So we've worked with real numbers like 3 and imaginary numbers like \(2i\) separately, but you're often going to see expressions that have these two numbers together, something like \(3+2i\). These are called complex numbers when a real number and an imaginary number are added together. Now, complex numbers are going to be really important for us throughout this course and have a ton of different uses. While they might sound a little complicated at first, I'm going to walk you through what they are and how we use them step by step. So let's get started.

A complex number has a standard form of \(a+bi\). In this complex number, \(a+bi\), \(a\) is called the real part of the number, and \(b\) is called the imaginary part because it's multiplying \(i\), our imaginary unit. Now, it's important to know that \(b\) by itself is the imaginary part. It's not the whole term \(bi\). When I'm identifying the real and the imaginary part in the number I have up here, this \(3+2i\), \(3\) is going to be the real part.

Let’s look at a couple of different examples of complex numbers and identify the real and imaginary parts of each of them. So first, I have this \(4 - 3i\). Looking at this number, the real part \(a\) is going to be this \(4\) because it's out there by itself. It's not multiplying my imaginary unit. This is going to be my real part \(a\). Then \(b\), so I want to look for what is multiplying \(i\), my imaginary unit. In this case, it is negative 3. Now, it's important to look at everything that's multiplying our imaginary unit. So if it's a negative number, if it's a square root, if it's a combination of a number and a square root, I want to get everything that's multiplying my imaginary unit. So in this case, \(b\) is going to be a negative 3. That's what's multiplying my imaginary unit \(i\).

Let's look at another example. So here I have \(0+7i\). Now, if I look for the real part of my number, what is not multiplying my imaginary unit, I have this as \(0\). So that means that, \(a\), my real part is going to be \(0\). Then, \(b\), my imaginary part, the part that's multiplying my imaginary unit \(i\), in this case is going to be positive \(7\). Now you might look at this number and think, couldn't you just write that number as \(7i\)? That zero isn't really doing anything. And you're right. I could just write this as \(7i\). But we still need to know that if we're looking at this as a complex number, it still has a real part. It's just 0.

Let’s look at another example. So I have \(2+0i\) over here. So what do you think the real part of this number is? Well, since this \(2\) is out here by itself, it's not multiplying my imaginary unit. \(2\) is going to be the real part of my number, \(a\). Then looking at \(b\), so the imaginary part, the part that is multiplying my imaginary unit, in this case is just 0. So, again, you might be looking at this number thinking, isn't that just a real number? Couldn't I just write that as \(2\)? And you're right. Again, I could just write this as \(2\). But remember, if we're looking at this as a complex number, it still has an imaginary part. It's just 0. So that's all for this one, and I'll see you in the next video.