# Multiplying Complex Numbers

- Author:
- Brian Sterr

- Topic:
- Complex Numbers, Numbers

This graph shows how we can interpret the multiplication of complex numbers geometrically.
Given two complex numbers:
, where
Consider their product

Focus on the two right triangles in the diagram:
, , .
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:
Key results:

1 | Dilate | |

2 | Rotate | |

3 | | Dilate |

4 | | Translate |

- The right triangle formed by
, and the positive real axis. - The right triangle formed by
, and

- 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