Complex to Polar Form Calculator
Verified Calculation Engine
The online Complex to Polar Form Calculator helps you calculate instantly and solve problems related to Complex Numbers. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Converts complex number from rectangular form (a + bi) to polar form r(cos θ + i sin θ) or r e^(iθ).
z = a + bi = r(cos θ + i sin θ) = r e^(iθ)
where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a)
Polar form expresses complex number using modulus and argument
Inputs
Results
Worked Examples
Example 1: Basic Conversion
Convert 3 + 4i to polar form
- real: 3
- imaginary: 4
- modulus: 5
- argument: 0.9273
- argumentDegrees: 53.13
- polarForm: 5(cos(0.9273) + i sin(0.9273))
- exponentialForm: 5e^(i0.9273)
Example 2: Board Level
Convert 1 + i to polar form
- real: 1
- imaginary: 1
- modulus: 1.4142
- argument: 0.7854
- argumentDegrees: 45
- polarForm: √2(cos(π/4) + i sin(π/4))
- exponentialForm: √2 e^(iπ/4)
About this calculator
Overview
Complex to Polar Form Calculator evaluates Complex to Polar Form for the math / complex-numbers library. Primary input cue: Real Part [required]: Real part a. Method anchor: z = a + bi = r(cos θ + i sin θ) = r e^(iθ) where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). Polar form expresses complex number using modulus and argumen.
When to use
Use Complex to Polar Form Calculator when you already know the inputs and need complex to polar form without setting up a spreadsheet.
Inputs explained
- Real Part [required]: Real part a
- Imaginary Part [required]: Imaginary part b
Formula / method
z = a + bi = r(cos θ + i sin θ) = r e^(iθ) where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). Polar form expresses complex number using modulus and argument (calculator id: complex-to-polar-form-calculator).
Worked example
Worked values: real = 3, imaginary = 4 → modulus = 5, argument = 0.9273, argumentDegrees = 53.13, polarForm = 5(cos(0.9273) + i sin(0.9273)), exponentialForm = 5e^(i0.9273)
Interpreting results
Cross-check with a simple numeric example before relying on extreme inputs for complex to polar form.
Assumptions
- Inputs for `complex-to-polar-form-calculator` follow the on-page labels; values/shapes must match this operation.
- Units (when shown) must be consistent across inputs for complex to polar form.
- Real-number arithmetic is used unless complex to polar form explicitly needs integers.
Limitations
- Complex to Polar Form Calculator cannot invent missing inputs; incomplete forms stop short of a forged complex to polar form value.
- Display rounding can differ slightly from a CAS/symbolic exact form of complex to polar form.
- Complex/branch cuts and undefined points are not always interactively annotated for complex to polar form.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
z = a + bi = r(cos θ + i sin θ) = r e^(iθ) where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). Polar form expresses complex number using modulus and argument (calculator id: complex-to-polar-form-calculator).
Example Calculation
Worked values: real = 3, imaginary = 4 → modulus = 5, argument = 0.9273, argumentDegrees = 53.13, polarForm = 5(cos(0.9273) + i sin(0.9273)), exponentialForm = 5e^(i0.9273)
Frequently Asked Questions
What is Complex to Polar Form Calculator?
- Complex to Polar Form Calculator evaluates Complex to Polar Form for the math / complex-numbers library. Primary input cue: Real Part [required]: Real part a. Method anchor: z = a + bi = r(cos θ + i sin θ) = r e^(iθ) where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). Polar form expresses complex number using modulus and argumen.
How does Complex to Polar Form Calculator work?
- z = a + bi = r(cos θ + i sin θ) = r e^(iθ) where r = |z| = √(a² + b²) and θ = arg(z) = tan⁻¹(b/a). Polar form expresses complex number using modulus and argument (calculator id: complex-to-polar-form-calculator).
Why use this Math Numbers calculator?
- Use Complex to Polar Form Calculator when you already know the inputs and need complex to polar form without setting up a spreadsheet.