1. What is Complex to Polar Conversion?

The polar form represents a complex number in terms of its magnitude (r) and angle (θ) rather than its real (a) and imaginary (b) components. This form is particularly useful in fields like electrical engineering and physics.

2. How Does the Calculator Work?

The calculator uses these conversion formulas:

Where:

• \( a \) — Real part of complex number (unitless)
• \( b \) — Imaginary part of complex number (unitless)
• \( r \) — Magnitude (always non-negative, unitless)
• \( \theta \) — Angle in radians (range: -π to π)

Explanation: The magnitude represents the distance from origin in the complex plane, while the angle represents the direction from the positive real axis.

3. Importance of Polar Form

Details: Polar form simplifies multiplication and division of complex numbers, and is essential for analyzing AC circuits, signal processing, and quantum mechanics.

4. Using the Calculator

Tips: Enter real and imaginary parts of your complex number. Both can be positive, negative, or zero (but not both zero). The angle will be in radians between -π and π.

5. Frequently Asked Questions (FAQ)

Q1: What if both a and b are zero?
A: The origin (0,0) has undefined polar form as the angle cannot be determined. The calculator will show no results in this case.

Q2: How do I convert back to rectangular form?
A: Use \( a = r \cos(\theta) \) and \( b = r \sin(\theta) \) to convert from polar back to rectangular form.

Q3: Why is my angle negative?
A: Angles are measured from the positive real axis. Negative angles represent clockwise rotation.

Q4: Can I get the angle in degrees?
A: Multiply the radian value by (180/π) to convert to degrees. 1 radian ≈ 57.2958 degrees.

Q5: What's the difference between atan and atan2?
A: This calculator uses atan2 which considers the signs of both components to determine the correct quadrant for the angle.