Summary of Complex Numbers

Introduction

So, guys, complex numbers are totally not for me. I can barely even read this.

Okay, so complex numbers, the kind of numbers that have both real and imaginary parts?

It's kinda cool. It's also a whole new world of math to navigate, where you have to deal with complex conjugates and the idea of roots of unity.

But don't let the math scare you! Even if you're a math nerd, you can still appreciate the beauty of complex numbers. It's like finding a secret code to unlock a whole new world.

Just remember that the real part is the actual number and the imaginary part is this strange "j" part that makes complex numbers special.

And yes, complex numbers are super useful, especially in fields like electrical engineering and signal processing.

They allow us to model oscillating waves, like electricity, with more precision and clarity than with just real numbers.

It's like having a Swiss Army knife for complex calculations.

Okay, so now you're hooked, right? You just love math, huh?

Because it's a whole different beast. It's more fun.

It's a whole other level of understanding the world.

Complex numbers, with all their conjugates and their powers, are like the Swiss Army knife of math, the go-to tool for anyone dealing with waves and oscillations.

Just remember the rules, but don't be afraid to get a little wobbly.

You can do it.

And maybe you can even impress your friends with your complex number knowledge.

Because honestly, complex numbers are just that – more complex.

More like "unusually complex".

Just because they're more complex, though, doesn't mean you can't find some sweet and simple math concepts.

Complex numbers can be intimidating, but that's because we're just getting used to it, not because complex numbers themselves are the enemy.

It's a new perspective, and new insights.

Just remember, even the most complex problems have simple solutions.

Just take it slow.

Break it down into bite-sized pieces.

Don't let the complexity intimidate you.

It's just math.

It's just you.

So, get comfortable with it.

Let your math be messy and wonderful.

Remember, math is a conversation, not an interrogation.

It's about asking the right questions and listening for the answer.

Okay, now you're on the right track.

Let's break it down together, step by step.

So, you have a complex number, represented by z = a + bi, where "a" is the real part and "b" is the imaginary part.

Now, we want to find the square root of this complex number.

This might sound crazy, but trust me, it's doable.

Don't worry about the imaginary part, the imaginary part doesn't make us squiggle like crazy.

It's like a regular square root, only with the square root of "i".

And remember, complex numbers are like a whole different universe, full of surprises and cool tricks.

Just don't forget that you're still a math student, okay?

Even when dealing with the imaginary numbers, you're still a math student.

Okay, let's see what you got so far.

Just kidding, it was a whole other world.

Complex numbers are just numbers with a real and an imaginary part, like z = a + bi.

Now we want to find the square root of that complex number, z = a + bi.

It's kinda weird, like finding the square root of an imaginary number.

Don't worry, I'll show you how it works.

It's all about finding a complex number that, when we square it, we get the original complex number z = a + bi.

This might sound a little crazy, but I'm going to show you how to do it, step by step.

So, we'll call our unknown complex number, x + yi, where x is the real part and y is the imaginary part.

Our goal is to find the values of x and y.

We're looking for a number that when we square it, we'll get a + bi back.

How do we do it, you ask? Well, we'll start by assuming x and y are real numbers and they might have some weird imaginary parts that are squared away.

Then we'll use the square of the numbers.

Let's start with squaring x and y, just to show that the process might work, it might work out to something that's easier.

Okay, let's start with x = 2.

Well, okay, this one looks good, the value we're finding for the real part of z, so the x, is 2.

So, that means that if x is 2, then when we square 2, we get a perfect number.

So, squaring x gives us the number a = 2, perfect.

Let's square the imaginary number y and add it to 2.

So, 2 + (yi)^2, this might sound like it'll result in 4 + y2i, but the point is, we need to find something that squares down to 4.

It's the only thing that needs to stay the same because when we square, the numbers on the other side should cancel out, it's just about the values.

It's not really going to work out like this, but maybe you're gonna look for other patterns.

If the numbers squared need to get back to where they are, the number a, this can only be a real number.

Because if we get an imaginary part in there, we can't be able to solve for a.

So, that tells me, let me try some other numbers for the real part, the number we are squaring.

Well, let's say we try x = -2.

Okay, it looks good, we'll end up with a real number.

But let's see what the imaginary part looks like.

Squaring a real number, like we squared 2, squaring it just gives us another number, but let me take y, the imaginary number, and square it, okay.

So, -2 + yi is a, that means squaring that gives us -2 + y2i, but we need the result to be equal to 4 + y2i.

Okay, let me see, it can't work because -2 plus y squared i's isn't going to look like it's getting rid of the -2.

Wait, not really, but that is not a real number either.

And that is not something that's equal to the thing we're trying to find, we want 4 + y2i.

