Probability: Cards vs Dice
(A new question of the week) A couple recent questions involved related subtleties in probability and combinatorics. Both were about apparent conflicts between similar problems involving cards and dice.
Roots of Complex Numbers, and Complex Roots of Real Numbers
We’ve looked at positive integer powers of complex numbers, and imaginary powers of e. Now let’s look at roots of complex numbers, which will bring us to the idea that led to this series, namely the roots of unity.
Euler’s Formula: Complex Numbers as Exponents
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”.
Complex Numbers: Multiplication and Rotation
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 …
Arithmetic with Complex Numbers
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
How Imaginary Numbers Became “Real”
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 …”.
Making Sense of Imaginary Numbers
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.
Partial Fractions: Complex and Trivial Cases
(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, …
Abraham Lincoln and the Rule of Three
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!
Many Ways to Solve a Proportion
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 …