Relationship-Mapping-Inverse (RMI)
(invented by Prof Xu Lizhi 徐利治 中国数学家 http://baike.baidu.com/view/6383.htm)

Find Z = a*b

By RMI Technique:
Let f Homomorphism: f(a*b) = f(a)+f(b)

Let f = log
log: R+ –> R
=> log (a*b) = log a + log b

1. Calculate log a (=X), log b (=Y)
2. X+Y = log (a*b)
3. Find Inverse log (a*b)
4. ANSWER: Z = a*b

Prove:

$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$

1. Take f = log for Mapping:
$latex \log\sqrt{2}^{\sqrt{2}^{\sqrt{2}}} $
$latex = \sqrt{2}\log\sqrt{2}^{\sqrt{2}}$
$latex = \sqrt{2}\sqrt{2}\log\sqrt{2} $
$latex = 2\log\sqrt{2} $
$latex = \log (\sqrt{2})^2 $
$latex = \log 2$

2. Inverse of log (bijective):
$latex \log \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= \log 2$
$latex \sqrt{2}^{\sqrt{2}^{\sqrt{2}}}= 2$

Prof Su 苏步青, the founding pioneer Math professor of the China’s top universities (Zhejiang 浙江大学 and Fudan 复旦大学), was one of the few mathematicians who had longevity above 100 years old (the other was French Mathematician Hadammard).

http://en.m.wikipedia.org/wiki/Su_Buqing

Two men A and B are 100 km apart, walking towards each other, A at speed 6 km/hour and B at 4 km/hour.
A brings a dog which runs at 10 km/hour between them,  starting from A towards B, upon reaching B it runs back to reach A, then back to B again, and so on…

Find total distance the dog has covered when A and B finally meet ?

[Note: the content of this post is standard number theoretic material that can be found in many textbooks (I am relying principally here on Iwaniec and Kowalski); I am not claiming any new progress on any version of the Riemann hypothesis here, but am simply arranging existing facts together.]

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function $latex {\zeta}&fg=000000$, defined by

$latex \displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}&fg=000000$

for $latex {\hbox{Re}(s)>1}&fg=000000$ and extended meromorphically to other values of $latex {s}&fg=000000$, and asserts that the only zeroes of $latex {\zeta}&fg=000000$ in the critical strip $latex {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}&fg=000000$ lie on the critical line $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.

One of the main reasons that the Riemann hypothesis is so important to number theory is that the zeroes of…

A collection of excellent free resources for demonstrating the various circle theorems:
Tim Devereux has created GeoGebra applets which allow exploration of the circle theorems. You can access each theorem from the menu on the left which includes a useful summary of all the theorems.

See alsothese excellent demonstrations.

Top >10 Mathematics Websites remains a very popular post on this blog.

I have read various ‘Top (insert number here) Mathematics Websites’ posts and all of them have left me with the thought that so many excellent sites are missing from such lists. Any post claiming top 10 or >10 in my case is clearly the author’s top 10, notthe top 10! These are my top >10 because I really do use them – a lot – in the classroom! For my own list, I have decided to include some categories as well as individual sites which gives me the excuse to mention far more than 10! Note that every site mentioned here is free to use.

A quick quiz: which of the following four words doesn’t fit with the others??

Massive/Open/Online/Courses

We are going to muse about MOOCs today, a hot and highly debated topic in higher education circles. Are these ambitious new approaches to delivering free high quality education through online videos and interactive participation over the web going to put traditional universities out of business, or are they just one in a long historical line of hyped technologies that get everyone excited, and then fail to deliver the goods? (Think of the radio, TV, correspondence courses, movies, the tape recorder, the computer; all of which held out some promise for getting us to learn more and learn better, mostly to little avail, although the jury is still out on the computer.)

It’s fun to speculate on future trends, because of the potential—indeed likelihood—0f embarrassment for false predictions. Here is the summary of my argument today:…

As you can see from the picture, it was a packed house! After waiting in line for fifteen minutes, I was so lucky (and excited) to get a seat to hear Jo Boaler speak, even if my seat was in the next to last row.

Jo opened the presentation with Dweck’s research on mindsets. “In the fixed mindset, people believe that their talents and abilities are fixed traits. They have a certain amount and that’s that; nothing can be done to change it. In the growth mindset, people believe that their talents and abilities can be developed through passion, education, and persistence.”

Jo states that the fixed mindset contributes to one of the biggest myths in mathematics: being good at math is a gift. She referenced her book, The Elephant in the Classroom (added it to my reading list) and showed the audience various television/movie clips that continue to perpetuate…

Last week a friend who is a fourth grade teacher came to me with a math problem.  The father of one of his students had showed him a trick for checking the result of a three-digit multiplication problem.  The father had learned the trick as a student himself, but he didn’t know why it worked.  My friend showed me the trick and asked if I had seen it before.  This post describes this check and explains why it works.

Suppose you want to multiply 231 $latex \times $ 243.  Working it out by hand, you get 56133.  Add the digits in the answer (5+6+1+3+3) to get 18.  Add the digits again to get 9.  Stop now that you have a single digit.

Alternatively, do this digit adding beforehand.  Adding the digits of 231 together, we get 6.  Adding the digits of 243 together, we get 9.  Multiply 6 $latex \times$ 9…

Having a blog gives me a chance to defend myself against a number of people who took issue with a passage in Mathematics, A Very Short Introduction, where I made the tentative suggestion that an abstract approach to mathematics could sometimes be better, pedagogically speaking, than a concrete one — even at school level. This was part of a general discussion about why many people come to hate mathematics.

