Complex Numbers Explained with Simple Animations

Welcome to the fascinating world of complex numbers!To understand complex numbers, let's see how our number system has evolved over time.Complex numbers represent a fundamental expansion of our number system, allowing us to solve equations that were previously impossible.Let's break down what makes up a complex number.To visualize complex numbers, we use the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.Every point on this plane represents a unique complex number. The horizontal position gives us the real part, and the vertical position gives us the imaginary part.This means that any point with coordinates (x, y) represents the complex number x plus y i.For example, to plot the complex number 3 plus 2i, we move 3 units right along the real axis, and then 2 units up along the imaginary axis.Now that we understand what complex numbers are and how to visualize them, we're ready to explore their properties further.The imaginary unit i is defined as the square root of negative one.When we multiply any number by i, it rotates that number 90 degrees counterclockwise in the complex plane.Multiplying by i again gives us i squared, which equals negative one, completing another 90 degree rotation.A third multiplication by i brings us to negative i, rotating another 90 degrees.And a fourth multiplication completes the full rotation, bringing us back to positive one.Let's examine the pattern of multiplying by i.This rotation pattern follows the unit circle, showing how i is fundamentally connected to circular motion and periodic behavior.This rotation property applies to any complex number. When we multiply a complex number by i, it rotates 90 degrees counterclockwise while maintaining its magnitude.Adding complex numbers is similar to vector addition, where we handle real and imaginary parts separately.Let's start with the complex number 3 plus 2i.We'll add this to 1 plus 4i.When adding complex numbers, we add the real parts and imaginary parts separately.Three plus one equals four for the real part.And two i plus four i equals six i for the imaginary part.This gives us our result: four plus six i.Now let's look at subtracting complex numbers.We start with two plus three i.And subtract one plus i.Subtracting complex numbers is like adding the negative. We subtract the real parts: two minus one equals one. And subtract the imaginary parts: three i minus i equals two i.To multiply complex numbers, we use the FOIL method, similar to multiplying binomials.Let's multiply two plus i times one plus three i.First, we multiply the outer terms: two times one.Next, we multiply the outer and inner terms, then combine like terms.Remember that i squared equals negative one.Finally, we combine all terms to get negative one plus seven i.Geometrically, when we multiply complex numbers, we multiply their magnitudes.And we add their angles. The angle of our result is the sum of the individual angles.Here's our final result: negative one plus seven i, shown in green.Complex numbers are essential in electrical engineering, especially when analyzing AC circuits.Circuit elements like resistors, capacitors, and inductors each have their own complex impedance.In computer graphics and game development, complex numbers provide an elegant way to rotate objects.In quantum mechanics, wave functions are described using complex numbers, with both real and imaginary components.These wave functions describe the probability of finding a particle in different locations.When we combine these waves, we get the probability distribution for the quantum system.