Dork Scratchings

A Pythagorean triple is a set of three non-zero integers $a,b,c$ satisfying Pythagoras' formula $$ c^2 = a^2 + b^2 $$ Pythagoras certainly wasn't the first to know that this formula applied to the sides of right angled triangles, or to compile lists of Pythagorean triples, but he may have been the first to present a proof .  In this post I'm going to show a simple method of finding all Pythagorean triples, using complex numbers. First we need to introduce Gaussian Integers .  These are complex numbers $z=a+ib$ where $a,b$ are integers.  If $z$ is any complex number then, since multiplication is commutative, $$ (zz^*)^2 = z^2(z^2)^* $$ However, in the case where $z$ is a Gaussian Integer the LHS is a square of an ordinary integer, and the RHS is the sum of two squares of integers.  For it to be a sum of two non-zero squares we need $z^2$ to be neither purely real or purely imaginary, or equivalently, that $z$ is neither purely real, purely imaginary, or on a diagonal $a =

Is it possible to bring a temperature below absolute zero? And if so, what does, say, -1K look like? Surprisingly, negative temperatures are in fact possible, but only in some circumstances. To understand how, we need to get a better idea of what temperature actually means from a statistical thermodynamics point of view. Suppose you have a system of $N$ particles$^{\dagger}$, each of which can have an energy level from a discrete list $$ 0 < E_1 < E_2 < E_3 < ... $$ Now suppose that there are $N_1$ particles in state $E_1$, $N_2$ in $E_2$ and so on. Using combinatorics$^{\dagger_2}$ we can see that the number of ways this particular configuration can be achieved is given by $$ \Omega = \frac{N!}{N_1!N_2!N_3!...} $$ The next question is: if we know the total energy of the system $E_{total}$ can we work out the distribution of particles across the energy states? Well, we know the most likely distribution is that which can be achieved in the greatest number of ways. I.e. we