Try an Example
Pick a scenario to see how the calculator works, then adjust the values
Signal Addition
Add two phasor signals in complex form to find the resultant.
Key values: z₁ = 3 + 4i · z₂ = 1 − 2i · Addition
Circuit Impedance
Divide two impedances to compute the transfer ratio.
Key values: z₁ = 3 + 4i · z₂ = 1 + 2i · Division
De Moivre Power
Raise a complex number to the 10th power using De Moivre's theorem.
Key values: z₁ = 1 + i · n = 10 · Power
Cube Roots of −8
Find all three cube roots of −8, visualized as equidistant points on a circle.
Key values: z₁ = −8 + 0i · n = 3 · nth Root
Complex Number Calculator
Enter complex numbers and choose an operation to see the result.
This calculator is also known as Rectangular to Polar Form Calculator.
Read the complete guideStep-by-Step Conversion
Given z = a + bi: (1) Compute r = √(a² + b²). (2) Compute θ = atan2(b, a) in radians. (3) Convert to degrees if needed: θ° = θ × 180/π. (4) Write in polar form: z = r∠θ° or z = r·cis(θ).
Examples
Convert −1 − √3·i to Polar Form
Find the polar form of z = −1 − √3·i.
r = √(1 + 3) = 2. θ = atan2(−√3, −1) = −2π/3 = −120°. So z = 2∠(−120°).
Key takeaway: Points in the third quadrant have negative arguments (below the real axis). The atan2 function handles this automatically.
Rectangular-to-Polar Tips
Common tips for form conversion:
- Memorize common values: |1+i| = √2, arg(1+i) = 45°; |1+√3·i| = 2, arg = 60°
- Use atan2 to avoid quadrant errors — it is built into every scientific calculator and programming language
- In exam settings, sketch the point on the Argand diagram first to verify the quadrant visually
Frequently Asked Questions about Rectangular to Polar Form Calculator
Why not use arctan(b/a) instead of atan2(b, a)?
arctan(b/a) only gives angles in (−π/2, π/2), which covers only the right half of the plane. atan2(b, a) returns the correct angle in (−π, π] for all four quadrants and handles a = 0.
How do I convert back to rectangular form?
Given r∠θ: a = r·cos(θ) and b = r·sin(θ). So z = r·cosθ + i·r·sinθ.
Specialized Calculators
Choose from 11 specialized versions of this calculator, each optimized for specific use cases and calculation methods.
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