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Quadrilateral family tree
I have always loved the naming of quadrilaterals, right from when I first heard about it in high school. I'm not entirely sure why, but some of it has to do with the nested nature of the definitions – I like that a square is a kind of rectangle and a rectangle is a kind of parallelogram.
One reason I'll still use pi
Every so often, someone brings up the thing with tau (τ) versus pi (π) as the fundamental circle constant. In general I find the discussion wearisome because it usually focuses on telling people they are stupid or wrong for choosing to use one constant or the other. There are more productive uses of your time, I think.
All dogs have tails
In maths, or at least university maths, there are a lot of statements that go like this: "If ...., then ..." or "Every ..., has ...." or "Every ..., is ...". For example, "Every rectangle has opposite sides parallel", "If two numbers are even, then their sum is even", "Every subspace contains the zero vector", "If a matrix has all distinct eigenvalues, then it is diagonalisable". Many students when faced with statements like these automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. This post is about a way of helping students see the problem.
Where the complex points are
When you first learn complex numbers, you find out that they give you ways to solve equations that were previously unsolvable. The classic example is the equation equation x^2 + 1 = 0, which if you're only using real numbers has no solutions, but with complex numbers has the solutions x=i and x=-i.
