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Magic Numbers

You learn something new every day. It seems that some strings of random numbers are more random than others. That's kind of interesting, but not really a surprise when you think about it. Whenever we look at the characterisitcs of a string of random digits occuring in Pi or e or some other irrational number, we are looking at only a tiny fraction of the digits. Actually, it may not even be accurate to describe it as a fraction.

The linked article describes how mathematician Steven Pincus made some interesting discoveries when looking at the randomness of the first 280,000 digits of Pi, the square root of 2, and several other irrational numbers. However, even 280,000 isn' t really a fraction of an infinite number, now is it? How many digits would it take before you had a representative sample of an infinite string? I'm not a mathematician, but I'm guessing it would take an infinite string.

But before you wrap your head too tightly around that, consider what Pincus observed when he started comparing these strings of digits: some have higher levels of entropy (randomness), some lower. Then he started looking for the same characteristic of entropy in real-world strings of numbers, such as you might get from tracking, say, the stock market. He discovered that the stock market hits its highest level of entropy right before a crash.

Pincus observes that entropy

appears to be a potentially useful marker of system stability, with rapid increases possibly foreshadowing significant changes in a financial variable.

He goes on to conclude:

Independent of whether one chooses technical analysis, fundamental analysis, or model building, a technology to directly quantify subtle changes in serial structure has considerable real-world utility, allowing an edge to be gained... And this applies whether the market is driven by earnings or by perceptions, for both sort- and long-term investments.

Expect to hear a lot more about entropy and financial markets in the near future. The movie Pi, which I thought was well-made and entertaining, but suffered from a silly premise, may just turn out to be prescient.

via GeekPress

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Comments

there are two flavors of infinity in mathematics: countable and uncountable.

for example, there is a countable infinity of whole numbers (..., -3, -2, -1, 0, 1, 2, 3, ...), whereas there is an uncountable infinity of real numbers. there's even an uncountable infinity of real numbers between say pi/2 and pi, or even between 1.2345 and 1.2346

still, as you say, 280,000/infinity is still zero.

if you've never read The Man Who Loved Only Numbers (about Paul Erdos), then put it on your reading list.

Chris

I will definitely add it to the list of must-read books. I am somewhat familiar with the countable/uncountable dichotomy thanks to a Martin Gardner book I read years ago. So if the digits in Pi are an uncountable infinity, could you get a "representative sample" of Pi using a countably infinite number?

And if so, what good would it do finite creatures like ourselves? ;-)

"Don't you know that you can remove an infinite set of numbers from an infinite set and still have an infinite set? God! My wife understood this on our very first date!"

-The Mirror Has Two Faces

Sure, Stephen. I believe we're talking about Barbra Streisand, here.

Clearly, she knows everything. :-)

if the digits in Pi are an uncountable infinity, could you get a "representative sample" of Pi using a countably infinite number?

Actually the digits of Pi are a countable infinity. You can think of this distinction like this: associate the first digit with the whole number 1, the second digit with the whole number 2, and so on. each digit can be associated with one of the whole numbers, thus they can be counted. there's an infinity of digits, because no matter how high you count, there's always more.

Having said that, i still don't know the answer to the question. My guess is that there is no representative sample. There are very likely strings of repeated sequences in the digits of Pi (or any other irrational number), and you could land on one of those strings and convince yourself that the sequence has terminated. I seem to recall that there is something like that fairly early in the sequence of Pi (or is it e?).

There's a strong sense in which many of the above numbers are nonrandom. If we look at a continued fractions representations we often see patterns. Ie, a number can be represented as a_0+1/(a1+1/(a2+...))) where a_, a_1,... are integers.

For example, the square root of 3, we get a repeating sequence of integers (1,2,1,2,...) after the first couple of terms. "e", the base of the natural log also has a very predictable pattern (2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,...). As far as I know, no known pattern of the continued fraction (at least patterns as obvious) exists for pi.

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