Complex Number Operations: A Reference Guide

This reference guide provides a concise overview of the fundamental operations involving complex numbers, essential for students, educators, and math enthusiasts.

Key Operations and Concepts

1. Representation: A complex number is denoted as \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the square root of -1.

Example: The number \(3 + 4i\) has a real part of 3 and an imaginary part of 4.

2. Addition:

\((a + bi) + (c + di) = (a + c) + (b + d)i\)

3. Subtraction:

\((a + bi) – (c + di) = (a – c) + (b – d)i\)

4. Multiplication:

\((a + bi) \times (c + di) = (ac – bd) + (ad + bc)i\)

5. Division:

\(\frac{a + bi}{c + di} = \frac{(a + bi) \times (c – di)}{c^2 + d^2}\)

6. Conjugate: The conjugate of \(a + bi\) is \(a – bi\).

7. Modulus: The modulus of a complex number \(a + bi\) is:

\(|a + bi| = \sqrt{a^2 + b^2}\)

8. Argument: The argument of a complex number is the angle it forms with the positive x-axis in the complex plane.

9. Polar Form: A complex number can be represented in polar form as:

\(r(\cos(\theta) + i\sin(\theta))\), where \( r \) is the modulus and \( \theta \) is the argument.

10. Euler’s Formula:

\(e^{i\theta} = \cos(\theta) + i\sin(\theta)\)

For more in-depth explanations and applications of complex numbers, consider exploring advanced mathematical resources or courses.