Yesterday, we identified that adding complex numbers is equivalent to translating a point and multiplying complex numbers is equivalent to dilating and rotating that point. Furthermore, we noted that while the conventional rectangular coordinates for identifying a complex number are useful for identifying the direction and distance of a translation, to properly understand the dilation/rotation we need the polar coordinates.
Today, we quickly reviewed converting between rectangular and polar form at the start of class, then spent a considerable amount of time in class practicing this conversion and identifying the transformations they refer to. This work was on the Complex Numbers Transformations sheet and together we finished some of the front and about half of the back. Your homework is to finish the front.
We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.
The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.
We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.