Monday 15 March 2010

A Guest Blog (Rant?) from Dave Renfro

I mention Dave a lot here because his regular supply of interesting journal articles from today back 200+ years has been a major source of my continuing education in the last few years... Dave sent me a copy of a recent set of remarks he had written about recent posts related to Pi day. Actually Dave calls it a "rant" but we should all keep in mind that he is NOT referring to MY pi-day post (That's right, isn't it Dave.....DAVE? Talk to me Dave?)
Anyway, here is Dave's guest post, and I greatly appreciate him letting me be the on-line voice for his ideas this time.

I've mostly tried to not read anything about pi day, but in two cases I failed, with predictable results.
In the March 8-12 issue of "The Chronicle of Higher Education" there is an article about pi day titled "A Small Number With a Big Following". At one point in the article I saw the following comment:

"pi--so tiny (it's closer to three than four) yet random and infinite (as far as anyone knows)."

A question immediately occurred to me. Given the readership audience, which consists mostly of academic Ph.D.'s, why say something as meaningless and silly as this?

"random" -- What does that mean? I guess when computers calculate the digits of pi they're just randomly guessing what the digits are and then someone comes along and checks (how, nobody knows) to see if the digits are correct. Also, there are plenty of clearly non-random-looking sequences of arithmetic operations that generate pi, such as 4 - 4/3 +
4/5 - 4/7 + 4/9 - ... Unlike the CHE writer, some of us don't think the only way to represent a real number is by (integer) + a/10 + b/100 + c/1000 + ...

"infinite" -- If they mean the decimal expansion is infinite, then so too is the expansion of 1/3, the expansion of 1/7, etc. Maybe they mean that pi can't be described in a computable way, like Chaitin's constant "omega", except everyone knows pi is computable in very simple ways (as far as computability theory measures of complexity go), or do they?
Perhaps not everyone can type "pi" into Google and skim the pi Wikipedia page (the top hit when I tried this).

I think I know what they wanted to say, which was that no one knows whether, for each n, all possible n-digit strings appear in the decimal expansion with the same limiting frequency, although all computer explorations into the digits of pi seem to suggest this. Of course, they might want to express it a little less mathematical than this, but still
get the point across. As for saying "infinite", that's silly and a waste of words. Besides, it's automatic if you say something along the lines of what I just said, not to mention that everyone learns in 10th grade geometry (probably in 7th or 8th grade math books by now) that pi is irrational. Personally, I think knowing pi is irrational and what that means regarding decimal expansions is at least equivalent to knowing some of the literary references and schools of thought that get mentioned in their other articles without batting an eye. But you see, it's O-K in educated circles to say "economic determinism", but you don't ever want to say "irrational number".
The other article I saw showed up on March 12 at the CNN internet news
The article is titled "On Pi Day, one number 'reeks of mystery'", and surprisingly it seems to be pitched at a higher mathematical level (probably 8th or 9th grade) than the Chronicle of Higher Education article (about 6th or 7th grade). Here is an excerpt from the CNN article:


Mathematicians know that pi is irrational -- it cannot be represented as one number divided by another -- and transcendental, meaning it is not algebraic. That means, theoretically, that its digits will continue on indefinitely without ending in repetition -- in other words, the digits won't suddenly continue infinitely as 5s after 3 trillion digits (Pi's digits were calculated out to a record 2.7 trillion places in December
by French computer scientist Fabrice Bellard).

That also means, mathematicians theorize, that any string of numbers you can imagine is somewhere in pi -- for instance, look for your birthday. Coincidentally, "360," the number of degrees in a circle, occurs at digits 358 to 360. (Pat cool, I did not know that... I know that it's a coincidence, but I love it)


Right off the bat, in the first sentence, we have something any good middle school student would question -- isn't pi equal to pi divided by 1, and hence pi can be represented as "one number divided by another"? O-K, so the editor was asleep on that part. Let's continue. What's with this "theoretically" part? The digits of any irrational number (it's
worded as if you need to know that pi is both irrational and transcendental to conclude this, which is also an editorial oversight)continue indefinitely without being periodic, period. Then, in the next paragraph, we have another editorial flop. It's written as if the fact that pi is an irrational number (and maybe also the fact that pi is a transcendental number, an ambiguity we're left to figure out on our own)might mean that pi contains every finite string of digits, which of course isn't true -- plenty of irrational (and even transcendental)numbers have this property and plenty don't. Mathematicians theorize that pi might have this property, and even the much stronger limiting frequency property of these digit strings that I mentioned earlier, but to say that mathematicians theorize this on the basis of pi being irrational is extremely misleading. I think the author just wanted to write "This also means" because it sounded like a good transitional phrase, without worrying about what the phrase actually meant, and apparently the editor didn't worry about what it actually meant either.

