Convert Imaginary to Polar Calculator

Conversion Formulas:

\[ r = \sqrt{a^2 + b^2} \] \[ \theta = \arctan\left(\frac{b}{a}\right) \]

Magnitude (r):

Angle (θ):

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1. What is Polar Conversion?

The polar form represents complex numbers using magnitude (r) and angle (θ) instead of real (a) and imaginary (b) components. This conversion is essential in many fields including electrical engineering, physics, and signal processing.

2. How Does the Calculator Work?

The calculator uses these formulas:

\[ r = \sqrt{a^2 + b^2} \] \[ \theta = \arctan\left(\frac{b}{a}\right) \]

Where:

\( a \) — Real part of the complex number
\( b \) — Imaginary part of the complex number
\( r \) — Magnitude (always positive)
\( \theta \) — Angle in radians (-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, wave functions, and rotational systems.

4. Using the Calculator

Tips: Enter both real and imaginary parts. The calculator handles all cases including when a=0 (which would normally make the denominator zero in the θ calculation).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between atan and atan2?
A: atan2 properly handles all quadrants and cases where a=0, giving angles in the correct range (-π to π).

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

Q3: What if both a and b are zero?
A: The origin (0,0) has r=0 but θ is undefined.

Q4: How do I get the angle in degrees?
A: Multiply radians by 180/π (approximately 57.2958).

Q5: What are common applications?
A: Used in impedance calculations, Fourier transforms, phasor analysis, and quantum mechanics.