Complex Multiplication Overview
Return to the complex plane unit description.
In this multi-part activity you will investigate the multiplication of two complex numbers `v` and `w` using a combination of algebra and geometry. First, remember that you can express `w` as `w=(x_w + i*y_w)`.
(Although one more commonly sees expressions like `v = (a + b*i)` or `w = (c + d*i)`, in the current explorations we choose to connect the real and imaginary components of a complex number to its horizontal and vertical coordinates on the complex plane. We will also make significant use of the modulus (distance from the origin) and the argument (angle from the positive real axis) of a complex number, which we will represent as `(r_w, theta_w)` to emphasize the connection to polar coordinates.)
Because `w=(x_w + i*y_w)`, you can use the distributive property to write `v*w = v*(x_w + i*y_w) = v*x_w + v*i*y_w`.
In this expanded form, `x_v` and `y_v` are both real numbers. This enables you to use geometry on the complex plane to multiply complex numbers, based on three geometric techniques:
- geometrically multiplying a complex number by `i` (to construct `v*i`),
- geometrically multiplying a complex number by a real number (to construct `v*x_w` and `v*i*y_w`), and
- geometrically adding two complex numbers (to construct the sum `v*x_w + v*i*y_w`).
In the following activities, you will develop these three techniques and use them to find an elegant way to multiply two complex numbers.
- Part 1: What happens when you dilate a complex number by a scale factor like `3`, `0.5`, or `-2`? Describe the result numerically.
- Part 2: What happens when you rotate a complex number by an angle such as `pi`/`2`, `pi`, or `3 pi`/2? Describe the result numerically. How can you use your results to multiply a complex number by `i`?
- Part 3: What happens when you translate a complex number by the vector corresponding to a second complex number? Describe the result numerically. [Not yet available]
- Part 4: Start with two complex numbers and apply what you’ve learned to construct their product: `v·w = v*x_w + v*i*y_w`
- Part 5: Starting with the construction of `v*w` in Part 4 based on algebra, find a much simpler way to multiply two complex numbers by using just two steps: the dilate-rotate method. Then investigate whether this method is commutative with respect to `v` and `w`? (That is, does `w*v = v*w` for complex numbers?) Second, is the composition itself commutative? (That is, does rotate-dilate give the same result as dilate-rotate?)
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