Multiplying Complex Numbers
This graph shows how we can interpret the multiplication of complex numbers geometrically.
Given two complex numbers:
, where
Consider their product
| 1 | | Dilate by a scale factor of |
| 2 | | Rotate by about |
| 3 | | Dilate by a scale factor of |
| 4 | | Translate by |
Focus on the two right triangles in the diagram:
- The right triangle formed by , and the positive real axis.
- The right triangle formed by , and
The first right triangle has sides of length: , , .
The second right triangle has sides of length , , and .
Since we have the proportion: , we can conclude the triangles are similar since two pairs of corresponding sides are proportional and their included angles (the right angles) are congruent.
This has two implications:
- The ratio of similitude is , which means that (this is an alternative to the algebraic proof you did for homework)
- The angle formed by , and is congruent to , since they are corresponding angles of similar triangles
This leads us to our other conclusion, that
Key results: