 So lots of useful insights in mathematics and in life can be gained by looking at things in a different way. So we've introduced complex numbers algebraically as solutions to quadratic equations, but can we look at complex numbers geometrically? And the answer is, well, I hope so because otherwise there's no point in the rest of this video. Now in order to look at complex numbers geometrically, we have to take a closer look at the number line. And the number line is one of the most useful ideas in mathematics, but it's very often misunderstood. We often use a number line, but we usually use it incorrectly. And the important thing to recognize here is that when we mark a number on a number line, the number does not correspond to the point. The number is not the point. The number corresponds to the distance from the origin. So we can apply a geometric transformation to distances on the number line. For example, x to x plus h, we're going to shift this distance h units to the right. Or we can take x and send it to cx, and this is going to be a stretch or a compression of the distance by a factor of c. And if we send x to minus x, this is going to be a reflection of this distance across the origin. Now let's think about breaking this down even further. What about a stretch like x to 4x? Well, on the one hand, we can view this as a stretch of a distance by a factor of 4, but we can also treat this as a sequence of two transformations. Since 4x is 2 times 2x, we can apply the transformation x goes to 2x. That's a stretch by a factor of 2. And then 2x goes to 2 times 2x, where we stretch again by a factor of 2. And so now let's make this connection. What if we tried the same with the reflection x to minus x? Now since i squared is equal to negative 1, we can apply two transformations, x to ix, where we stretch by a factor of i. But let's not worry about what that means quite yet. Then our second transformation, ix, goes to i squared x, where we stretch by another factor of i. So what we've done is we've taken this single transformation, x to minus x, and rewritten it as two transformations. In this case, stretching by a factor of i, and then stretching again by a factor of i. And that works out great, except we have to ask ourselves, what does a stretch by a factor of i mean? And again, if we hope to get any insight into this, we should consider another viewpoint. Consider the geometric transformation of multiplying by minus 1. We can actually view this in two ways. We can view it as a reflection across x equals 0, but we might also consider this to be a 180-degree rotation around x equals 0. And so we have this algebraic task, multiplying by negative 1, and we can view this in two different ways geometrically, either as a reflection or a rotation. But here's the important thing. We can't break a reflection down into pieces. There is nothing we can do twice to get a reflection. On the other hand, we can break the rotation down into smaller pieces. So let's think about that. A 180-degree rotation can be viewed as a 90-degree rotation, followed by another 90-degree rotation. So let's put these two ideas together. First, multiplying by i twice produces a 180-degree rotation. On the other hand, rotating by 90 degrees twice produces a 180-degree rotation. Now they produce the same effect, a 180-degree rotation, and so that suggests that multiplying by i and rotating by 90 degrees really is the same thing. And so this suggests that the multiplication by i can be interpreted as a 90-degree rotation around x equals 0. So for example, let's consider the graph on the number line 5, negative 3, and 2i. So remember when we graph on the number line, we're really graphing the distance from the origin. So we'll put down our origin, and if it's not written down, it didn't happen, we should label. 5 is going to correspond to a 0.5 unit to the right of the origin. Now since the distance from the origin is the thing that's actually important here, let's go ahead and indicate that length and label. Minus 3 is going to correspond to a 0.3 unit to the left of the origin. And again, we'll mark out the distance and label the point. What about 2i? We can view 2i as the point that's 2 units to the right of the origin, which is then rotated by 90 degrees around x equals 0. And what's interesting here is that takes us off the number line. And this leads us to the notion of graphing in the complex plane, and we'll graph real numbers, our along a horizontal number line, and pure imaginaries, our along a vertical line. What about a number like 4 plus 3i? We can graph this by graphing the point 3 units to the right of the origin, rotating this by 90 degrees to get the point 3i, and then shifting this right 4 units to get 4 plus 3i. Now, we've graphed this point 4 plus 3i over here someplace, but this should look awfully familiar to you. And this process suggests the following. The graph of the complex number x plus iy corresponds to the point xy. And so we might try to graph the points 3 plus 4i, 4 plus 3i, and 3 minus 4i. So the real part of our complex number gives us the x coordinate, the imaginary part of the complex number gives us the y coordinate. So 3 plus 4i is going to correspond to the point 3, 4. Meanwhile, 4 plus 3i has real part 4, so that's our x coordinate, and complex part 3i, so our y coordinate is 3. And 3 minus 4i has x coordinate 3, y coordinate negative 4. And again, it's important to remember that when we graph a point, we're not associating the point with the number, but rather the distance of the point from the origin. So the number 3 plus 4i really corresponds to this distance. 4 plus 3i corresponds to this distance, and 3 minus 4i corresponds to this distance.