Math Blog

Mathematics Comedy Videos

Mathematical humour is quite difficult to do. I have previously blogged about the book Comic Sections and there are other books like Carl Linderholm’s Mathematics made difficult and Ian Stewart’s books which contain much humour; a cartoon series Ian drew is the inspiration for this

Mathematical humorous videos are even rarer. Here are a couple. The first is very well-known and has been written about all over the net for a long time. It is Finite Simple Group of Order Two by the Klein Four Group and involves very clever use of mathematical terms. The other is G103 which is described as a (surreal) day in the life of an undergraduate on the 4-year MMath degree at the University of Warwick. Anyone who has experience of a pure mathematics degree will recognise the accuracy of the amusing observations it makes. There’s more about the film at the G103 site.

There was also a previous humorous video made at Warwick University called Maths Club. Unfortunately, it’s not available on that page or anywhere else as far as I can see. Does anyone know if it’s still available?

Links to other amusing mathematics videos are very welcome.

PS The Unapologetic Mathematician links to a spoof basic mathematics tutorial produced by the BBC called Look Around You - 1 - Maths.

Carnivals of Mathematics

The Carnival of Mathematics is a fortnightly look at mathematical blogs. As I have commented before the growth in such blogs has been phenomenal recently which is very welcome. The carnivals are thus a gateway to the treasures that are out there, so I think it worth listing the carnivals so far (all numbers are approximate, mathematicians can count badly ):

Inaugural Edition hosted by Abstract Nonsense has 19 assorted links ranging from quotes exhorting people to study mathematics to algebraic topology.
The Second Carnival Of Mathematics: The Math Geeks are Coming to Town! hosted by Good Math, Bad Math has 27 fascinating assorted links including the Halting problem, Using math for astronomy and Rubik’s Magic Cube.
Carnival Of Mathematics #3 hosted by Michi’s Blog has 20 articles grouped into five ‘halves’: didactic, financial mathematics, humour, dimensions, and number theory, geometry, topology, algebra.
Carnival of Mathematics Number 4 hosted by EvolutionBlog has 21 links from the Bernouilli process to finding the last two digits of 31000.
Carnival of Mathematics, Ordinal 5 hosted by Science and Reason is dedicated to the memory of Paul J. Cohen with tributes and discussion of his work. There’s at least 30 links to other topics including tilings, Lie groups etc and surreal numbers.
The Carnival of Mathematics Sixth Edition hosted by Modulo Errors has 19 links with many presenting problems of various difficulty.
Carnival of Mathematics Edition #7 hosted by nOnoscience has 28 links from Euler to German bloggers and includes a new improved number system and a calculus paradox.
8th carnival of mathematics hosted by The Geomblog has 20 links with “time to revisit, reflect, and ponder on things we think we already know” so has an educational section and is the first carnival to have a cartoon.
Carnival of Mathematics IXhosted by JD2718 is an alphabetical list of 36 links that appeal to the school teacher blogger.
The next carnival is due on 15th June at MathNotations. Carnival 11 is due to be hosted by Grey Matters on 29th June and Carnival 12 on 13th July by Vedic Maths Forum. Do let them know if you have anything you wish to be included.

That’s over 200 links (not all mutually exclusive) - mathematical blogging is alive and well.

Refresh your High School Math skills

In my first article “The most enlightening Calculus books”, I argued the importance of maintaining high standards for mathematics education and suggested deep and inspiring calculus books for those of you who are interested in pursuing the joy of learning mathematics. The feedback has been overwhelming and I wish to follow up with an article that addresses a couple of remarks that I’ve received by email.

