1. What is Phase of a Complex Number?

The phase (or argument) of a complex number is the angle between the positive real axis and the vector representing the complex number in the complex plane. For a complex number a + bi, the phase φ is given by φ = arctan(b/a).

2. How Does the Calculator Work?

The calculator uses the arctangent function:

Where:

• \( a \) — Real part of complex number
• \( b \) — Imaginary part of complex number
• \( \phi \) — Phase angle in radians

Explanation: The calculator uses the two-argument arctangent function (atan2) to properly handle all quadrants and special cases (like when a = 0).

3. Importance of Phase Calculation

Details: Phase is crucial in signal processing, control systems, and quantum mechanics. It represents timing relationships in signals and wavefunctions.

4. Using the Calculator

Tips: Enter both real and imaginary parts. The calculator handles all cases including pure real (b=0) and pure imaginary (a=0) numbers.

5. Frequently Asked Questions (FAQ)

Q1: What's the range of phase values?
A: The phase ranges from -π to π radians (-180° to 180°), with positive angles counter-clockwise from the positive real axis.

Q2: How is phase different for a=0?
A: When a=0, phase is π/2 (90°) for positive b and -π/2 (-90°) for negative b.

Q3: What's the phase of zero?
A: The phase of 0+0i is technically undefined, though sometimes considered 0 by convention.

Q4: Why use atan2 instead of simple arctan?
A: atan2 correctly determines the quadrant and handles cases where a=0, unlike simple arctan(b/a).

Q5: How is phase used in practical applications?
A: Phase is essential in AC circuit analysis, signal processing (Fourier transforms), and quantum mechanics (wavefunction phase).