Tag Archives: pi

The experts list, or: how can a journalist find out how to compute pi to high precision?

A reporter at the Wisconsin State Journal called me the other day with a really good question.  He had heard that pi had been computed to ten trillion decimal places.  And he wanted to know:  how could you possibly measure a circle that precisely?

So how did he know to call me?  Because I’m on the experts list, which UW-Madison’s public relations office set up to give journalists the opportunity to consult a Wisconsin professor on just about any subject.  Topics of current news interest get promoted to the front: on today’s front page we’ve got the professor who can talk about Kim Jong-Il, the professor who can talk about the Supreme Court’s decision to take up the Arizona immigration law, and the professor who can talk about the Scott Walker recall.  (I have a feeling that last guy is going to be on the front page for a while.)

Such a simple idea, but such a good one!  The UW-Madison ought to be a resource for Wisconsin journalists — and everybody else in Wisconsin, for that matter.  Good for the PR office for making it as easy as possible to reach faculty who want to face the public and share what they know.

Oh:  here’s the article about pi in the WSJ, by Dave Tenenbaum.  I thought it came out well!

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I talk about Pi Day on Wisconsin Public Radio

Tomorrow (Monday) morning on Morning Edition at 88.7FM, aired at 6:35 and again at 8:35.  Available online at Wisconsin Life.

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Efficient markets and the digits of pi

John Quiggin had a good post yesterday on Crooked Timber about the various flavors of the Efficient Market Hypothesis, which according to Quiggin is a piece of shuffling zombie social science that won’t die no matter how many flaming sticks you jam through its skull.

The weakest form of the EMH is the so-called “random walk hypothesis” that the future behavior of a stock price is independent from its prior behavior.  If that’s the case, no amount of staring at charts is going to help you beat the market.  The random walk hypothesis is pretty well-supported by the data we have; a really nice popular account is Burton Malkiel’s A Random Walk Down Wall Street, one of the mathiest bestsellers I know of.  It makes a great present for anyone in need of a good rationale for not paying attention to their investments.

Quiggin writes

The success of the random walk hypothesis showed that the existence of predictable price patterns in markets with rational and well-informed traders was logically self-contradictory.

But this doesn’t seem quite right.  The EMF, even in its weakest form, holds that the current price of a stock is a best estimate arrived at by an aggregate of profit-maximizing investors with knowledge of the stock’s previous price:  if there were a reliable way to use previous prices to determine that tomorrow’s price would be Y, then the investors would figure that out, and today’s going price would be Y as well.

But the random walk hypothesis is much weaker.  It makes no claim about the etiology of stock prices.  It’s compatible with EMF, but it’s compatible with plenty of other models too — for instance, the one in which stock prices really are a random walk, where the price at time t+1 is the price at time t plus, let’s say, a normally distributed random variable X.  Such a market would be as impossible to beat as a roulette wheel — but evidently not because stock prices are best estimates of the future price of the stock, or of the underlying value of the company, or of anything at all.  (Fellow mathematician David Speyer makes a similar point in the comments at CT.)

You can dress this up a bit:  suppose price(t+1) – price(t) is the random variable X + [(.001) x t’th digit of pi] – (.045).   Unless something very weird is going on with the digits of pi, this version of the stock market would also satisfy the random walk hypothesis.  But unlike the pure random walk it’s a market where you can make some money; if I’m lucky enough to have access to the “digits of pi” rule, I can make a small average profit.  So the random walk hypothesis can hold for markets that are neither efficient nor unbeatable.

Stronger versions of the EMH hold that the market price already takes into account, not only the previous prices of the stock, but also publicly available information about the stock.  So what would happen if I let my secret investing strategy slip out?  This isn’t a rhetorical question; I’m authentically curious about what the rational-investor model would say about a market in which prices are publically revealed to have been governed, up until now, by some completely deterministic but “random-looking” sequence like the digits of pi.

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The Vilna Gaon approximates pi

Michael Coen, last seen here, told me this amusing story.  In the Torah, there’s a verse that describes a certain pool in the temple as being 10 cubits across and 30 cubits in circumference — in other words, pi = 3.  On the other hand, as the Vilna Gaon observed, the word meaning “circumference” here is spelled “kuf vav hay,” where it would normally be spelled “kuf vav.”  The standard spelling has an alphanumeric value of 106, but the variant used in the verse has a slightly larger value, 111.  If you “correct” the estimate 3 by a factor of 111/106, you get 333/106 = 3.1415…, the third continued fraction approximant to pi.

Clever dudes, those Torah scholars.

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