 The last thing I want to say about complex numbers is that it's very common for us to identify complex numbers with points on the plane. That is, we will identify the complex number, say z equals a plus bi, with the point a comma b. We could call this the graph of the complex number, where the real part of the complex number we associate with the x-coordinate. And because of that, we often refer to the x-axis in this context as the real axis. We're also going to identify the imaginary part of the complex number with the y-coordinate of the point. And for this reason, the y-axis, when we're graphing complex numbers, is often referred to as the imaginary axis. And so when you put complex numbers in the plane and you label your axes as the real axis and the imaginary axis, you put this together and often refer to this as the complex plane. And it's this observation, which is why I say the complex numbers, or we might say the complex plane, this is why we refer to them as two-dimensional numbers, because we really can't visualize these as points in a two-dimensional plane. So if we were going to do an example of such a thing, maybe we want to graph the complex number 2 plus 4i. What that means is since the real part is 2, we'll go 2 along the x-axis. And since the imaginary part is 4, we go 1, 2, 3, 4 up the imaginary axis. And we find this point 2 comma 4, but instead of kind of 2 comma 4, we call it the complex number 2 plus 4i. That's what it is for graphing a complex number. If we were to do, say, the additive inverse of this complex number, that is, we times it by negative 1, graphing this complex number would look like your x-coordinate is negative 2 and your y-coordinate is 1, 2, 3, negative 4 down there. And so you plot the point right here, negative 2, negative 4, and we associate with this point the complex number, negative 2 minus 4i. Now this is the additive inverse of our original point, and you'll notice that when it comes to complex numbers, its additive inverse that is multiplied by negative 1 reflects it through the origin. We see a similar thing when we look at the complex conjugate of the number, right? If you take the conjugate of 2 plus 4i, that becomes 2 minus 4i. So you're still going to go 2 along the x-axis for the real part, and you'll go down the y-axis by 4 for its imaginary part. So you find the points 2 comma negative 4, which gives us the complex number 2 minus negative 4 right there, and this would then be the graph of the complex conjugate of 2 plus 4i. Notice that the conjugate is actually the reflection of the complex number across the x-axis or the so-called real axis in the situation. And so this actually gives us a way of representing complex numbers in a geometric manner for which then we can start analyzing complex numbers using geometry and trigonometry and such. And if I went much deeper into this, this really would start getting into a trigonometry notion, which is a great topic you can learn about in Math 1060 trigonometry, but one we won't delve too much more into right now. And so this will conclude Lecture 16, which gives us an introduction to complex numbers. We do have to use complex numbers as we work with quadratic equations, and so this was an important primer that we do as we start studying complex numbers, sorry, as we start studying quadratic equations starting in the next lecture.