1. What is a Complex Number?

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit that satisfies the equation i² = -1. Complex numbers extend the concept of one-dimensional real numbers to two dimensions.

2. How Does the Calculator Work?

The calculator uses the standard complex number formula:

Where:

• \( a \) — Real part (unitless)
• \( b \) — Imaginary coefficient (unitless)
• \( i \) — Imaginary unit (√-1)

Explanation: The calculator combines the real and imaginary parts to form a standard complex number representation.

3. Importance of Complex Numbers

Details: Complex numbers are fundamental in mathematics, physics, and engineering. They are used in signal processing, control theory, electromagnetism, quantum physics, and many other fields.

4. Using the Calculator

Tips: Enter the real part (a) and imaginary coefficient (b) as real numbers. The calculator will format them properly into complex number notation.

5. Frequently Asked Questions (FAQ)

Q1: What does 'i' represent in complex numbers?
A: 'i' is the imaginary unit, defined as the square root of -1 (i² = -1).

Q2: Can b be zero in a complex number?
A: Yes, when b=0 the number is purely real (a + 0i = a).

Q3: Can a be zero in a complex number?
A: Yes, when a=0 the number is purely imaginary (0 + bi = bi).

Q4: What are some applications of complex numbers?
A: They're used in electrical engineering (AC circuits), quantum mechanics, fluid dynamics, and signal processing.

Q5: How are complex numbers represented graphically?
A: On a complex plane with the real part on the x-axis and imaginary part on the y-axis.