One person commented on the blog, and another wrote me privately, to express their concern that “harder books are not necessarily better books” and that teaching which is geared towards only the smartest kids is a mistake. I want to point out that I’m in no way advocating teaching for the brightest minds only. Wide access to mathematics is something that should be encouraged all over the world, and I’m pretty sure it will help take society in a better direction. In fact, education in general - and mathematics, technical and scientific education in particular - are key for the development of every country and ultimately for good of humankind. However my point was that with wider access to higher education mathematics, we should not reduce the expected and established testing standards. In other words, there is a fair level of understanding that we should expect from people who major in math or from students who strongly depend on mathematics for their future careers. Furthermore, the textbooks adopted should be mathematically sound and provide the right intellectual stimulation for those who could use it. That said, there is nothing wrong with teachers trying to use different styles of teaching to reach a wider audience, or for students who struggle with the level of math presented in the textbook, to supplement it with simpler books in order to get an easier start. Hence, it’s perfectly OK for a student (who for example is taking an undergraduate class in programming in C) to read C For Dummies if The C Programming Language by K&R is too hard for them off the bat. But that does not imply that the class should adopt “C for Dummies” as their textbook nor that the examination should be based on such a book. So to summarize this point, feel free to study any number of introductory books, as long as you know that if you plan to be serious about mathematics, you should be able to eventually read and understand standard books and be able to solve most of the exercises put forward in them. Having clarified the first concern, I’d like to provide an answer for the second point, which actually interests me the most. A few readers wrote me emails about how they feel enthusiastic about the post and the opportunity to study mathematics again, but how those books are way too advanced for them, because they simply forgot all the mathematics taught at a high school level. So I’ve received a few “how can I get a refresher of high school math?” type of questions. The mathematics that you learned in high school is classified as pre-calculus, and as you can expect it is propaedeutic to learn math at an higher level. It is normal that you forgot quite a few formulas, but having a good grasp of the essentials of precalculus can make a big difference when trying to master calculus. You should have a decent knowledge of basic algebra, trigonometry, exponential, logarithmic, and analytic geometry. Calculus itself will provide you with a refresher of some of these topics and give you a deeper understanding not only of “how” but rather “why”. That said, Calculus without a decent precalculus base can be a big challenge for most people. Before proceeding to suggest a few resources, let’s try to establish if you actually need a refresher course or not. Here is a (simple and of course incomplete) list of some basic exercises. If you haven’t a clue or struggle to find a lot of the solutions for them, a refresher may be in order.

Diagonal Stripes in Group Table

One of my student’s attention was drawn to the fact, that in cyclic groups of order 4 and 5, it is possible to arrange the elements so that the transverse diagonals (that is those perpendicular to the leading diagonal) of the group (Cayley) table consist of equal elements. The groups in question were $\mathbb{Z}^*_5$ with multiplication modulo 5 and $\mathbb{Z}_5$ with addition modulo 5. Thus you get the following diagonal patterns highlighted by the colours:

$\begin{array}{c|cccc} \times_5 & 1 & 2 & 4 & 3 \\ \hline 1 & 1 & \color{green}2 & \color{red}4 & \color{blue}3\ 2 & \color{green}2 & \color{red}4 & \color{blue}3 & 1\ 4 & \color{red}4 &\color{blue}3 & 1 & \color{green}2\ 3 & \color{blue}3 & 1 & \color{green}2 & \color{red}4\ \end{array} \qquad\qquad \begin{array}{c|cccccc} +_5 & 0 & 1& 2 & 3 & 4 \\ \hline 0 & 0 & \color{green}1 & \color{red}2 & \color{blue}3 & \color{magenta}4 \ 1 & \color{green}1 & \color{red}2 & \color{blue}3 & \color{magenta}4 & 0 \ 2 & \color{red}2 &\color{blue}3 & \color{magenta}4 & 0 & \color{green}1\ 3 & \color{blue}3 & \color{magenta}4 & 0 & \color{green}1 & \color{red}2\ 4 & \color{magenta}4 & 0 & \color{green}1 & \color{red}2 & \color{blue}3 \end{array}$

He asked 2 questions:

1. Can the elements of all (finite) cyclic groups be arranged to give these diagonal stripes?

If you take a finite cyclic group generated by the element (we will use multiplication for the binary operation) then the natural ordering $e, a ,a^2 , a^3, \dots, a^n$ will show this pattern:

$\begin{array}{c|llllllll} & e & a & a^2 & & & a^{i-1} & a^i & \\ \hline e & e & \color{green}a & \color{red}a^2 & & & & \color{blue}a^i \ a & \color{green}a & \color{red}a^2 & & & & \color{blue}a^i \ a^2 & \color{red}a^2 & & & &\color{blue}a^i\ \ a^{i-1} & & \color{blue}a^i\ a^i & \color{blue}a^i\ \ a^{j-1} & & & & & & & \color{magenta}a^{j-1+i}\ a^j & & & & & & \color{magenta}a^{j+i-1}\ \end{array}$

2. Are the cyclic groups the only ones that generate these patterns?

The answer is yes, but is not quite so obvious although not difficult to prove. Suppose a (finite) group G arranged as $G=\{e,a_1,a_2,\dots,a_n\}$ exhibits the diagonal stripes. Use induction. Let a_1=a and suppose also that a_j=a^j for $1 \le j<i$ . Then we get:
$\begin{array}{c|lllllll} & e & a & & & a_{i-1} & a_i & \\ \hline e & & & & & & \color{blue}a_i & \ a & & & & & \color{blue}aa_{i-1} & \ \end{array}$
It follows that $\color{blue}a_i=aa_{i-1}$ so by induction assumption, $a_i=aa^{i-1}=a^i$ and hence is the cyclic group generated by .

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