And finally, what's up with saying "transcendental" means "not algebraic"? Does the author really think anyone who doesn't know what a transcendental number is will be helped by saying this is a number that isn't algebraic? I found this especially puzzling in view of the fact that in practically every single news article I've ever come across in which the term "light year" is used, the author seems compelled to state that a light year is the distance that light travels in 1 year (and then the author usually gives the equivalent in miles), and yet here "algebraic number" is thrown in without comment. I'd be willing to bet almost anything that far more people know what a light year is than what an algebraic number is. If I were editing the article I would have suggested saying something like
"although mathematicians have known that pi is irrational since the late 1700s, and transcendental--a certain extreme way that a number can be irrational--since the late 1800s, to this day no one knows . . ."There's no need in an article like this to define transcendental, but one should probably use the word since it's so well connected with pi
that it would seem strange to knowledgeable readers to not use the word.
If all this careful language analysis sounds unfair, ask yourself if being this sloppy with language usage would be accepted in an article about a bank robber (oops, I mean an alleged bank robber) or in an article about world affairs. No, it wouldn't. But it's O-K in math for some reason, and why this is allowed without much criticism is rather curious for a society that is so science-math-technology based.
I hope I got all that right, and if not, it was almost certainly my cutting and pasting that created the problem. Thanks Dave, for bringing a little class to my blog... (hey, since I wrote part of this, does this qualify as co-publishing??? I'm ready to stretch the rules where needed...)

And Dave, If you ever have something else to share with my (somewhat limited) audience, I would love to host you again.

Sunday 14 March 2010

Good Sense

from Hit & Run by Katherine Mangu-Ward... just talked to my stats kids about this a couple of days ago....
A great post from Robert Wright at the New York Times about why the disproportionate attention paid to Toyota recalls is worrisome and innumerate:

if you drive one of the Toyotas recalled for acceleration problems and don’t bother to comply with the recall, your chances of being involved in a fatal accident over the next two years because of the unfixed problem are a bit worse than one in a million—2.8 in a million, to be more exact. Meanwhile, your chances of being killed in a car accident during the next two years just by virtue of being an American are one in 5,244.

So driving one of these suspect Toyotas raises your chances of dying in a car crash over the next two years from .01907 percent (that’s 19 one-thousandths of 1 percent, when rounded off) to .01935 percent (also 19 one-thousandths of one percent). I can live with those odds....

But it worries me that this Toyota thing worries us so much. We live in a world where responding irrationally to risk (say, the risk of a terrorist attack) can lead us to make mistakes (say, invading Iraq). So the Toyota story is a kind of test of our terrorism-fighting capacity—our ability to keep our wits about us when things seem spooky.

Passing the test depends on lots of things. It depends on politicians resisting the temptation to score cheap points via the exploitation of irrational fear. It depends on journalists doing the same. And it depends on Americans in general keeping cool, notwithstanding the likely failure of many politicians and journalists to do their part.

If you're curious about how he did the math, go here and scroll down. If you want to see a bunch of commenters miss the point, keep scrolling on that same page.

But WHY?

The image is a pumpkin carved a few Halloweens ago by Sonja L. One of my Stats/Calc students (and a really good Bassoonist, Bassooner, Bassoon-enough)

Given the day, you may figure out this one...but the question above is the real issue. This was in an article from L. Short and J.P. Melville of the Napier University in Edinburgh, Scotland (which is, at least partly, on Napier's old estate)...

Take a unit square. Now step by step, create a unit rectangle on its right side (this is another square), then create a unit rectangle (2 x 1/2) on the top. Now go back and create a unit rectangle on the right side again (this one will be 2/3 by 3/2) then back to create on on the top.. Continue this forever and the sequence of ratios of the length to height of the total rectangle formed will converge to a Limit... find it...

and then explain WHY that is the limit... (blatant confession... I have no idea why this happens geometrically, and don't know anyone who does...)

Some more stuff from an older post on Pi I wrote a while back,,, after all, today is the day for it....
I know there are lots of infinite products that are equal to pi, and always thought
Wallis' expansion for pi/2 was beautiful,

And of course it is not as pretty, at least not to me, but since it was the first infinite product in math, I feel compelled to mention that Viete gave one for the reciprocal, 2/pi:

The question about this one for the really clever student is what does it have to do with the old calculus limit of Sin(theta)/theta... or the half angle formulas?

