Magnitude (r): Always non-negative. For z = 3 + 4i, r = √(9 + 16) = 5.
• Argument (θ): For z = 3 + 4i, θ = atan2(4, 3) ≈ 53.13° or 0.9273 radians.

Worked Examples

Example 1: First Quadrant (a > 0, b > 0)

Convert z = 3 + 4i. Magnitude: r = √(9 + 16) = 5. Argument: θ = atan2(4, 3) ≈ 53.13° (0.9273 rad). Polar form: 5∠53.13°.

Example 2: Second Quadrant (a < 0, b > 0)

Convert z = −1 + √3·i. Magnitude: r = √(1 + 3) = 2. Argument: θ = atan2(√3, −1) = 120° (2π/3 rad). Polar form: 2∠120°.

Example 3: Fourth Quadrant (a > 0, b < 0)

Convert z = 1 − i. Magnitude: r = √(1 + 1) = √2 ≈ 1.414. Argument: θ = atan2(−1, 1) = −45° (−π/4 rad). Polar form: √2∠−45°.

Real-World Applications

Polar form is indispensable across applied science, as explained in Khan Academy's treatment of polar and rectangular forms:

• Electrical Engineering: AC impedance Z = R + jX converts to polar form to yield the impedance magnitude and phase angle used directly in phasor analysis and power factor calculations.
• Signal Processing: Fourier transform coefficients are complex numbers; their polar form separates the amplitude spectrum from the phase spectrum.
• Control Systems: Bode and Nyquist plots graph |H(jω)| and ∠H(jω) — the polar components of a transfer function evaluated on the imaginary axis.
• Quantum Mechanics: Probability amplitudes are complex numbers; r² gives the probability and θ carries the quantum phase.

Degrees vs. Radians

Select the output unit to match the downstream application. Radians appear directly in Euler's formula e^iθ and are standard in calculus and physics. Degrees are conventional in many engineering contexts. Conversion is straightforward: multiply radians by 180/π to obtain degrees, or multiply degrees by π/180 to obtain radians. For example, π/3 radians equals exactly 60°, and 1 radian ≈ 57.296°.