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 \(i^2 = -1\).
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!
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, \((2 + 3i)(2 - 3i) = 4 - 9i^2 = 4 + 9 = 13\).
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.
Test your knowledge of all complex number concepts.