1. What is Polar Form?

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

2. How Does the Calculator Work?

The calculator uses the polar form equation:

Where:

• \( z \) — Complex number (unitless)
• \( r \) — Magnitude (unitless)
• \( \theta \) — Angle in radians
• \( e \) — Euler's number (~2.71828)
• \( i \) — Imaginary unit (√-1)

Explanation: The equation converts between polar and rectangular forms using Euler's formula \( e^{iθ} = \cosθ + i\sinθ \).

3. Importance of Polar Form

Details: Polar form simplifies complex number operations, especially multiplication (multiply magnitudes, add angles) and division (divide magnitudes, subtract angles).

4. Using the Calculator

Tips: Enter magnitude (r ≥ 0) and angle in radians. The calculator will display both polar and rectangular (a+bi) forms.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between polar and rectangular form?
A: Polar uses magnitude/angle (r,θ) while rectangular uses real/imaginary parts (a+bi). They represent the same complex number differently.

Q2: How do I convert degrees to radians?
A: Multiply degrees by π/180. Many calculators have a degree-to-radian conversion function.

Q3: Can the magnitude be negative?
A: No, magnitude (r) is always non-negative. Negative values are handled by adjusting the angle by π radians.

Q4: What's the principal value of the angle?
A: Typically between -π and π (or 0 to 2π), but angles outside this range are equivalent (periodic with 2π).

Q5: When is polar form most useful?
A: Particularly valuable for complex number multiplication, division, powers, and roots, as well as in signal processing and physics applications.