1. What is Polar Form of Complex Numbers?

The polar form represents complex numbers using their magnitude (distance from origin) and angle (with positive real axis). It's especially useful for multiplication, division, and powers of complex numbers.

2. How Does the Calculator Work?

The calculator converts between polar and rectangular forms using these equations:

Where:

• \( r \) — Magnitude (modulus) of the complex number
• \( θ \) — Angle (argument) in degrees
• \( a \) — Real part in rectangular form
• \( b \) — Imaginary part in rectangular form

3. Importance of Polar Representation

Details: Polar form simplifies complex number operations. Multiplication becomes multiplying magnitudes and adding angles, while division becomes dividing magnitudes and subtracting angles.

4. Using the Calculator

Tips: Enter magnitude (must be ≥ 0) and angle in degrees. The calculator will display both the polar form and equivalent rectangular form.

5. Frequently Asked Questions (FAQ)

Q1: What's the range for angle θ?
A: While any angle is valid, typically -180° to 180° or 0° to 360° is used for principal value.

Q2: How is polar form different from rectangular form?
A: Rectangular uses x+yi format while polar uses magnitude and angle, better representing rotational operations.

Q3: Can I enter negative magnitude?
A: Magnitude (r) must be ≥ 0. Negative r would be equivalent to adding 180° to the angle.

Q4: What's Euler's formula relation to polar form?
A: Polar form can also be written as \( re^{iθ} \) using Euler's formula \( e^{iθ} = \cosθ + i\sinθ \).

Q5: When is polar form most useful?
A: Particularly valuable for complex number multiplication, division, roots, and powers, as well as in electrical engineering for AC circuit analysis.