1. What is Phase and Magnitude?

Phase and magnitude are fundamental properties of complex numbers that describe their behavior in the complex plane. The magnitude represents the distance from the origin, while the phase (or argument) represents the angle from the positive real axis.

2. How Does the Calculator Work?

The calculator uses the following formulas:

Where:

• \( Real \) — Real part of the complex number
• \( Imag \) — Imaginary part of the complex number
• \( Magnitude \) — Distance from origin in complex plane
• \( Phase \) — Angle in radians from positive real axis

Explanation: These formulas convert complex numbers from rectangular coordinates (Real, Imag) to polar coordinates (Magnitude, Phase).

3. Importance of Complex Number Analysis

Details: Phase and magnitude calculations are essential in signal processing, electrical engineering, control systems, and quantum mechanics. They help analyze frequency response, filter design, and system stability.

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. Both values can be positive, negative, or zero. The calculator will compute the magnitude and phase in radians.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of phase values?
A: Phase typically ranges from -π to π radians (-180° to 180°), with positive angles counterclockwise from the real axis.

Q2: How is phase calculated when real part is zero?
A: When real part is zero, phase is π/2 (90°) for positive imaginary and -π/2 (-90°) for negative imaginary.

Q3: Can magnitude be negative?
A: No, magnitude is always non-negative as it represents a distance in the complex plane.

Q4: What is the difference between atan and atan2?
A: atan2 considers the signs of both arguments to determine the correct quadrant, while atan only considers the ratio.

Q5: How are these calculations used in engineering?
A: In electrical engineering, magnitude represents amplitude and phase represents phase shift in AC circuits and signals.