I was trying to solve an olympiad type problem involving a nested radical of the form
$\sqrt{a+b\sqrt{r}}.$
I had managed to discover that $\sqrt{a^2- b^2r}$ is an integer but it turned out the trick is to rewrite $\sqrt{a+b\sqrt{r}} = c + d\sqrt{r}.$

Of course, one naturally asks if this is a specific incident or is there a general theory. This lead to digging up an article that I painstakingly photocopied from the library from back when photocopying was the norm. The article in question is by Susan Landau from 1994 in the Math. Intelligencer titled “How to tangle with a nested radical.”

A simplified version of Theorem 1 is this:
Let $k$ be a field extension of the rational numbers and $a, b, r \in k$ but $\sqrt{r} \notin k$. Then
$\sqrt{a^2- b^2r} \in k$ is equivalent to $\sqrt{a+b\sqrt{r}} \in k(\sqrt{s}, \sqrt{r})$ for some $0\ne s \in k$.

For example, one may check that
$\sqrt{5+2\sqrt{6}} =\sqrt{2}+\sqrt{3}$.

Millenium Bookball

is a pretty cool mathematical sculpture by George Hart. Gone with the Wind, Charlie and the Chocolate Factory, The Cat in the Hat, Green Eggs and Ham. What is not to like? Speaking of which, I still cannot believe that it was only because of my son who is now four that I read Dr Seuss for the first time in my life. Growing up with parents who do not speak English, I only started reading English books in Primary School. I still remember the joy when I did well in school examinations and the prize was to go to the school hall and select one free book for keeps.

On practice

I’m sure I am interpreting this in a context different from Vygotsky who was quoting Lenin

Man’s practice, repeated a billion times anchors the figure of logic in his consciousness.

Math videos

“Speaker, I’d like to talk about twin prime numbers …” goes McNerney in the US Congress. This took place on 11 Feb 2014. More details may be found here. There is still hope in politics afterall. I would love to see the day when mathematics is discussed in our parliament.

I’ve longed known about the discussion on the Monty Hall problem in the movie 21. But it was preceded by a short take on Newton’s or Newton-Raphson method. See the clip here at Mathematics in Movies site.

Prime Mystery

A nice puzzle from Aziz Inan on plus.maths.org

The number N represents the first 6 digits of a special number. N consists of three prime numbers put side by side. These three prime numbers come xth, yth and zth on the list of primes, where x, y and z are themselves three consecutive primes (for example, x, y and z could be 3, 5 and 7, in which case we’d be looking at the third, fifth and seventh prime numbers). In addition, if N is split in the middle into two separate numbers, the prime factors of the left part of N add up to its right part. What is N? And what is the special numbers whose first 6 digits it forms?

10 surprising things about our brains

an article from huffington. Worth the read.

“Gods” make Comeback at Toyota

is the headline of this bloomberg piece that reports the strange phenomena that human beings are replacing robots in the manufacturing lines. The point is that by automating some of these production process, much of the skill learnt by the human worker is lost. Without these knowledge, the humans cannot innovate and improve on the current process.

I see a close parallel between this and how students are losing their mathematical/computational ability because they are doing all their calculations on a calculator. Sure, if all the mathematics that you will do in future is to add some numbers on your bills, then by all means do that. But if you hope to develop a more mathematical/algebraic thinking then the hours of doing addition and multiplication sums are not something you should dispense with.

Come Together

is not the beatles song but the title of episode 16 of a tv program called discover science. I happened to catch it on local tv and the combination of quirky Japanese humour and the clever introduction to mathematics and science caught my attention. A preview of the episode is available at the above link but you need to watch the whole episode to find the link to parabolas.

Mathematics and poetry

A nice article by Ornes on slate and a paper by Glaz.

My favourites has to be the Fermat Last Theorem Poetry Challenge which I have read about previously elsewhere. Here’s one entry by
E. Howe, H. Lenstra, D. Moulton:
“My butter, garcon, is writ large in!”
a diner was heard to be chargin’.
exclaimed waiter Pierre,
“I couldn’t find room in the margarine.”

