1. What is an Imaginary Number?

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i, where i is defined by its property i² = -1. Complex numbers have the form a + b i, where a is the real part and b is the imaginary part.

2. How Does the Calculator Work?

The calculator performs operations on complex numbers in the form:

Available operations:

• Complex Conjugate: Changes the sign of the imaginary part (a - b i)
• Magnitude: Calculates the distance from origin in complex plane (√(a² + b²))
• Angle (Phase): Calculates the angle with the real axis in degrees (atan2(b, a))

3. Importance of Complex Numbers

Applications: Complex numbers are fundamental in electrical engineering, quantum physics, signal processing, and many areas of mathematics. They provide a powerful way to represent oscillations, waves, and rotations.

4. Using the Calculator

Instructions: Enter the real and imaginary parts of your complex number, select the operation you want to perform, and click "Calculate". The result will be displayed in the result box.

5. Frequently Asked Questions (FAQ)

Q1: What is the complex conjugate used for?
A: The complex conjugate is used in division of complex numbers, finding magnitudes, and in quantum mechanics for probability amplitudes.

Q2: What does the magnitude represent?
A: The magnitude represents the distance of the complex number from the origin in the complex plane, analogous to the absolute value of real numbers.

Q3: What is the angle (phase) of a complex number?
A: The angle represents the direction of the complex number relative to the positive real axis in the complex plane.

Q4: Can I enter zero for both parts?
A: Yes, but the result will be trivial (0 for magnitude, undefined angle for 0+0i).

Q5: Are there other operations you could add?
A: Future versions might include addition, subtraction, multiplication, and division of two complex numbers.