1. What Is Complex Number Magnitude And Angle?

The magnitude and angle of a complex number describe its position in the complex plane. The magnitude represents the distance from the origin, while the angle (argument) represents the direction from the positive real axis.

2. How Does The Calculator Work?

The calculator uses the following formulas:

Where:

• \( M \) — Magnitude (distance from origin)
• \( \theta \) — Angle in radians
• \( a \) — Real part of complex number
• \( b \) — Imaginary part of complex number

Explanation: The magnitude is calculated using the Pythagorean theorem, while the angle is determined using the arctangent function, providing the polar coordinates representation of the complex number.

3. Importance Of Complex Number Analysis

Details: Understanding magnitude and angle is crucial in electrical engineering, signal processing, quantum mechanics, and control systems. Polar form simplifies multiplication and division of complex numbers.

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 angle in radians.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of the angle θ?
A: The angle ranges from -π to π radians (-180° to 180°), covering all four quadrants of the complex plane.

Q2: How is the angle calculated when a = 0?
A: When a = 0, the angle is π/2 (90°) for positive b and -π/2 (-90°) for negative b. When both are zero, the angle is undefined.

Q3: What is the polar form of a complex number?
A: Polar form represents a complex number as \( z = M(\cos\theta + i\sin\theta) \) or \( z = Me^{i\theta} \) using Euler's formula.

Q4: Can I convert the angle to degrees?
A: Yes, multiply radians by 180/π (approximately 57.2958) to convert to degrees.

Q5: What applications use complex number magnitude and angle?
A: Used in AC circuit analysis, Fourier transforms, impedance calculations, vibration analysis, and quantum state representations.