We are still alive. We are currently in the process of exporting each post to our new site (by hand). The end date is ∞ days away! Pythagoras' theorem is both the oldest and the most important non-trivial theorem in mathematics. This is the first part of the first lecture of a course on the History of Mathematics, by N J Wildberger, the discoverer of Rational Trigonometry. We will follow John Stillwell's text Mathematics and its History (Springer, 3rd ed). Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. A few historical tidbits will be thrown in too... In this first lecture (with two parts) we first give a very rough outline of world history from a mathematical point of view, position the work of the ancient Greeks as following from Egyptian and Babylonian influences, and introduce the most important theorem in all of mathematics: Pythagoras' theorem. 2048 is a single-player online and mobile game created in March 2014 by 19-year-old Italian web developer Gabriele Cirulli, in which the objective is to slide numbered tiles on a grid to combine them and create a tile with the number 2048. It can be regarded as a type of sliding block puzzle, and is very similar to the Threes! app released a month earlier. As some of us have figured out, this is a very addicting game. It is available as a online game as well as a mobile game. To help feed people's addiction, we are proud to present another place where you can play it! Want a non-flash version? If so, check outhttp://gabrielecirulli.github.io/2048/ for a JavaScript version. On a mobile device? If so, check out the web-based version at http://git.io/2048 or use an app like the ones listed below! Android
iOS
Windows Phone
The Count Dracula awakes on night 0 and his evil after-life begins. On night 1, Dracula starts biting people. Every bitten human becomes a vampire and bites people as well. Let $v(n)$ be the vampire population after $n^\text{th}$ night. Investigate the function $v(n)$ in the following cases: a) Every vampire bites one human every night starting the first night after the initiation. Write down a recurrent and an explicit formula for $v(n)$. b) Every vampire bites one human every night starting the second night after the initiation. Write down a recurrent formula for $v(n)$. Prove that \[v\left ( n \right )=\frac{\left ( \frac{1+\sqrt{5}}{2} \right )^{n+2}-\left ( \frac{1-\sqrt{5}}{2} \right )^{n+2}}{\sqrt{5}}\] c) Every vampire bites $m$ humans on the $n^\text{th}$ night after the initiation. Write down a recurrent and an explicit formula for $v(n)$. |