About

Converting a complex number z = a + bi from rectangular to polar form requires computing two quantities: the modulus r and the argument θ. Errors in quadrant selection are the most common source of incorrect results. The two-argument arctangent function atan2(b, a) resolves this by mapping all four quadrants and the axes correctly, unlike the single-argument arctan which cannot distinguish between quadrants II and IV. This calculator uses IEEE 754-compliant atan2 and returns the principal argument in the range (−π, π].

At the origin (a = 0, b = 0), the modulus is 0 and the argument is undefined. This tool flags that condition explicitly rather than returning a misleading value. Results are provided in three equivalent representations: polar coordinates (r, θ), trigonometric form r(cos θ + i sin θ), and Euler form re^iθ.

Formulas

A complex number in rectangular form is written as:

z = a + bi

The modulus (magnitude) is computed as:

r = |z| = √a² + b²

The argument (phase angle) uses the two-argument arctangent to resolve quadrant ambiguity:

θ = atan2(b, a)

Conversion between radians and degrees:

θ_deg = θ_rad × 180π

The three equivalent polar representations are:

z = r ∠ θ

z = r(cos θ + i sin θ)

z = re^iθ

Where a = real part, b = imaginary part, r = modulus (always ≥ 0), θ = argument (principal value in (−π, π]), i = imaginary unit where i² = −1, and e = Euler's number ≈ 2.71828.

Reference Data

Complex Number	Modulus r	Argument (deg)	Argument (rad)	Quadrant
1 + i	√2 ≈ 1.4142	45°	π/4	I
−1 + i	√2 ≈ 1.4142	135°	3π/4	II
−1 − i	√2 ≈ 1.4142	−135°	−3π/4	III
1 − i	√2 ≈ 1.4142	−45°	−π/4	IV
3 + 4i	5	53.13°	0.9273	I
−3 + 4i	5	126.87°	2.2143	II
−3 − 4i	5	−126.87°	−2.2143	III
3 − 4i	5	−53.13°	−0.9273	IV
5	5	0°	0	Positive real axis
−5	5	180°	π	Negative real axis
5i	5	90°	π/2	Positive imaginary axis
−5i	5	−90°	−π/2	Negative imaginary axis
0	0	Undefined	Undefined	Origin
1 + √3i	2	60°	π/3	I
√3 + i	2	30°	π/6	I
−1	1	180°	π	Negative real axis
0.6 + 0.8i	1	53.13°	0.9273	I
7 + 24i	25	73.74°	1.2870	I
−12 + 5i	13	157.38°	2.7468	II
8 − 15i	17	−61.93°	−1.0808	IV

Frequently Asked Questions

The single-argument arctan(b/a) returns values only in (−π/2, π/2), which covers quadrants I and IV only. It cannot distinguish between, for example, 1 + i (quadrant I) and −1 − i (quadrant III) because both yield arctan(1) = 45°. The two-argument atan2(b, a) examines the signs of both a and b independently and returns the correct angle in (−π, π].

The modulus r equals 0. The argument θ is mathematically undefined because the origin has no direction. IEEE 754 atan2(0, 0) returns 0 by convention, but this is an implementation artifact. This calculator explicitly reports the argument as undefined in that case.

Add 2π (or 360°) to any negative argument. For example, θ = −135° becomes 360° − 135° = 225°. This calculator uses the principal value convention (−π, π], which is the standard in most mathematics and engineering textbooks.

By the principal value convention, pure negative real numbers (e.g., −5 + 0i) have an argument of π (or 180°), not −π. The interval is half-open: (−π, π], so π is included but −π is not. JavaScript's Math.atan2(0, −1) correctly returns π.

JavaScript uses 64-bit IEEE 754 double-precision floats, providing approximately 15 - 16 significant decimal digits. For most practical inputs, this is more than sufficient. Precision loss can appear when a and b differ by many orders of magnitude (e.g., a = 1e15, b = 1). This calculator displays results to 6 decimal places by default, which stays well within the reliable precision range.

Yes. Polar form makes multiplication trivial: multiply the moduli and add the arguments. For z₁ ⋅ z₂ = r₁r₂ ∠ (θ₁ + θ₂). De Moivre's theorem extends this to integer powers: zⁿ = rⁿ ∠ nθ. This is why polar form is preferred in signal processing, AC circuit analysis, and quantum mechanics.