Introduction to Complex Numbers and Vector Fields
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit satisfying the equation i^2 = -1. Complex numbers extend the one-dimensional real number line into a two-dimensional complex plane (called the Argand diagram), where the real part is plotted on the horizontal axis and the imaginary part is plotted on the vertical axis. This two-dimensional representation makes complex numbers exceptionally useful in geometry, physics, and engineering, particularly for modeling rotating vectors and alternating current (AC) circuits.
In electrical engineering, complex numbers are used as phasors to represent sinusoids, allowing circuit analysis involving resistors, capacitors, and inductors to be solved using algebraic equations rather than differential equations. The magnitude of the complex number represents the amplitude of the signal, while the phase angle represents the time shift or phase.
This calculator performs arithmetic operations (addition, subtraction, multiplication, and division) on two complex numbers entered in rectangular form, and separately finds the n-th roots of the first number using De Moivre's Theorem. Every result is automatically converted to polar form (magnitude and angle) alongside the rectangular answer, with the full step-by-step working shown.