So, let me try another number.

Well, okay, 3? Let's see what happens.

Let me take y, let's say it's a negative 3, so our number is -3 + yi.

Okay, let me square the numbers to find out.

So, if we square y and take that and add it to 3, we get something.

Okay, let me start over, but I'll be careful.

We know that squaring 3 gives us 32.

So, it's going to give us a perfect square when squared, so 3 + (-y)^2 is going to equal 4, let me see, well it's the same.

If y is the square of 4.

Let me check my notes.

Let's say it's 4, 4 is the square root of 4, let's check it, and 2 squared is 4, okay, so I'll use this.

It'll be the answer, 2.

Let's use 2 to check it out, so let me use the two numbers and check to make sure that that's correct.

Let me start over.

So, 3 + 2i times 2 + 2i, is it equal to 4 + 2i, let's check to see what happened.

Okay, we know that 2 is the square root of 4, but what's it doing with 4 + 2i, that's a real number.

Now we have the sum, and now let's square the imaginary part.

If we square 3, let me just try 2 again.

Well, I can take that from my notes, and then, that will be our square, okay?

Okay, let me use 4 in here.

Okay, it'll be the square, but 2, because let's try the perfect square, that's the square, it's going to look good.

So, I think, okay, if 3 times 3, it's the perfect square, but if y is, say 3, 4 is a square of 2.

Let me square 4 again to make it look nice.

Okay, so if we square y, we get the result we're looking for.

So, let me check, what if 4, but we don't square that.

Wait, it's squared to 4, we need it squared to get us back to where it starts.

So, 2 + 2i, I think we're gonna go for it.

It's going to get rid of the -2, 2 is the number that we start with.

It's the one that squares back into itself.

Okay, if we square 2 again, we get 4 + 2i.

It's equal to 4 plus 2i, that is what we need.

Well, this works because we're just getting the correct numbers, right, the -2, that we subtract it from the other, and the other, 3 plus 2i.

Okay, let's try to prove that.

Well, it works for x.

Okay, so now, we are gonna do it for y, y times y, that is equal to 4 plus y2i.

If y times y equals 4, okay.

Now, let me start over, so 2 is our original number and 4, I've just added y2, okay, we have the equation that's going to let us know, okay, is we got, and we want the result that is, is 4 plus y2, so what happens if y is 0, okay, then 0 squared is zero, so we've got a number, that's 2 + 2i, which is our result, so it's gonna work for 0.

Let's take 1.

Let me square 1, okay, square 1.

Okay, let's add it to 2 and y, that is, add it to 3, we have 3 + 1 + i times 1.

So, y, so, it's just the one.

We'll do it with y, 2 plus 2, that is equal to 4, perfect.

Let me see if there are any patterns for that, y.

It looks like there are a pattern here that's going to be great for it.

It's a whole different whole pattern here, it's all going to be interesting.

Getting Serious

That was a joke. The real reason we use complex numbers is to simplify equations like this one. Consider the function:

What’s this going on?

First, we get all of the terms on one side. Then we get all the variables on the opposite side.

Let’s see what happens when we move that first side to the right:

Then we have just an equation of a simple linear form. We could also say this function is of a degree 1:

Notice that our new equation still has only two unknowns:

In the linear case, this is nice, right?

Wrong! That’s not the full story.

Remember that in a complex analysis, you have two parts, or pieces of information, about an unknown. If we call them

Why is there a different “form” for the above example?

If we have a solution x in a set , we have that x is an element of the set , and we can find the “multiple” of x in a set :

Now, our solution must also be in the set , right?

Not if it doesn’t contain a square root.

In the real numbers, that’s the end.

In the complex numbers, it’s not quite that simple.

You see, you can take a complex number in the form:

You might be surprised to learn that the number we write down on the screen is actually the product of the complex number we’ve just given you and the imaginary number 1:

Okay, so that’s all the imaginary part we need.

But that leaves us with the complex part to be more careful.

Remember what we just said, we were taking a complex number from the set and multiplying it by a complex number from the set .

So, let’s just have that and set it in our work, right?

Isn’t it a bad idea to work with 2 unknowns on both sides of an equation? No problem, because if we’re using the standard linear equation technique, we have:

Now we need to move everything with respect to the first unknown on both sides, right?

Let’s try the same approach.

Now that we have 2 equations and 2 variables, we can find all of the possible values of . But what about that? The solution to the real numbers is 4. We have four choices of.

Okay, let’s make some decisions.

First off, let’s say we don’t really care about the solution .

What can we say about the imaginary part of that solution?

Well, there are two possibilities:

• Well, this isn’t very useful.

• Oh no!