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.


keninwa said...

Just curious, is this the same Dave Renfro who was a professor at Central Michigan University for a time?

Sue VanHattum said...

Thanks for posting that. As I read it, I realized that my post on Pi day stayed at the basic end, and my favorite pi connection recently was trying to work through the proof at The Math Less Traveled, that pi is irrational. I've edited my Pi Day post. (Maybe no one will read the edit, but it's there for next year.)

Pat's Blog said...


He is indeed the same Dave Renfro... were you a colleague or a student?

and Sue, I think Dave is not trying to limit discourse, he just thinks the news media ought to set as high a standard for editing math articles as anything else... (can't you see the ex-math phobic editorial staff pushing math/science articles back and forth, almost bragging about their math illiteracy... ) But it is a good reminder to all of us to work on effective communication...

keninwa said...

I was a student in his Modern Algebra 1 course a few years back. Please tell him "hi" for me. On a side note, he offered a paradox to the class one day regarding distances between 2 points that I continue to revisit on occasion. I think I have almost figured it out...

Dave L. Renfro said...

"... he offered a paradox to the class one day regarding distances between 2 points that I continue to revisit on occasion."

Pat told me about your comment just now, although I would have seen it at some point since I visit his blog every week or two (but probably every day for the next few days . . .)

So, what's this paradox? I don't remember it right now, unless it's this problem:

Find all points in xyz-space that are equi-distant from the three points (1,0,0), (0,1,0), and (0,0,1). Hint: There is more than one such point.

For more about this problem, see the following two posts. The first gives its history relative to me and the second gives a solution.

Dave L. Renfro said...

Oh, wait. I bet I know what it was. The length of the diagonal of a unit-length-edge cube in R^n is sqrt(n) (apply the R^n distance formula to (0,0,...,0) and (1,1,...,1)). Thus, in a high enough dimension, you can shine a flashlight from one corner of a unit cube to the furthest opposite corner and it will take over a year for the light to get to that furthest opposite corner.

If that's not it, then it might be the "Losing your marbles in hyperspace" puzzle (google this phrase).

keninwa said...

The paradox (which I think I have solved, but haven’t had time to verify my solution; and I have thought that I had solved it in the past and others have always found errors in my logic) goes like this: A person starts at (0,0) and wants to travel to (1,1) but can only move in right angles. If they travel one unit over then one up, they travel 2 units. If they go .5 units over, then .5 up, then .5 over, then .5 up, they travel 2 units. If they continue to decrease the distance traveled before turning by a factor of .5, to the point where the distance traveled before turning is at about 0 (limit as d -> o and as # of turns -> infinity) then the distance traveled suddenly becomes sqrt (2).

Dave L. Renfro said...

Oh, that paradox! The underlying issue is that arc length is not a continuous function (in a suitably defined context). The limit of the lengths of the curves is not always equal to the length of the limit of the curves. However, the situation is not totally chaotic.

Let C_n be a sequence of curves that converge to a curve C, and let L(C_n) and L(C) be their lengths. Then while in general we don't have

LIM{(C_n)} = C implies LIM{L(C_n)} = L(C)

(note this has the same form as saying that a_n --> a implies f(a_n) --> f(a), which is the sequence formulation of continuity of f(x) at x = a), we do have

LIM{(C_n)} = C implies LIM{L(C_n)} >= L(C),

a property that is called "lower semicontinuous". For more about this curve paradox and lower semicontinuity, see

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