# 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:
1. The right triangle formed by , and the positive real axis.
2. 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:
1. The ratio of similitude is , which means that (this is an alternative to the algebraic proof you did for homework)
2. 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: ﻿