What is polar form of a complex number?

z = r(cos θ + i sin θ) where r = |z| and θ = arg(z).

How do I convert from rectangular to polar form?

r = √(a²+b²), θ = atan2(b,a).

Why is atan2 used instead of arctan?

atan2 handles all quadrants correctly.

Complex Number Polar Form Calculator

Complex Number Polar Form Calculator — Convert Between Rectangular & Polar

Convert complex numbers between rectangular (a + bi) and polar form (r∠θ) with step-by-step explanation. Learn Euler's formula and geometric interpretation. For related tools, see our complex modulus calculator.

1What Is Polar Form?
2Rectangular to Polar Conversion
3Polar to Rectangular Conversion
4Step-by-Step Examples
5More Conversion Examples
6Euler's Formula: r·e^(iθ)
7Argument and Quadrants
8Multiplication in Polar Form
9Common Mistakes to Avoid
10Real-World Applications
11Practice Problems with Solutions
12Tips for Mastering Polar Form
13Final Thoughts
14FAQ: Common Questions

Convert Between Rectangular and Polar Form

Enter a complex number in rectangular form (a + bi) or polar coordinates ("r, θ"). Click Convert for step-by-step working.

Complex Number (a + bi) or "r, θ" for polar Angle Unit

Result will appear here.

Note: Polar form uses modulus r and argument θ: r(cos θ + i sin θ) = r∠θ = r·e^(iθ).

Complex Number Polar Form – Complete Explanation

Polar form represents a complex number using modulus r and argument θ. It simplifies multiplication, division, and exponentiation.

Polar Form Representations

z = r(cos θ + i sin θ)    Trigonometric
z = r∠θ                    Angle notation
z = r·e^iθ                 Exponential (Euler)

1. What Is Polar Form?

Polar form answers: "What are the magnitude and direction of this complex number?"

2. Rectangular to Polar Conversion

r = √(a² + b²).
θ = atan2(b, a) — handles all quadrants.

3. Polar to Rectangular Conversion

a = r cos θ.
b = r sin θ.

4. Step-by-Step Examples

Example 1: 3 + 4i to Polar

r = √(9+16) = 5
θ = atan2(4,3) = 53.13°
Polar: 5(cos 53.13° + i sin 53.13°) = 5∠53.13°

Example 2: -1 + i to Polar

r = √(1+1) = √2
θ = atan2(1,-1) = 135°
Polar: √2(cos 135° + i sin 135°) = √2∠135°

5. More Conversion Examples

5 → 5∠0° (real number)

-3i → 3∠-90° (pure imaginary)

-2-2i → 2√2∠-135°

1+i√3 → 2∠60°

6. Euler's Formula: r·e^(iθ)

Euler's Formula

e^iθ = cos θ + i sin θ

e^(i0) = 1 | e^(iπ/2) = i | e^(iπ) = -1 | e^(i3π/2) = -i | e^(i2π) = 1

7. Argument and Quadrants

QI: θ = arctan(b/a) | QII: θ = arctan(b/a)+180° | QIII: θ = -180° to -90° | QIV: θ = -90° to 0°

8. Multiplication in Polar Form

Multiplication Rule

r₁e^iθ₁ × r₂e^iθ₂ = r₁r₂ e^{i(θ₁+θ₂)}

Multiply moduli, add arguments.

9. Common Mistakes to Avoid

Using arctan instead of atan2.
Forgetting angle units.
Negative modulus.
Confusing forms.
Not simplifying angle.

10. Real-World Applications

Electrical Engineering: Phasors.
Signal Processing: Fourier transform.
Control Systems: Bode plots.
Computer Graphics: 2D rotation.

11. Practice Problems with Solutions

Problem 1: 1+i → √2∠45°

Problem 2: 5∠53.13° → 3+4i

Problem 3: 2e^(iπ/3) → 1+i√3

Problem 4: 2∠30°×3∠45° → 6∠75°

12. Tips for Mastering Polar Form

Use atan2 always.
Draw a sketch to verify quadrant.
Multiplication: multiply r, add θ.
Division: divide r, subtract θ.
Memorize Euler's identity.

13. Final Thoughts

Polar form reveals the geometry of complex numbers. Use this calculator to verify your work, but practice until conversions become natural.

14. Frequently Asked Questions

Polar form represents z = a+bi using modulus r = √(a²+b²) and argument θ = atan2(b,a): z = r(cos θ + i sin θ) = r∠θ = r·e^(iθ).

Calculate r = √(a²+b²) and θ = atan2(b,a). Then write z = r(cos θ + i sin θ) or r∠θ.

Calculate a = r cos θ and b = r sin θ. Then write z = a + bi.

e^(iθ) = cos θ + i sin θ. This connects complex exponentials to trigonometry. Euler's identity e^(iπ) + 1 = 0 is a special case.

atan2(b,a) correctly identifies the quadrant. Ordinary arctan(b/a) only gives angles in (-90°, 90°) and cannot distinguish between QI and QIII, or QII and QIV.

Multiply the moduli and add the arguments: r₁e^(iθ₁) × r₂e^(iθ₂) = r₁r₂e^(i(θ₁+θ₂)). This represents scaling and rotation.

No. The modulus is a distance and is always ≥ 0. If you get negative r, take its absolute value and add 180° to the argument.

Explore Math Topics

Continue your learning journey with our other math guides

🔢 Basic Arithmetic 📐 Pre-Algebra 📊 Algebra 📏 Geometry 📈 Statistics

Related Calculators

Explore these other math tools that complement polar form:

🔹 Complex Modulus Calculator — Find |z|.
🔹 Complex Conjugate Calculator — Find z̄.
🔹 Complex Number Multiplication — Multiply complex numbers.
🔹 Complex Number Division — Divide complex numbers.
🔹 Complex Number Addition — Add complex numbers.

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