Rectangular To Polar Complex Number Calculator

Rectangular to Polar Conversion:

\[ z = x + y i \Rightarrow r e^{i\theta} \] \[ r = \sqrt{x^2 + y^2} \] \[ \theta = \arctan\left(\frac{y}{x}\right) \]

Magnitude (r):

Angle (θ) in radians:

Angle (θ) in degrees:

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1. What is Rectangular to Polar Conversion?

The rectangular to polar conversion transforms a complex number from its rectangular form (x + yi) to its polar form (re^iθ). This representation is particularly useful in fields like electrical engineering, physics, and signal processing.

2. How Does the Calculator Work?

The calculator uses the following formulas:

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

Where:

\( x \) — Real part of the complex number
\( y \) — Imaginary part of the complex number
\( r \) — Magnitude (modulus) of the complex number
\( \theta \) — 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 Complex Number Representations

Details: Polar form is particularly useful for multiplication and division of complex numbers, as well as for analyzing periodic phenomena like alternating currents or wave functions.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between rectangular and polar forms?
A: Rectangular form uses Cartesian coordinates (x + yi), while polar form uses magnitude and angle (re^iθ).

Q2: How do I convert back from polar to rectangular?
A: Use x = r·cos(θ) and y = r·sin(θ) to convert back to rectangular form.

Q3: What's the range for the angle θ?
A: The calculator returns θ in the range (-π, π] radians or (-180°, 180°].

Q4: What happens when x = 0?
A: When x = 0, θ will be π/2 (90°) if y > 0 or -π/2 (-90°) if y < 0.

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