Complex Number to Polar Form Calculator
This calculator converts a complex number from rectangular form (a + bi) to polar form (r∠θ), the format engineers, electricians and physics students often need for AC circuit analysis, signal processing and rotations on the complex plane. You enter the real part (a) and the imaginary part (b), then choose whether the angle should be shown in degrees or radians. The tool instantly returns the modulus (r), worked out as the square root of a² + b² using the Pythagorean theorem, and the argument (θ), worked out using the atan2 function so the correct angle comes back in every quadrant, not just the first and fourth. Results appear as a full polar display of r∠θ alongside the equivalent Euler form, and result tiles break the argument down into degrees, radians and a fraction of pi. Below that sits a complete step-by-step working through the calculation, plus a summary of alternative notations including phasor form, trigonometric form, the modulus squared, and the complex conjugate. The tool also shows which quadrant your number falls in, based on the signs of a and b, and gives an input summary in standard a + bi notation. Type in your own real and imaginary parts and every figure updates immediately, letting you check working by hand or convert several numbers in a row without redoing the square roots and inverse tangents yourself.
1. Enter the Complex Number
2. Input Summary
Step-by-Step Working
Alternative Notations
What Is Polar Form?
Every complex number z = a + bi can be represented as a point on the complex plane, with the real part a on the horizontal axis and the imaginary part b on the vertical axis. Polar form describes this same point using two values: the distance from the origin (the modulus r) and the angle from the positive real axis (the argument θ).
Polar form is written as z = r∠θ, or equivalently as z = r(cos θ + i sin θ), or in Euler notation as z = re^(iθ). All three forms are equivalent and interchangeable.
The Conversion Formulas
To convert from rectangular form (a + bi) to polar form (r∠θ):
| Quantity | Formula | Description |
|---|---|---|
| Modulus (r) | r = √(a² + b²) | Distance from the origin; always ≥ 0 |
| Argument (θ) in radians | θ = atan2(b, a) | Angle from positive real axis, range (−π, π] |
| Argument (θ) in degrees | θ = atan2(b, a) × (180/π) | Range (−180°, 180°] |
The atan2(b, a) function is used rather than plain arctan(b/a) because it correctly handles all four quadrants, including when a is zero or negative.
Worked Example
Convert z = 3 + 4i to polar form.
- Calculate a² = 3² = 9
- Calculate b² = 4² = 16
- Sum: a² + b² = 9 + 16 = 25
- Modulus: r = √25 = 5
- Argument: θ = atan2(4, 3) ≈ 0.9273 radians ≈ 53.13°
- Polar form: z = 5∠53.13° (or 5ei0.9273)
With the default inputs (a = 3, b = 4), this calculator gives exactly these results: modulus 5 and argument 53.13°.
Quadrant Reference
| Quadrant | Signs of a, b | Angle range (degrees) |
|---|---|---|
| I | a > 0, b > 0 | 0° to 90° |
| II | a < 0, b > 0 | 90° to 180° |
| III | a < 0, b < 0 | -180° to -90° |
| IV | a > 0, b < 0 | -90° to 0° |
Why Use Polar Form?
Polar form makes certain operations with complex numbers much simpler. Multiplication of two complex numbers in polar form is carried out by multiplying the moduli and adding the arguments: (r1∠θ1) × (r2∠θ2) = r1r2∠(θ1 + θ2). Division is equally straightforward: divide the moduli and subtract the arguments. Raising a complex number to a power uses De Moivre's theorem: (r∠θ)^n = r^n ∠ nθ. These operations would require expanding brackets in rectangular form, making polar form far more efficient for engineering, signal processing, and physics applications.
Related Calculators
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Method: Modulus calculated as r = sqrt(a² + b²) using the Pythagorean theorem. Argument calculated using JavaScript Math.atan2(b, a) which correctly handles all quadrants and returns values in the range (-π, π]. Degrees converted from radians by multiplying by 180/π.
This calculator computes the exact modulus and argument for any complex number with real and imaginary parts you enter. Results are displayed to 4 decimal places. For very large or very small numbers, the result is computed using standard IEEE 754 double-precision floating point arithmetic.