And Leibniz (you remember, the guy who invented Calculus if you DON'T live in England, just Kidding folks, it was Newton all the way, good job Ike) wrote one that Clifford Pickover calls "eye candy for pi";
pi/4 = 1 - 1/3 + 1/5 - 1/7+ 1/9....

except, it takes forever to converge.... the sum of the first 250 terms is not accurate to the second decimal place...

Ok, but the point of all this........

But today I came across one I had never seen, from the master of us all, Euler. Euler, it seems wrote pi/2 as an infinite product of fractions in which the numerators were all prime and the denominators were all even numbers excluding multiples of four. What appears from what I see is that each denominator is one more or less than the prime in the numerator, but always avoiding the one which would be a multiple of four... (Ok, now how do you write that as in product notation??).

pi/2 = 3/2 ( 5/6)(7/6)(11/10) (13/14)(17/18)

I looked for this a little and could not come up with a reference. If anyone knows where Euler wrote this, please advise.

Franz Gnaedinger who wrote the post where I picked this up, also pointed out that
"The analogous infinite product using all odd numbers
in the numerator seems to approximate the natural
logarithm of 2:

ln2 = 1/2 x 3/2 x 5/6 x 7/6 x 9/10 x 11/10 ..."
And there may be a clue in all this for the sequence that started all this off at the top...

Saturday 13 March 2010

Neighboring Binomial Equalities

Quick problem, not easy so plan some time.... Can you find a pair of neighboring combinations so that (hey, you can't use comb(2,0)... Good Luck, and if you find one quickly (tell me how, I would be amazed) find the next.

Irrational Birthday Celebrations

Dave Richeson from Dickenson College has a nice blog at his Division by Zero blog that is good news (of a sort) for anyone born in 1980 (if you live long enough).
"De Morgan was always interested in odd numerical facts and writing in 1864 he noted that he had the distinction of being {x} years old in the year {x^{2}} (He was 43 in 1849). Anyone born in 1980 can claim the same distinction."
If you live to your 45th birthday in that year, you too can say you are x years old in the year x^2. Overcome by curiosity, he searched out and found the last few times this has happened
Here are a few other people who could have made De Morgan’s claim (listed by year of birth). 1892: J.R.R. Tolkein was 44 in 1936 1806: John Stuart Mill was 43 in 1849 1722: Samuel Adams was 42 in 1764 1640: Bernard Lamy, the mathematician, was 41 in 1681 1560: Annibale Carracci, Italian painter was 40 in 1600 1482: Maria of Aragon and Castile, queen of Portugal would have been 39 in the year 1521 (she died in 1517) 1122: Eleanor of Aquitaine was 34 in the year 1156
Dave's list of notables was a little "unimaginative" for a mathematician (sorry Dave) so I would alter his list to : 1892: Stefan Banach was 44 in 1936 1806: Isambard Kingdom Brunel was 43 in 1849 (and amazingly, the Brooklyn Bridge Engineer, John Roebling was also(good year for Engineers?) 1722: Samuel Adams was 42 in 1764 *(OK, I like Adams, and I could not find one mathematician on the St Andrews Math History site who was born in 1722... my closest choice for a math related person would be French astronomer Abbé Jean Chappe d'Auteroche. 1640: Bernard Lamy, the mathematician, was 41 in 1681*** wonderful, especially right before Pi Day...Nice one Dave. 1560: Thomas Harriot (Solving equations by factoring is Still called "Harriot's method by some (me)", 40 in 1600, and he turned a telescope to the stars before Gallileo. 1482: The first printed version of Euclids Elements, printed in Venice, was 39 in the year 1521 1122: Ok, by this time I have developed a new appreciation for the difficulty Dave had coming up with names...You would think some mathematician/scientist/engineer would have been 34 in the year 1156, but they are hard to come up with... Ok, submit your list.. who would be the best "square" for each year. In exchange, I propose that we do not allow people who were integrally blessed to steal all the glory...If you were born anytime after 1936, you can share this property sometime during your 44th year. For instance, in 1910,  if the 302nd day of your 45th year (you would have been 44.83302354.. years old) you qualify.. and if you are too young, in 2011 on the 308th day you turn 44.844... years old, you can join this illustrious group...   alas, my time has passed, but I can say with DeMorgan, that I turned x years old in the year x, it just was an irrational x, but heck, it was an irrational year when it happened. .

Sunday 7 March 2010

We badly need a Hippocratic Oath for schooling:

"Above All Else, Do No Harm"

Sunday morning... put on the coffee, turn on the feed from KSCS in Fort Worth (Ok, you got me..I'm country) and start flipping through blogs I haven't had time to read all week... and the one that most caught my eye had the quote above.

