polar and rectangular calculator

What This Calculator Does

This tool helps you switch between the two most common ways to represent a complex number: rectangular coordinates and polar coordinates. If you work in electrical engineering, control systems, signal processing, physics, or mathematics, this conversion is something you do constantly.

In rectangular form, a complex number is written as x + yi, where x is the real part and y is the imaginary part. In polar form, the same number is written as r∠θ, where r is the magnitude and θ is the angle.

Rectangular vs. Polar Form

Rectangular Form (x + yi)

Rectangular form is ideal when you need to add or subtract complex numbers. You can combine real parts with real parts and imaginary parts with imaginary parts.

Polar Form (r∠θ)

Polar form is convenient for multiplication, division, powers, and roots. Magnitudes multiply/divide naturally, and angles add/subtract.

Core Conversion Formulas

• Magnitude: r = √(x² + y²)
• Angle: θ = atan2(y, x)
• Real part: x = r cos(θ)
• Imaginary part: y = r sin(θ)

The atan2(y, x) function is important because it returns the correct angle quadrant automatically.

Quick Example

Rectangular to Polar

For z = 3 + 4i:

• r = √(3² + 4²) = 5
• θ = atan2(4, 3) ≈ 53.13°

So the polar form is approximately 5∠53.13°.

Polar to Rectangular

For z = 10∠30°:

• x = 10 cos(30°) ≈ 8.6603
• y = 10 sin(30°) = 5

So the rectangular form is approximately 8.6603 + 5i.

Common Mistakes to Avoid

• Mixing up degrees and radians.
• Using arctan(y/x) instead of atan2(y, x), which can give wrong quadrants.
• Forgetting that multiple angles can represent the same direction (e.g., 30° and 390°).
• Rounding too early, especially in multi-step calculations.

When You Should Use Each Form

• Use rectangular: addition, subtraction, plotting on Cartesian axes.
• Use polar: multiplication, division, powers, roots, and phase analysis.

Final Note

A strong understanding of both formats makes complex-number work much faster and less error-prone. Use the calculator above to check homework, validate engineering calculations, or build intuition with your own examples.