Simple complex numbers

For a recent project I’ve found myself having to use complex numbers since the formula I relied on uses them.

The only problem was that I had no idea what a complex number is (feel free to shame me). I started looking for explanations and definitions and they all looked the same:

A complex number is a number of the form a + bi, where a and b are real numbers, and i is an indeterminate satisfying i2 = −1.

… Uhu. Imaginary, you say. Tell me more.

The value of i = (√-1). For example, 2+3i is a complex number, where 2 is a real number (Re) and 3i is an imaginary number (Im)

Okay this isn’t going to help me. At all.

So to help future me, and perhaps some unlucky randos who find themselves here, I decided to try and simplify.

Complex numbers for simpletons

Step back, I’ll now attempt to explain complex numbers in a way that is easier to understand. The only prior knowledge required is some basic coding since I’ll be using TypeScript to illustrate some things.

Now, let’s piss off some mathematicians! Here’s the dumbest explanation I could come up with:

A complex number is not a number. It’s a dictionary of two values that adhere to some constraints.

In code:

interface ComplexNumber {
	a: number; // The "real" number
	bi: number; // The "Imaginary" number
}

Only constant i can make imaginary numbers.

A complex number is a number with 2 components: real, and imaginary.

• The real number represents the amplitude
• The imaginary number represents the phase or timing information

It essentially holds 2 values that can be used by a function to do stuff.

Imagine a good ol’ coordinate plane with a circle on it. The x-axis represents the real values, while the y-axis represents the imaginary values. The origin (0, 0) is at the center of the circle. Each point on the circle corresponds to a complex number.

Now, that a + bi equation I mentioned earlier? Here’s what it means in our world. The “a” value is the x-coordinate on the circle, indicating the real part of the complex number. The “bi” value is the y-coordinate, representing the imaginary part. By plotting these coordinates, we can visualize complex numbers as points on our coordinate plane.

But hold on, why should we bother with these complex numbers? Well, they have some nifty properties that come in handy in various fields like math, physics, and engineering. They help us solve problems involving waves (that’s what I am using them for), rotations, electrical circuits, and more. So, they’re not just some fancy mathematical voodoo; they have practical applications.

In a nutshell, complex numbers aren’t as daunting as they first appear. They’re just a mix of real and imaginary parts, nicely packaged as a + bi. Understanding them opens up a world of possibilities, letting us tackle complex problems with elegance and efficiency.

So, let’s embrace the power of complex numbers and let them be our guide through the fascinating realms of math and beyond.

And also let’s all collectively dunk on the mathematicians who went with “imaginary” and “complex” as names for these numbers.