I Will STILL Derive

OK, after all those years of teaching calculus, it seems I've been saying it all wrong, or at least that is the opinion of several people on the AP calculus list who think that the verb "Derive" is not acceptable for "differentiate" which they suggest is better. I have even posted videos "I will Derive" parodies of the old "I will survive" song.

Ok, maybe they have a point...but I'm a little confused. If they want to get pedantic about not using "derive" then they certainly must avoid "derivative" (that which is derived, to obtain or receive from a source)..If you differentiate, then you get a differential; yes??? So if you want to find the derivative... well, you get my drift.

Derive is kind of a great word, it comes directly from the word river (which once meant the banks and not the river, hence the Riviera is the coast). In days of yore the draining of water out of the river and into the fields for irrigation was something like "derivering" and then worked its way into "deriving" (get it, to draw from a source). We use derive in that sense a lot in math for any type of deductive reasoning. To me it seems perfectly logical to talk about deriving the function for the slope from the original function, and calling the result of that derivation, the derivative.

Sorry guys, but I'm afraid that I've been doing it way to long to break the habit now, so I will continue to derive, but if I can remember I will try to tell my students that they may encounter a future professor who may dislike the usage, and they should be prepared for such an event... but honest, I don't much care which word they use, I just wish they would remember to apply the chain rule.

What! You Don't Believe in Aliens?

Casey, one of my Stats students, didn't believe in the evidence of alien intelligence on Earth. I decided that I would have to convince him with the one most certain piece of evidence imaginable, so I asked, "You know what a thremos bottle is, right; so what happens if I put hot coffee in a thermos?"

Casey: "It stays hot." almost as a question

ME: "Yes.... and what if I put cold milk in the thermos,,,, what then?"

Casey, uncertainly: "Ummm... well, ..it stays cold."

ME: " Un Huh... (in triumph) and HOW does the thermos know which I put in????" with a slow wink and a finger tapping the side of the head

I'm sure that Casey now believes, but if you are one of those people who STILL doesn't believe there is other intelligent life in the universe, here is the incontrovertable proof you have been waiting for from John Hodgman. A story about aliens, physics, time, space and the way all of these somehow contribute to a sweet, perfect memory of falling in love.

Searching for Snarks

The happy band of mathematical warriors above joined me on Thursday night as we set out to the Center for Mathematical Sciences at Cambridge. Our quest was to discover all that Emeritus Gresham College Professor of Geometry Robin Wilson might know about "Lewis Caroll in Numberland." The lecture, which can only be described as "math-lite" was entertaining none the less, and made even better by the good companions who joined me.

Wilson is also, not by coincidence, the author of a book on the topic that is soon to be released in the US (and can be purchased in advance from Amazon at a healthy discount) entitled, Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life.

Wilson explained that if Dodgeson (the real name of Lewis Carroll) had not written the "Alice" stories for which he is so well remembered, he might well be remembered for being one of the pioneering child photographers of the 19th century. And if he had not done either, he might be remembered as an accomplished mathematician and teacher who made contributions in the areas of Logic, algebra, geometry, and the mathematics of elections. Wilson points out that:
"Yet another interest of his was the study of voting patterns. Some of his recommendations were adopted in England, such as the rule that allows no results to be announced until all the voting booths have closed. Others, such as his various methods of proportional representation, were not. As the philosopher Sir Michael Dummett later remarked:

It is a matter for the deepest regret that Dodgson never completed the book he planned to write on this subject. Such was the lucidity of his exposition and mastery of this topic that it seems possible that, had he published it, the political history of Britain would have been significantly different."

He also credits Carroll with the invention of the modern method of seeding tennis matches:
Another interest of Dodgson's was the analysis of tennis tournaments:"At a lawn tennis tournament where I chanced to be a spectator, the present method of assigning prizes was brought to my notice by the lamentations of one player who had been beaten early in the contest, and who had the mortification of seeing the second prize carried off by a player whom he knew to be quite inferior to himself.
Let us take sixteen players, for example, ranked in order of merit, and let us organise a tournament with 1 playing 2, 3 playing 4, and so on. Then the winners of the first round will be 1, 3, 5, and so on; those of the second round will be 1, 5, 9 and 13; the final will then be won by player 1, defeating player 9 who wins the second prize but actually started in the lower half of the ranking.
To avoid this difficulty, he managed to devise a method for re-scheduling all the rounds so that the first three prizes go to the best three players, which presaged the present system of seeding.

For a brief discription of some of his work in Logic diagrams see My page here and for A different version of the Wilson's talk on Lewis Carroll given at Gresham College, look here

More on Price matrices for Primitive Pythagorean triples.

For the lack of a better term, I will call the 2x2 matrices used to describe Primitive Pythagorean Triples in my last blog as "the matrices".

One of the consequences of the way they are constructed is that the two numbers in the right column represent the difference and sum of the two numbers in the left column.

This allows you to construct the full 2x2 matrix when any two of the elements are known. This, in conjunction with the knowledge that each offspring on the Barning tree has an incenter that was one of the three excenters of the parent, allow us to produce the tree without the use of matrix transformations.

If we start with the 3,4,5 triangle, which has a matrix ) we have already established that the products of the bottom row and the two diagonals give the radii of the three excircles (1x3=3, 1x2=2, 2x3=6)... So we can find the three offspring of the 3,4,5 triangle by promoting each of these excircles to an incircle, as shown here:

Now by using the fact that the right hand column is made up of the difference and sum of the left hand values, we can complete the two missing values to find the descendants of the parent matrix.

But we don't really want to know the matrices, we want the Pythagorean triples. Fortunately it is easy to find the triple associated with any of the matrices. Multiply down the left column and double to get the even leg; multiply down the right column to get the odd leg, and then sum the products of the two diagonals to get the hypotenuse. So the matrix on the lower left with rows of 2,1 and 3,5 will have an even leg that is twice 2x3, or 12. Its odd leg is 1x5 = 5 and the hypotenuse is 1x5 + 1x3 = 13. That means one descendant of the 3,4,5 triangle is the 5, 12, 13. You can work out the other two.

If you are paying careful attention, we just added the excenters of the two legs to get the hypotenuse of the triangle... did you know that worked that way? How about this, you also get the hypotenuse if you subtract the top row product from the bottom one. Geometrically that would be subtracting the incenter from the excenter on the hypotenuse... Now go convince yourself that makes perfect sense.

If you find the determinant of a matrix, but ignore the signs and add everything (like we did when we added the diagonals of the 2x2) it is called the Permanent of the matrix. I had never seen this word before. Jeff Miller's web site on the first use of math words provided a little detail. The term seems to have been created by Cauchy. "In his book Permanents [9] H. Minc mentions that the name permanent is essentially due to Cauchy (1812) although the word as such was first used by Muir in 1882. Nevertheless a referee of one of Minc's earlier papers admonished him for inventing this ludicrous name!"