Phase Calculator Complex Number

Phase Calculation Formula:

\[ \text{Phase} = \arctan\left(\frac{\text{Imag}}{\text{Real}}\right) \]

Phase Result:

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1. What is Phase Calculation?

Phase calculation determines the angle or argument of a complex number in the complex plane. It represents the direction of the complex vector from the origin and is a fundamental concept in complex analysis and signal processing.

2. How Does the Calculator Work?

The calculator uses the phase calculation formula:

\[ \text{Phase} = \arctan\left(\frac{\text{Imag}}{\text{Real}}\right) \]

Where:

\( \text{Real} \) — Real part of the complex number
\( \text{Imag} \) — Imaginary part of the complex number
\( \arctan \) — Inverse tangent function (atan2 is used for proper quadrant determination)

Explanation: The phase is calculated using the two-argument arctangent function (atan2) which properly handles all four quadrants and special cases where the real part is zero.

3. Importance of Phase Calculation

Details: Phase calculation is essential in various fields including electrical engineering (AC circuit analysis), signal processing (Fourier analysis), control systems, quantum mechanics, and many other applications involving complex numbers and waveforms.

4. Using the Calculator

Tips: Enter both real and imaginary parts of the complex number. The calculator will compute the phase angle in both degrees and radians. Avoid entering zero for both parts simultaneously.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of phase values?
A: Phase values typically range from -180° to +180° (-π to +π radians), covering all four quadrants of the complex plane.

Q2: How does atan2 differ from regular arctan?
A: atan2(y, x) properly determines the quadrant of the result and handles cases where x = 0, unlike atan(y/x) which can produce ambiguous results.

Q3: What happens when both real and imaginary parts are zero?
A: The phase is undefined (singularity) since the complex number is at the origin with no defined direction.

Q4: How is phase used in practical applications?
A: Phase is crucial for analyzing alternating currents, signal processing filters, communication systems, and understanding wave interference patterns.

Q5: Can phase be negative?
A: Yes, phase can be negative, representing angles measured clockwise from the positive real axis in the complex plane.