Maths & Fashion

That headline caught my eye and earned the newspaper article a more detailed look. It’s about a sub brand from Issey Miyake called 132.5, which apparently stands for 1 piece of material which forms a 3d dress but yet can be folded flat into 2d. The meaning of the .5 wasn’t that clear to me.

Fashion’s not my thing but this is certainly cool enough to merit a link.

http://www.isseymiyake.com/en/brands/132_5.html

The Princeton Companion to Mathematics contains a short section of sound advice to young mathematicans by Sir Micheal Atiyah and others. It is certainly worth reading.

– a proof due to Zolotarev and its connection to dealing cards.

A list of 240 and counting published proofs of the theorema aureum.

Of word problems and place values

If you google “Abbott & Costello 7 x 13 = 28″, you can get a funny sketch of how to show 7 x 13 = 28 by dividing, multiplying and adding. There are a couple of versions, and in one of them, Abbott ask Costello

A: “did you ever go to school stupid?”
C: “Yes sir, and I came out the same way!”

Another classic one goes like this

A captain owns 26 sheep and 10 goats. How old is the captain?

According to the book Making Sense of Word Problems by Verschaffel, Greer and de Corte, children from different parts of the world have actually offered answers to the question.

The broken stick problem

A slick solution to the broken stick problem or otherwise known as the uncooked spaghetti problem

That’s an emphatic NO!

all 49% of it.

Seen via math-fail.com

To restore our sanity, here’s a neat problem from the 2012 AMC 8:
Let R be a set of 9 distinct integers. Six of the elements are 2, 3, 4, 6, 9 and 14. What is the number of possible values of the median of R.

Polynomial Puzzle and Lill’s Method

I have in mind a polynomial with nonnegative integer coefficients. Can you determine completely the polynomial by just asking two questions?

I forgot where I saw this from but it was interesting enough to spur a few lunch time discussions with colleagues. I finally saw a reference in Dan Kalman’s book “Polynomia and Related Realms.” It was asked by I. B. Keene on the College Mathematics Journal, vol 36 (2005) page 100 and answered on page 159.

2005? I suspect the problem is much older than that.

Kalman’s book is a gem. In the first few pages alone, I learned about the Horner form of a polynomial and the fascinating Lill’s method for geometric visualization of the real roots of a polynomial.

Mathematical Induction

Apparently the French terminology for mathematical induction is “raisonnement par recurrence” or reasoning by recurrence. That makes so much more sense.

According to David Reid and Christine Knipping*

“Mathematical induction” is based on deductive reasoning, not inductive reasoning. The confusing terminology comes about because reasoning by recurrence makes use of specific cases, and as we saw above inductive reasoning is sometimes defined as reasoning beginning from specific cases. Reasoning by recurrence, however, also makes use of a general rule …”

Although, I have doubts whether the typical local student has any clear idea about what is deduction and what is induction.

*Proof in Mathematics Education, Research, Learning and Teaching. Sense Publishers, The Netherlands, 2010. (page 99)

Ranking Journals

has always been a task destined for controversy. Most academics have an implicit ranking of journals in their heads. When considering where to publish, the first consideration is usually appropriateness of the journal, followed by publishing in as good a journal as their paper is worthy of. Because of inherent differences in opinions, an explicit listing of journals is always subject to criticism. On the other hand, administrators want an explicit list so that they can rank academics according on a scorecard.

The only explicit and open list that I know of is the one published by the Australian Mathematics Society. My former university also had an internal distribution only list for the purpose of work review. The European Society for Research in Mathematics Education (ERME) has recently announced a ranking (of 17 journals) of mathematics education journal.

A combinatorialist worth his salt

Just as every analyst automatically interchanges the order of integration, and every combinatorialist worth his salt double counts at the drop of a hat, Erdös was constantly on the lookout for opportunities to apply the probabilistic method. [emphasis mine]

Taken from Bollabas’ article on Paul Erdös. A few months late but still, happy 100th birthday!

Source: Bollabas, B. To Prove and Conjecture: Paul Erdos and His Mathematics,
The American Mathematical Monthly, Vol. 105, No. 3 (Mar., 1998), pp. 209-237.