A total ordering $\lt$ on a set $S$ is called dense if for every two $x,y\in S$ with $x \lt y$, there exists a $z\in S$ such that $x\lt z\lt y$. A total ordering is said to be without endpoints if for every $x\in S$ there exists $y,z\in S$ such that $y \lt x\lt z$. […]
Series hold endless fascination. To converge or not to converge? That is the question. Let's take the series $1 + 1/2 + 1/3 + \cdots$. It's called the harmonic series, and it diverges. That's because it is greater than the series $$1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + […]
Did you know that the closed interval $[0,1]$ cannot be partitioned into two sets $A$ and $B$ such that $B = A + t$ for some real number $t$? Of course, the half-open interval $[0,1)$ can so be partitioned: $A = [0,1/2)$ and $t = 1/2$. Why is this? I will leave the full details […]
A class of fractals known as Mandelbrot sets, named after Benoit Mandelbrot, have pervaded popular culture and are now controlling us. Well, perhaps not quite, but have you ever wondered how they are drawn? Here is an approximation of one: From now on, Mandelbrot set will refer to the following set: for any complex number […]