Rectangular to Polar Calculator Imaginary

Rectangular to Polar Conversion:

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

Magnitude (r):

Angle (θ):

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Rectangular to Polar Conversion?

Rectangular to polar conversion transforms complex numbers from rectangular form (a + bi) to polar form (r∠θ). This is particularly useful in 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 (absolute value)
\( \theta \) — Angle (phase) 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 is particularly useful for multiplication and division of complex numbers, as well as for analyzing AC circuits and wave functions.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will compute both the magnitude and angle (in radians and degrees).

5. Frequently Asked Questions (FAQ)

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

Q2: How is the angle determined?
A: The angle is calculated using the arctangent of (b/a), adjusted for the correct quadrant using atan2 function.

Q3: What happens when a = 0?
A: When a=0, the angle is π/2 (90°) if b>0 or -π/2 (-90°) if b<0.

Q4: Can I convert back to rectangular form?
A: Yes, using a = r·cos(θ) and b = r·sin(θ).

Q5: Why is polar form useful?
A: It simplifies multiplication (multiply magnitudes, add angles) and division (divide magnitudes, subtract angles) of complex numbers.