Last week we explored how the polar form of complex numbers gives multiplication a simple geometric meaning. Here we’ll go one more step, and express polar form exponentially, which makes DeMoivre’s theorem trivial, and gives us a simple notation to replace “cis”.
Having looked at the idea of complex numbers and how to perform basic operations on them, we are ready for one of the most important features for applications: their relationship to rotation. We’ll see this first in describing complex numbers by a length and an angle (polar form), then by discovering the meaning of multiplication …
We’ve seen what complex numbers are; now we can look at what we can do with them. The basic operations are not hard, but have a few interesting features related to graphs. So that’s where we’ll start
Last week we started a series on complex numbers, looking at how we introduce the concept. This time I want to look more at the actual history of the idea, leading to how mathematicians were able to define complex numbers without saying “Just suppose …”.
Several recent questions (including last week’s post) involved complex numbers, and made me realize we haven’t yet talked about them here. So let’s start a series on the topic, beginning with how we talk about them to students who are just meeting the idea for the first time, or are troubled by it.
(A new question of the week) We discussed four years ago how to make a partial fraction decomposition of a rational function, and why it can always be done; a question from mid-May brings up two side issues: when you can factor the denominator, and whether a trivial decomposition, which takes no work at all, …
For the last two weeks, we have examined new and old ways to think about proportions. This time, we’ll look at an old method called the Rule of Three (both “single” and “double”), and how you might have learned to solve these problems 200 years ago without algebra. Be prepared for a deep dive!
Last week we looked at a set of special rules for working with proportions, which have been largely replaced by the more general “tool” of algebra (the “Swiss army knife” of problem solving, which can do the job of many specialized tools), though the latter can still be useful. We still find that many students …
A new question of the week A simple question about the Basic Proportionality Theorem (which goes by several other names, including Thales’ Theorem) and its relationship to similar triangles leads to some helpful ideas about how to choose a suitable manipulation at each step in a proof, a skill central to good problem solving. In …
Last week we looked at some recent questions about limits, where we focused first on what limits are, in terms of graphs or tables, and then on finding them by algebraic simplification. This week, we’ll look at two old questions about a trigonometric limit that can’t be determined that way: sin(x)/x, as x approaches zero.