The example I chose was logarithms and exponentials. The traditional method of teaching them, I would suggest, is to explain what they mean and then derive their properties from this basic meaning. So, for example, to justify the rule that x^a+b=x^ax^b one would say something like that if you have a xs followed by b xs and you multiply them all together then you are multiplying a+b xs all together.

Today I had an experience that I have had many times before, and so, I imagine, has almost everybody (at least if they are old enough to be the kind of person who might conceivably read this blog post). I was in a queue in a chemist (=pharmacy=drugstore), and I knew that my particular item would be quick and easy to deal with. But I had to wait a while because in front of me was someone who had an item that was much more complicated and time-consuming. In this instance the complexity of the items was not due to their sizes, but a more common occurrence of the phenomenon is something that often happens to me in a local grocery: I want to buy just a pint of milk, say, and I find myself behind somebody who has a big basket of things, several of which have to be…

Now that the project to upgrade my old multiple choice applet to a more modern and collaborative format is underway (see this server-side demo and this javascript/wiki demo, as well as the discussion here), I thought it would be a good time to collect my own personal opinions and thoughts regarding how multiple choice quizzes are currently used in teaching mathematics, and on the potential ways they could be used in the future.  The short version of my opinions is that multiple choice quizzes have significant limitations when used in the traditional classroom setting, but have a lot of interesting and underexplored potential when used as a self-assessment tool.

I’m in the happy state of just having finished marking exams for this year. There is very little of interest to say about the week that was removed from my life: it would be fun to talk about particularly bizarre mistakes, but I can’t really do that, especially as the results are not yet known (or even fully decided). However, one general theme emerged that made no difference to anybody’s marks. There seems to be a common misconception amongst many Cambridge undergraduates that I’d like to discuss here in the hope that I can clear things up for a few people. (It is an issue that I have discussed already on my web page, but rather than turning that into a blog post I’m starting again.)

The question where the misconception made itself felt was one about functions, injections, surjections, etc. I noticed that a lot of people wrote things…

I’ve received quite a lot of inquiries regarding a recent article in the New York Times, so I am borrowing some space on this blog to respond to some of the more common of these, and also to initiate a discussion on maths education, which was briefly touched upon in the article.

Firstly, some links:

• The video for the talk “Structure and randomness in the prime numbers” mentioned in the article can be found here (requires RealPlayer). The slides can be found here. My other expository and research material on number theory can be found here.
• I don’t have any specific advice regarding gifted education, though some articles on my own experiences can be found here. I do however have some thoughts on career advice at the undergraduate level and beyond.
• I have some responses to several other common queries (e.g. regarding books…

As the previous discussion on displaying mathematics on the web has become quite lengthy, I am opening a fresh post to continue the topic.  I’m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion.

It’s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathML, jsMath, MathJax, OpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g.  LaTeX2HTML, LaTeXMathML, LaTeX2WP, Windows 7 Math Input, itex2MML, Ritex, Gellmu, mathTeX, WP-LaTeX, TeX4ht, blahtex, plastex, TtH, WebEQ, techexplorer…

High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write $latex {\frac{1,572,342}{4,124}}&fg=000000$ as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve…

Michael Gove, the UK’s Secretary of State for Education, has expressed a wish to see almost all school pupils studying mathematics in one form or another up to the age of 18. An obvious question follows. At the moment, there are large numbers of people who give up mathematics after GCSE (the exam that is usually taken at the age of 16) with great relief and go through the rest of their lives saying, without any obvious regret, how bad they were at it. What should such people study if mathematics becomes virtually compulsory for two more years?

A couple of years ago there was an attempt to create a new mathematics A-level called Use of Mathematics. I criticized it heavily in a blog post, and stand by those criticisms, though interestingly it isn’t so much the syllabus that bothers me as the awful exam questions. One might…

In the diagram, the circumference of the external large circle is
1) longer, or
2) shorter, or
3) equal to,
the sum of the circumferences of all inner circles centered on the common diameter, tangent to each other.

Answer: 3) equal

circumference = π. diameter

Let d be the diameter of the external large circle C
Let dj be the diameter of the inner circle Cj

$latex \displaystyle d = \sum_{j} d_j$
$latex \displaystyle \pi. d = \pi. \sum_{j} d_j= \sum_{j}\pi.d_j$

Circumference of the external circle
= sum of circumferences of all inner circles

I had a mathematical conversation yesterday with a 17-year-old boy who is in his second year of doing maths A-level. Although a sample of size 1 should be treated with caution, I’m pretty sure that the boy in question, who is very intelligent and is expected to get at least an A grade, has been taught as well as the vast majority of A-level mathematicians. If this is right, then what I discovered from talking to him was quite worrying.

The purpose of the conversation was to help him catch up with some work that he had missed through illness. The particular topics he wanted me to cover were integrating $latex \log x$, or $latex \ln x$ as he called it, and integration by parts. (Actually, after I had explained integration by parts to him, he told me that that hadn’t been what he had meant, but I don’t think…

Shawn asked a good question in class yesterday about the differences between stratified sampling and quota sampling. In terms of sampling mechanism (i.e. the actual process by which cases are chosen from the population), it is clear that these two samples are different. Unclear, however, is why they would lead to different results.

Recall that stratified sampling is conducted by dividing a population into two or more strata by virtue of some characteristic, and taking random samples from each strata. This is done when a simple random sample of an entire population will likely not generate enough analyzable cases for a given group of particular interest.

Let’s say we want to study the income differences between blacks and whites in the United States. Unfortunately, we only have enough funding to distribute 500 questionnaires. Given that 10% of the population is black (made up, but reasonably approximate), a simple random…