Those who have read my blog for awhile know that I occasionally set aside my blogs on math to comment on education, my grand kids, or whatever happens to catch my interest.

This Sunday it was a month old blog at the Daily Kos (I have no idea what that means) by teacher Ken. It is long, but good enough to take the time..

The title is "students should graduate with a résumé, not a transcript", and is taken from a quote by Arnold Packer, who was the principal author of the SCANS report, from the Dept of Labor. Much of the blog is based on an article by Grant Wiggins, a regular proponent of "Authentic Education." (Sort of like "Pro-life", who could be against that?)
Here is the opener from the Wiggins piece:

Imagine the following HS requirements being recommend to the School Board:
• 3 years of economics and business
• 2 courses in philosophy – one in logic, the other in ethics
• 2 years of psychology, with special emphasis on child development and family relations
• 2 years of mathematics, focusing on probability and statistics
• 4 years of Language Arts, but with a major focus on semiotics and oral proficiency
• US and World history, taught as Current Events - backwards from the present
• 1 Year of Graphics Design, Desktop Publishing, and Multimedia presentation

Outrageous? Hardly – if we do an analysis of what most graduates actually need and will use in professional, civic, and personal life. How odd it is that we do not require oral proficiency when every graduate will need the ability. How absurd it is in this day and age that students aren’t required to understand the capitalist system. How sad it is that physics is viewed as more important than psychology, as parents struggle to raise children wisely and families work hard to understand one another. Requirements based on pre-modern academic priorities and schooling predicated on the old view that few people would graduate and fewer still would go on to college make no sense. Ask any adult: how much algebra did you use this past week?

Ok, that last line struck a nerve that I come back to sometimes....... If someone took a survey in 1900 of the frequency with which people used automobiles, television, airplanes, (complete with your own examples) no organization could use it to justify research on the basis of that survey. And the adult who has NO training in algebra will have only the most superficial understanding of probability and statistics from his two years of mathematics in those areas. ...and how much do you learn in a course in semiotics in the absence of algebraic reasoning??? consider this trilogy of the elements of the subject in a online "introduction"....

* semantics: the relationship of signs to what they stand for;
* syntactics (or syntax): the formal or structural relations between signs;
* pragmatics: the relation of signs to interpreters
remember, you are teaching this to the same kids who have trouble figuring out how to add fractions, or find the area of a triangle, or explain what a polynomial is

But given that big objection I like the idea of a more student tailored curriculum. I like the idea of a student getting the education they want, getting a recognition of the level of achievement they have reached, and let business decide (and test to see) if they have the credentials that the business needs.. I had a parent recently say she would be having her kid skip a day of school to go to Cambridge and participate in the science day there... Another is missing a week to go to Crete with her father who just returned from six months being deployed in Iraq. I hate for kids to miss my class, but not much I will do in a day or a week will be as life changing as a day with world class scientists designing and explaining the future of an area of physics that one kid is interested in, or spending a week on a beautiful island with family to recover from some of the sacrifice of an extended absence (he didn't just miss seeing her spike a volley ball, he missed her first real date, falling totally in love for the very first time, and her first heartbreak six weeks later when the boy decided they should "just be friends" all of which she went through missing only a moment or two of classes for a trip to the hall when tears flowed). These kinds of events should be part of the experience of high school. I believe schools ought to have more independent explorations in subjects the student is interested in... Bright kids take calculus because it is the next course in the "college-intending" sequence, but some (many) would be much better served with a course in Statistics, and although I teach it, AP Statistics is not necessarily the best syllabus for that learning for lots of them. And I have several bright students who have accelerated right through all the pre-calculus curriculum and never saw a Moebius Strip, never played with a bent nail puzzle, and have never heard of Fibonacci, or a thousand other things that give math life and energy.

So Let's offer kids the option to tailor their own education... really empower students, and you Mr. Taxpayer, will need to cough up several more dollars each year because to do that schools will need to have lots more small classes (hence more teachers) and the technology to do more independent study .. What? oh, on second thought you don't like it as much... Yeah, let's just go back to blaming all our problems on the teachers and the unions.... Did you hear, there is a school over in Whatsit Town that let a private company take over the school system and saved thousands of dollars a semester. They can manage with just one certified teacher in each subject to design lessons and they hire aides to manage classes and the kids learn independently.... and their state test scores hardly dropped at all this year.....

Saturday 6 March 2010

Some Proofs are Prettier Than Others

Just read a post about Nichomachus's Theorem at as a guest blog on Loren Shure's Matlab blog. It pointed out that the theorem was cleverly proven by the simple picture below.
The idea of a proof without words is that you can "see" what it shows.

