Example 1.2.14 (d): Multiplying geometrically

Compute in your head, then confirm your answer.
Context Context

To do this in our head we look at radius and angle separately:

  • (blue) |i| = 1 and Arg(i) = /2
  • (red) |1+i| = and Arg(1+i) = /4
  • (green) |i-1| = and Arg(i) = 3/4
  • (gold) |1-i| = and Arg(i) = -/4

According to our results on multiplying and dividing geometrically we have for the lengths:

length of = (1)/() = 1

As for the angles we have:

angle of = /2 + /4 + 3/4 - 2(-/4) = 2

Thus:

= 1 cis(2) = 1

Verifying this algebraically is work - so it's left to you.

Interactive Complex Analysis, ver. 1.0.0
(c) 2006-2007, Bert G. Wachsmuth
Page last modified: May 29, 2007