How Risks Add Up

(A new question of the week) Real life questions of probability often require information that we don’t have – they become a job for statistics instead. But sometimes just trying some plausible numbers, as in a Fermi problem, can yield interesting results. Here we consider the probability of an injury when kids play near a …

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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.

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”.

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.