Complex Number Polar Form Converter

A complex number plots as a point (a, b); polar form stores the same information as radius and angle. For z = 3+4i: the modulus is r = √(9+16) = 5 and the argument is θ = atan(4/3) ≈ 53.13° (0.9273 radians), so z = 5(cos 53.13° + i sin 53.13°), or 5e^{0.9273i} in Euler form. The calculator shows the triangle (a, b, r), uses atan2 to place the angle in the correct quadrant (−1−i has θ = 225°, not 45°), and converts back: r(cos θ) = 3, r(sin θ) = 4. The payoff is arithmetic: multiplication multiplies moduli and adds angles — (5∠53.13°)(2∠30°) = 10∠83.13° — and powers follow De Moivre, zⁿ = rⁿ∠nθ, so (1+i)⁴ = (√2∠45°)⁴ = 4∠180° = −4.

1. 1

   Enter a + bi

   Real and imaginary parts, e.g. 3 and 4

2. 2

   Compute modulus and argument

   r = √(a²+b²); θ = atan2(b,a) in the right quadrant

3. 3

   Write the polar forms

   r(cosθ+isinθ) and Euler re^{iθ} appear together

4. 4

   Convert back on demand

   a = r·cosθ, b = r·sinθ verified

• 1

  Multiply: moduli multiply, angles add

• 2

  atan2 fixes the quadrant that plain atan misses

• 3

  Euler form re^{iθ} makes powers and rotations trivial

• Using atan(b/a) blindly — −1−i lands in the third quadrant (225°), not 45°

• Forgetting radians vs degrees when reading θ

• Adding angles during multiplication but forgetting to multiply moduli

• Reporting r negative; take r ≥ 0 and adjust θ by 180° instead