A general cubic polynomial has the form \[ ax^3+bx^2+cx+d \] but a general cubic equation can have the form \[ x^3+ax^2+bx+c=0. \] We can always divide through by the coefficient of \(x^3\) (assuming it to be non-zero) to obtain a monic equation; that is, with leading coefficient of 1. We can now remove the \(x^2\) …

The date January 26 is one of immense current debate in Australia. Officially it's the date of Australia Day, which supposedly celebrates the founding of Australia. To Aboriginal peoples it is a day of deep mourning and sadness, as the date commemorates over two centuries of oppression, bloodshed, and dispossession. To …

Recently we have seen senators behaving in ways that seem stupid, or contrary to accepted public opinion. And then people will start jumping up and down and complaining that such a senator only got a tiny number of first preference votes. One commentator said that one senator, with 19 first preference votes, …

This evening I saw the Australia Brandenburg Orchestra with guest soloist Lixsania Fernandez, a virtuoso player of the viola da gamba, from Cuba. (Although she studied, and now lives, in Spain.) Lixsania is quite amazing: tall, statuesque, quite absurdly beautiful, and plays with a technique that encompasses the …

Here's an example of a transportation problem, with information given as a table: Demands 300 360 280 340 220 750 100 150 200 140 35 Supplies  400 50 70 80 65 80 350 40 90 100 150 130 This is an example of a balanced, non-degenerate transportation problem. It is balanced since the sum of supplies equals the sum of …

For my elementary linear programming subject, the students (who are all pre-service teachers) use Excel and its Solver as the computational tool of choice. We do this for several reasons: Excel is software with which they're likely to have had some experience, also it's used in schools; it also means we don't have to …

Here's an example of a coloured tetrahedron: hello <div oncontextmenu="return false;" id="viewerContext" style = "width:640px;height:470px;" design-url="tetrahedron.jscad"></div> <div id="tail" style="display: none;"> <div …

There's a celebrated elementary result which claims that: There are irrational numbers \(x\) and \(y\) for which \(x^y\) is rational. The standard proof goes like this. Now, we know that \(\sqrt{2}\) is irrational, so let's consider \(r=\sqrt{2}^\sqrt{2}\). Either \(r\) is rational, or it is not. If it is rational, …

For years I have been running a blog and other web apps on a VPS running Ubuntu 14.04 and Apache - a standard LAMP system. However, after experimenting with some apps - temporarily installing them and testing them, only to discard them, the system was becoming a total mess. Worst of all, various MySQL files were …

These are a class of root-finding methods; that is, for the numerical solution of a single nonlinear equation, developed by Alston Scott Householder in 1970. They may be considered a generalisation of the well known Newton-Raphson method (also known more simply as Newton's method) defined by \[ x\leftarrow …