1. What is Polar Form?

The polar form represents a complex number using its magnitude (r) and angle (θ) rather than its real and imaginary components. It's particularly useful for multiplication and division of complex numbers.

2. How Does the Calculator Work?

The calculator converts from rectangular form (a + bi) to polar form:

Where:

• \( r = \sqrt{a^2 + b^2} \) — Magnitude (modulus)
• \( \theta = \arctan(b/a) \) — Angle (argument) in radians

Explanation: The magnitude represents the distance from the 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 complex number operations, especially multiplication and division, and is essential in fields like electrical engineering and physics.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will compute the magnitude and angle, then display the polar form representation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rectangular and polar form?
A: Rectangular form uses x and y coordinates (a + bi), while polar form uses magnitude and angle (r∠θ).

Q2: Can the angle be in degrees instead of radians?
A: Yes, but the standard mathematical convention uses radians. Multiply by 180/π to convert to degrees.

Q3: What happens when both a and b are zero?
A: The polar form would be 0 (cos(0) + i sin(0)), but the angle is technically undefined.

Q4: How is the angle determined?
A: The atan2 function is used to determine the correct quadrant for the angle.

Q5: What's Euler's formula relation to polar form?
A: Euler's formula \( e^{iθ} = \cosθ + i\sinθ \) allows writing polar form as \( z = re^{iθ} \).