Category Archives: elementary

What is a Liouville number?

An irrational number $r$ is called a Liouville number if for every positive integer $n$ there exists integers $p$ and $q \gt 1$ such that $$| r – p/q | \lt 1/q^n.$$ The requirement that $q \gt 1$ is crucial because otherwise, all irrational numbers would satisfy this definition. Liouville numbers are transcendental. In an […]


All set endomorphisms of a finite field are polynomial

Let $F$ be a finite field. Did you know that given any function $\varphi:F\to F$, there exists a polynomial $p\in F[x]$ such that $\varphi(a) = p(a)$ for all $a\in F$? It's not hard to produce such the required polynomial: $$ p(x) = \sum_{a\in F} \left( \varphi(a)\prod_{b\not= a}(x – b)\prod_{b\not=a}(a-b)^{-1} \prod \right)$$ This works because every […]


Harmonic Numbers

The $n$th harmonic number $h(n)$ is defined as $$h(n) = \sum_{i=1}^n 1/i$$ The harmonic series is the associated series $\sum_{i=1}^\infty 1/i$ and it diverges. There are probably quite a few interesting ways to see this. My favourite is a simple comparison test: $$1/1 + 1/2 + 1/3 + \cdots\\ \geq 1/2 + 1/2 + 1/4 […]


Switching the order of summation

Characteristic functions have magical properties. For example, consider a double summation: $$\sum_{k=1}^M\sum_{r=1}^k a_{r,k}.$$ How do you switch the order of summation here? A geometric way to think of this is arrange the terms out and "see" that this sum must be equal to $$\sum_{r=1}^M\sum_{k=r}^M a_{r,k}.$$ I find this unsatisfactory because the whole point of good […]


When Is Squaring and Cubing a Group Homomorphism?

Let $G$ be a group. Define $\varphi:G\to G$ by $\varphi(x) = x^2$. When is $\varphi$ a homomorphism? Clearly, $\varphi$ is a homomorphism whenever $G$ is abelian. Conversely, if $\varphi$ is a homomorphism then for any $x,y\in G$, we get $xyxy = \varphi(xy) = \varphi(x)\varphi(y) = x^2y^2$. So, $xyxy = xxyy$. Canceling the $x$ from the […]



Graphing the Mandelbrot Set

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 […]


60 Linear (Matrix) Algebra Questions

I sometimes am a teaching assistant for MATH 133 at McGill, and introductory linear algebra course that covers linear systems, diagonalisation, geometry in two and three dimensional Euclidean space, and the vector space $ \mathbb{R}^n$, and I've collected a few theoretical questions here that I like to use in the hope that they may be […]


One Fair Coin and Three Choices

A few nights ago as I was drifting off to sleep I thought of the following puzzle: suppose you go out for ice cream and there are three flavours to choose from: passionfruit, coconut, and squid ink. You like all three equally, but can only choose one, and so you decide you want to make […]