Complex Numbers

Basic Operations with Complex Numbers

A complex number is a number in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit defined as \(i^2 = -1\).

Basic Operations

For complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\):

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

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

Multiplication: \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\)

Division: \(\frac{a + bi}{c + di} = \frac{(ac + bd)}{c^2 + d^2} + \frac{(bc - ad)}{c^2 + d^2}i\)

Interactive Complex Plane: Visualize complex numbers and operations!

Complex Conjugates

The complex conjugate of a complex number \(a + bi\) is \(a - bi\).

The product of a complex number and its complex conjugate is a real number.

For example, \((2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13\).

Magnitude and Absolute Value

The magnitude (or absolute value) of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\).

This represents the distance from the origin to the point (a, b) in the complex plane.

Ready to Practice?

Test your knowledge of all complex number concepts.