The theorem is usually presented in pre-calculus classes when they cover sequence and series. In simple words it says the square of the nth triangular number is equal to the sum of the cubes of the first n integers. Cool math language to replace all that verbiage is .

Nicomachus lived in what would now be part of Jordan around 100 AD, and was a follower of the Pythagorean Cult. He wrote about arithmetic and his works were translated into Latin by Boetheus. His book on music and its relation to math is the earliest source of the story that Pythagoras came up with the idea of harmonic tones when he walked past a blacksmith pounding on an anvil. His "Art of Arithmetic" contains the equality above.

I like the "proof without words", but I think my favorite was by Charles Wheatstone. I first heard of Wheatstone as a young electronic trainee in the Air Force. He didn't invent, but did improve a device for measuring the resistance of ..well, almost anything, which is now called the Wheatstone Bridge. He also invented the Playfair cipher (math names are confusing, but Lord Playfair was a heavy promoter of the cipher), the concertina and the stereoscope (if you are older, you looked at cards at your grandmothers house through one of these that gave a 3-D view of the Taj Mahal or other beautiful scenery. He also came up with the following really clever proof of Nicomachus's theorem.

1 + 8 + 27 + 64 + 125 + ...
= (1) + (3 + 5) + (7 + 9 + 11) + (13 + 15 + 17 + 19) + (21 + 23 + 25 + 27 + 29) + ...
= 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23 + 25 + 27 + 29 ...

and since the sum of the first n odd integers is equal to n^2, we only have to see that by the method he has broken up the cubes there will be 1+2+3+...+n odd integers for the first n cubes... which seems really nice to me.

Monday 1 March 2010

A Different Prime Sieve....

Sadly, or perhaps happily for internet security, there is no explicit formula that will generate the prime numbers. Many students have heard about Euler's interesting formula n2 + n + 41 which will generate 40 consecutive prime numbers when n=0 to 39 are entered. The first few are 41, 43, 47, 53, 61... (I think I remember reading that there is no polynomial that will produce a longer string of primes for consecutive arguments) but if you want to find ALL the primes, some form of sieve is required.

Sieve is not a well known word to many students. They are more likely to know the kitchen implement as a sifter. The two synonymous nouns describe a tool which, in the words of one dictionary, "separates wanted elements from unwanted material using a filter such as a mesh or net. " A prime sieve then, separates out primes from non-primes. Most students encounter the sieve of Eratosthenes in middle school. You take a list of integers and circle two, and then cross off every 2nd number after that as multiples of two, then circle three and cross out every third number (some of which will be already crossed out). Each time you return to the beginning of the list, you circle the next unmarked number n, and then crossing out every nth number following.

Around 1930, a little known or remembered Indian mathematician named S.P. Sundaram came up with a different sieve. It operates on simple arithmetic sequences.

Start with 4 and create an arithmetic sequence by repeatedly adding three... 4, 7, 10, 13, 16, 19, 22...
In the second row, start with seven, and add five each time 7, 12, 17, 22, 27,....
continue starting with each number in the first sequence as the initial term, and to each sequence add the next consecutive odd number...

It looks like this
4 7 10 13 16 19 22 25 28
7 12 17 22 27 32 37 42 47
10 17 24 31 38 45 52 59 66
13 22 31 40 49 58 67 76 85
16 27 38 49 60 71 82 93 104

Ok, some are prime, some are not....what's up.... Take any number that appears in the list, multiply by two and add one.... Now check, Is it prime??
Try another... and time after time it turns out the number is NOT prime. 17 is in the numbers, and 2(17)+1 = 35, which is not prime...
But now find a number that does not show up in the list... five is not there, neither is six, or eight, or lots of others. Repeat the 2n+1 idea and Voila..primes emerge.

Cute, but the most intriguing sieve I have ever seen is called the visual sieve, and uses a parabola and a straight edge. Start with the simple parabola x=y2 and label the point for each integral point, (1,1); (4,2) etc... with the absolute value of its y-coordinate.. It will look sort of like this. Now connect the point at (4,2) with all the points below the x-axis. You should get something that looks like this:
Notice that each point on the x-axis that is crossed out is a composite number. Repeat for every number above the x-axis and you will eventually get something that looks like this,

it actually looks a little like a sieve. Notice that we are left with only the prime numbers on the x-axis.

I came across this on the Plus Math web magazine article "Catching primes" by Abigail Kirk. Even better for students, they have a really nice section called "Why does that work?" They are also the source of all the colorful pictures I just used, and actually they have several more that make it even easier for a student to follow... pass this on to your students.