Wednesday, March 4, 2009

Irrational Numbers Metaphor

This is kind of a whimsical post, perhaps you may find it of very questionable relevance to a psychiatry blog.

I invite some input from any number theory experts out there, perhaps some of my thinking about the following subject is erroneous.

Irrational numbers are numbers which cannot be expressed as a ratio of integers. So, for example, the square root of 2 is irrational (it is approximately, but not exactly 1.414; it can be visualized as the distance diagonally across a square which has each side of length=1). The number pi (the ratio of a circle's circumference to its diameter) is irrational, approximately 3.14. The natural exponential base e is irrational, approximately 2.7. If we attempt to express an irrational number in decimal form, we can only ever get an approximation. The digits will keep on going forever, in a non-repeating fashion.

A hypothesis I have about the digit expansion of an irrational number is that the sequence represents a form of true randomness. At one point I did plot out the frequencies of digits in an expansion of pi to a million digits or so, then performed some statistical tests on this, and determined that the results are consistent with random ordering. They MUST be "random", for if they weren't, the number could not be irrational. I would invite a number theorist to show me a proof of this. My idea about randomness invites a philosophical, or mathematical, discussion, about what the meaning of true randomness really is.

But the digits of irrational numbers are calculable. That is, the millionth, or trillionth, digit, in the decimal expansion of pi, can be determined, systematically, through various algorithms. The number e can be calculated in a number of ways (this is a way I discovered as a child, playing with my calculator: take (1+1/n) multiplied by itself n times, with the calculation becoming more and more accurate as n grows larger--only perfectly accurate, though, when n reaches infinity).

So, I am claiming that the digits are calculable, yet randomly ordered. This is a seeming contradiction.

However, I believe there is no simple formula for the "nth" digit of pi. In order to get the "nth" digit, at least n arithmetic steps must be taken. That is, computational work must be done in order to do the calculation, and more computational work is required in order to reach a more precise result, which is at least linearly proportional to the level of precision desired.

Since all computational work requires energy, and there is a finite amount of energy available in the universe, let us suppose that we use all the mass-energy of the universe to perform computational work to determine as many digits of pi, for example, as is possible. (this would involve, in our thought experiment, harnessing all of the great nuclear energies from the stars, etc. to power a computational device just for this task)

Now, having generated all of these digits (I suspect there would be over 10^1000 digits generated, using all the energy of the universe efficiently for this task), we still only have an approximation to the number pi. The NEXT digits of pi are theoretically calculable, but cannot be calculated or known, because we have used all available computational energy.

Thus, we have calculable digits, which yet cannot be known, because there is not enough energy in the universe to do the calculations to know them.

There is something almost mystical about this: any sequence of digits, for example, randomly conceived in the mind, must correspond to a sequence of digits in the unknowable expansion of pi (in that realm over 10^1000 digits into the expansion), based on the laws of probability.

Something that we can prove is outside the realm of human knowledge is actually part of the ordinary daily products of our imagination.

As an added concept related to this, imagine what your entire life history would look like, translated into a sequence of digits -- perhaps this would include a few thousand pages of text, a few million images, together with the entire sequence of your genome, all transformed into a digit sequence, maybe a few trillion digits long.

It can be shown that this sequence -- an intimate representation of your identity -- must occur at some point in the decimal expansion of all irrational numbers, including pi. (suppose the sequence representing your life story is 10 trillion digits long; then the probability of your sequence occurring starting at or before the nth position in pi's expansion is 1-(1-1/(10 trillion))^n, assuming that pi's digit expansion behaves as a random sequence. With this assumption, once you are into pi's expansion by 10 trillion digits, there's a 63% chance that your sequence will have shown up (interestingly, this probability is approximately 1-1/e). And the more digits you go into pi's expansion, the more likely it is that "your" sequence will show up; this probability converges towards 100% as the number of digits approaches infinity. Actually, we could go on to say that "your" sequence actually recurs, an infinite number of times, in pi's expansion!

In our imagination, we can conceive an ideal circle, and we can imagine the ratio between its circumference and its diameter. That is pi exactly. We have imaginatively visualized something, with perfect precision, something that cannot be expressed logically with perfect precision.

There is a life lesson in this, I think. Be open to possibility. That which is seemingly impossible may require an imaginative re-framing to see that it was always in front of you, available to you, part of "ordinary" daily life. And there can be more to simple relationships than meets the eye -- dividing a circumference by a diameter yields a number which contains information paralleling all known information in the universe, including the story of yourself.

3 comments:

XiXiDu said...

Great post, thanks! :-)

Anonymous said...

Counterexample:

0.101001000100001000001...

is irrational - it cannot be represented as a ratio of integers - but the distribution of digits in the decimal expansion is highly nonrandom. Not only are there no twos or nines, there aren't even any sequential ones. It's pretty easy to represent as a series, though:

Sum for n = 1 to infinity
1/10^((n^2 + n) / 2)

Please pardon my notational barbarity :)

The property you're looking for is normality, rather than irrationality. We know that almost all numbers are irrational, and that almost all irrational numbers are normal, but we don't know how to prove that a given number's normal, in almost any case. Even a constant as familiar as pi has yet to be proved.

That's a bit of a fib, as pretty much everyone believes that pi's normal, and there's some good statistical, inductive evidence (like the research you did prior to this post) - but no deductive proof. So do let everyone know if you come up with one - seriously!

Have you seen the Gregory-Leibniz series for pi, by the way? It's even prettier when written out longhand:

4/1 - 4/3 + 4/5 - 4/7 + 4/9...

Doesn't look random at all, does it :) But it probably is.

Anyway, be careful when you're working with a uniform distribution over an infinite set (such as the decimal expansion of an irrational number). Just because you come up with a 100% chance that some kind of pattern occurs doesn't mean it's ever there at all. In mathematical jargon, that's "almost surely", and it doesn't mean the same thing that "surely" does when applied to an infinite set. You really can't say that a given sequence of digits in the decimal expansion of pi shows up at all, let alone an infinite number of times.

The above may very well sound nitpicky and tendentious in light of the conclusions you're aiming for - my apologies for that. You're certainly not wrong when you say an imaginary circle encodes a cosmos worth of information. It'd just require some kind of deity to unspool all of that meaning. And deities are sorely lacking in the local mathematics department.

- Beth (not a number theorist)

GK said...

Thanks Beth, what a great comment!

Nice to hear from a mathematician!

(you're evidently more a number theorist than I)

I think we're stuck with some phenomena that remain mysterious and subject to philosophical debate (also there are some unproven--and possibly unprovable-- theorems alluded to here).

In terms of the "deities" required for unspooling meaning, I suspect we all underestimate ourselves in this regard --

I think a common personal or psychological problem is feeling detached from the delightful, meaningful, mysterious, or sublime aspects of our daily activities, rather than recognizing them with joy and satisfaction. These aspects I consider to be "deific". (it is important to know that I borrow this religious terminology, and apply it in a metaphorical way). I think people working in mathematics departments in particular are full of "deific" characteristics. I think it is healthy to acknowledge, nurture, and share this part of ourselves expansively. I have often wished there was more whimsical dialog between mathematics professors, other scholars (of all types, from English literature to physics to anthropology, etc.), and the public.