Complex numbers are numbers that include the imaginary unit \( i \), where \( i = \sqrt{-1} \). These numbers can be represented as \( a + bi \), where:
In the Argand Plane (also called the complex plane), the real part \( a \) is plotted on the x-axis, and the imaginary part \( b \) is plotted on the y-axis. This gives you a 2D representation of complex numbers.
For example, the complex number \( 3 + 4i \) is represented as a point at coordinates (3, 4) on the Argand Plane.
Multiplying complex numbers geometrically is like applying a rotation and scaling in the plane. For example, multiplying by \( e^{ix} \) rotates a point by an angle \( x \).
Euler's formula shows this relationship:
\[ e^{ix} = \cos x + i \sin x \]
Complex numbers are essential in fields like electrical engineering, physics, and signal processing. For instance, they help us analyze alternating current (AC) circuits and waveforms.