Technologically Embodied
Geometric Functions
Complex Multiplication Part 4: Multiply Two Complex Numbers

Return to the complex multiplication overview.

In this 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)`.

Using this formula for `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_w` and `y_w` are both real numbers. This enables you to use geometry on the complex plane to multiply the complex numbers `v·w`, based on three geometric techniques to:

  • geometrically multiply a complex number by `i` (to construct `v*i`), and
  • geometrically multiply a complex number by a real number (to construct `v*x_w` and `v*i*y_w`),
  • geometrically add two complex numbers (to construct the sum `v*x_w + v*i*y_w`).

On pages 1–3, you'll review these three techniques.

On page 4 you'll use the techniques to multiply two complex numbers `v` and `w`, and you'll investigate the geometric relationships connecting the complex numbers `v`, `w`, and `v·w`.


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Update History:

18 October 2016: Created this page.
16 November 2016: Clarified some directions and reordered the tools. Still need to make hint videos.