Addition
(a+bi) + (c+di) = (a+c) + (b+d)i
Multiplication
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Division
(a+bi)/(c+di) = [(ac+bd)+(bc−ad)i]/(c²+d²)
Magnitude
|z| = √(a² + b²)
z = r(cos θ + i sin θ)
or z = r·e^(iθ)
r = magnitude = √(a² + b²)
θ = argument = arctan(b/a)
Complex numbers are numbers that extend the real number system by including an imaginary unit, denoted as "i", which is defined by the property i² = −1. A complex number is written in the form z = a + bi, where "a" is the real part and "b" is the imaginary part. Complex numbers are essential in many areas of mathematics, physics, and engineering.
The complex plane (also called the Argand diagram) is a geometric representation where the real part is plotted on the horizontal axis and the imaginary part on the vertical axis. This visualization helps understand operations like addition (vector addition) and multiplication (rotation and scaling).
Complex numbers have wide-ranging applications across science and engineering. In electrical engineering, they represent alternating current (AC) circuits where the real part represents resistance and the imaginary part represents reactance. In signal processing, the Fourier transform uses complex exponentials to analyze frequency components.
In quantum mechanics, complex numbers are fundamental to wave functions. In control systems, they help analyze stability. Even in fractal geometry, the famous Mandelbrot set is defined using complex number iterations. Understanding complex arithmetic is essential for advanced mathematics and physics.
Note: Complex number calculations follow standard mathematical formulas. Results depend on correct input and selected operation. Angles are displayed in degrees for readability.