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 satisfying the equation \( i^2 = -1 \). Complex numbers extend the concept of one-dimensional real numbers to the two-dimensional complex plane.

2. Complex Number Representation

The standard form of a complex number is:

Where:

• \( a \) — Real part (unitless)
• \( b \) — Imaginary part (unitless)
• \( i \) — Imaginary unit (\( \sqrt{-1} \))

Explanation: The real part represents the horizontal axis, while the imaginary part represents the vertical axis in the complex plane.

3. Operations with Complex Numbers

Details: Complex numbers support various operations including addition, subtraction, multiplication, division, and more. They follow algebraic rules while accounting for \( i^2 = -1 \).

4. Using the Calculator

Tips: Enter the real and imaginary parts of your complex number. The calculator will display the properly formatted complex number in standard form.

5. Frequently Asked Questions (FAQ)

Q1: What is the imaginary unit i?
A: The imaginary unit \( i \) is defined by the property that \( i^2 = -1 \). It extends real numbers to complex numbers.

Q2: Can complex numbers be plotted?
A: Yes, complex numbers can be plotted on a 2D plane (Argand diagram) with real part on x-axis and imaginary part on y-axis.

Q3: What are practical applications of complex numbers?
A: Complex numbers are used in electrical engineering, quantum physics, signal processing, and many other fields.

Q4: How are complex numbers added or multiplied?
A: Addition is component-wise, while multiplication follows distributive property with \( i^2 = -1 \).

Q5: What is the complex conjugate?
A: The complex conjugate of \( a + bi \) is \( a - bi \). It's useful for division and finding magnitudes.