eon

Posted by tpc at August 26th, 2010

A problem from MAA website dating back to Fibonacci.

Four men already having denari found a purse of denari; the first man said that if he would have the denari from the purse, then he would have twice as many as the second. The second, if he would have the purse, then would have three times as many as the third, and the third, if he would have it, then he would have four times as many as the fourth. The fourth,if he would have it, five times as many as the first. How much denari does each man have? (Fibonacci, Liber Abaci, 1202)

Posted by tpc at May 29th, 2010

Picked up a copy of Lewis Carroll in Numberland by Robin Wilson from the library. I must admit that I browsed through it instead of reading it, picking up bits and pieces that I find interesting. For example, it is told of how Queen Victoria was charmed by Alice’s Adventures in Wonderland that she demanded:
Send me the next book Mr Carroll produces …
And the next book that arrived was … An Elementary Treatise on Determinants.

Ok, the story is not true but how cool would it be if it had been. Another fun tidbit is the following game. Starting from the number 1, A and B take turns adding a number from 1 to 10 to the running total. Whoever gets to 100 wins. What is a winning strategy? Working backwards, for B to be certain of winning, he would need to get to 89. That way, A can’t win with one turn but whatever number he picked, B can go for the win. Inductively, to get to 89, B needs to first get to 78, 67, 56, 45, 34, 23, 12 — steps of 11.

Posted by tpc at September 28th, 2008

Any rigid body displacement where a point is fixed is equivalent to a rotation. I saw this neat proof from Don Koks’ Explorations in Mathematical Physics.

By the hypothesis

Hence is orthogonal and But should vary continuously from the identity transformation, allowing us to conclude So


since we are working in 3D. We are forced to conclude
for some .
Thus has an eigenvector with eigenvalue 1. The transformation is thus the rotation about the direction .

Posted by tpc at April 9th, 2006

Another gem from The Lady Tasting Tea.

The mathematical theory of input-output analysis requires that the matrix that describes the economy have a unique inverse … Leontief’s initial set of sectors led to a 12 x 12 matrix, and Jerry Cornfield proceeded to invert that … It took him about a week, and the end result was the conclusion that the number of sectors had to be expanded. So, with trepidation, Cornfield and Leontief began subdividing the sectors until they ended with the simplest matrix they thought would be feasible, a 24 x 24 matrix … During World War II, Harvard University had developed one of the first, very primitive computers … Cornfield and Leontief decided to send their 24 x 24 matrix to Harvard … When they sought to pay for this project, the process was stopped by the accounting officer of the Bureau of Labor Statistics. The government had a policy at that time; it would pay for goods but not for services. The theory was that the government had all kinds of experts working for it. If something had to be done, there should be someone in government who could do it. They explained to the government accountant that, while this was theoretically something that a person could do, no would be able to live long enough to do it. The accountant was sympathetic, but he could not see a way around the regulation. Cornfield then made a suggestion. As a result, the bureau issued a purchase order for capital goods. What capital goods? The invoice called for the bureau to purchase from Harvard “one matrix, inverted.”

Posted by tpc at November 12th, 2005

If you go to the zoo, the chimpanzee will wave to you. I do not understand their psyche, but I presume they do not understand what they are doing, except when they wave, they get food. I still remember seeing a tourist throw an ice-cream into the enclosure to the adorable chimp.

It’s sad when students are taught monkey tricks, i.e. techniques to solve problems without understanding. One example is at A level when they are required to diagonalize a 3×3 matrix (with distinct eigenvalues.) Some schools actually teach a method to find eigenvectors by using cross products. I know this because a few students asked me whether they can use that method in our course. I’m not sure how it goes, but I guess it’s this:

To find the basis for the nullspace of , take two rows and compute the cross product. This works because the rowspace is the orthogonal complement of the nullspace and nullity=1.

The problem with this method is that it only works when A is 3×3 and the eigenspace is one dimensional, but I have a good feeling the students don’t know this. I recall a friend (JC teacher) who tested his students by asking them to diagonalize a 2×2 matrix and some of them who knew how to do it for 3×3 matrices couldn’t do it in this simpler case!

Posted by tpc at September 15th, 2005

I just read (from a text by K Hardy) that the Vandermonde determinant was named after Vandermonde by Henri Lebesgue, and was not ever recorded in Vandermonde’s work.

The list of other Misnomers

Posted by tpc at June 5th, 2005

I will be teaching Linear Algebra next semester and so I now have two copies of the same textbook with me. When I learnt the subject, we used the 7th edition, this time round, I will be teaching from the 8th edition. Frankly, I don’t see any difference in the two. (The latest news is that the 9th edition will be released this year.)

This edited picture sums up my feelings exactly. It’s taken from this link.
Disclaimer: I have not seen that particular calculus textbook and have nothing against the authors whatsoever.

Back to my linear algebra text, which had a new edition published about every 4 years. Now, let’s recall what wonderful breakthrough in the field of linear algebra has occurred over the past 25 years that required the author to keep up to date. Did I hear nothing? Mr Schmidt of Gram-Schmidt died in 1959.