 We can interpret the real numbers as distances along a number line. Is it possible to interpret complex numbers geometrically? What approach is the argon diagram, named after a 19th century French mathematician? To motivate the argon diagram, consider the real number line. There's an origin, and we label points based on the distance from the origin. Since distance is a real number, the labels are also real numbers. To avoid duplicate labels, points to the right are assigned positive distances, while points to the left are assigned negative distances. The number line gives us a geometric model for the real numbers. What about a geometric model for arithmetic operations? We can represent multiplication by a positive number as a scaling, and we can represent multiplication by a negative number as a scaling, followed by flipping the point across the origin. Let's interpret 2 times 3 and 2 times negative 3 on the number line. So 2 times 3 suggests we might start with the point 2 units to the right of the origin, and then scaling the distance by a factor of 3. Similarly, 2 times negative 3 suggests starting with the point 2 units to the right of the origin, scaling the distance by a factor of 3, and then flipping this point across the origin. Now consider a product like 2i. We begin with the point that's 2 units to the right of the origin, then we'll lerb it. To decide what lerb is, it's helpful to keep in mind things that do the same thing are the same thing. To decide what lerb is, let's use the fact that i squared is equal to negative 1. Since i squared equals negative 1, this means that if we lerb then lerb again, we get the same result as multiplying by negative 1. So lerb-ing twice is the same as flipping a point across the origin. So what can we do twice that's the same as flipping a point around the origin? One possible interpretation of lerb is that it's a 90 degree rotation. Wait, remember once you start calculus, degrees don't exist. So we'll make that a rotation by the angle of pi-habs radians. We'll use our standard convention and assume it's a counterclockwise rotation around the origin. So we can draw a representation of 2i. If we view this 2i as 2 times i, this corresponds to the real number 2, rotated by pi-habs radians around the origin. What about our other i-arthetic operations like addition? On the real number line, numbers like a and b correspond to points a and b units from the origin and the sum a plus b can be viewed as joining the two lined segments and the beginning, the lined segment from origin to a and the lined segment from the origin to b. So what about something like negative 3 plus 2i? So we've determined that negative 3 corresponds to a point 3 unit to the left of the origin and 2i corresponds to a point 2 units above the origin. Drawing the lined segments from the origin to each point, then joining them end to beginning gives us a representation of negative 3 plus 2i. The proceedings suggest that a complex number x plus iy can be viewed as joining a horizontal lined segment of length x and a vertical lined segment of length y. But this is exactly what we do when we plot the point with coordinates x, y. So we can represent complex numbers as points in the plane. And in particular, the complex number x plus iy can be identified with the point xy. And since xy are the rectangular coordinates of a point, we refer to x plus iy as the rectangular form of the complex number.