200 million pieces of Pi

Pi – the ratio of the circumference of a circle to its diameter – is an irrational number, meaning its decimal expansion never ends or repeats. As such, it could never be written out in full. If Pi is also a normal number, roughly meaning that the value of an arbitrary digit is random, an interesting property arises. Specifically, that every possible string of digits will be located in it somewhere. Given that any text can be perfectly converted into a number and any image or sound can be very well approximated by a number, this means that every possible written document, painting, photograph, symphony, and lecture can conceivably be located somewhere within that endless string of digits.

Quite a while ago, I had the idea that you could refer to any information in terms of a ‘Pi address’ – where to look within Pi to find the desired data. It would work for anything from the newest Tori Amos album to the first edition of the Oxford English Dictionary. The biggest problem is that the address would, in most cases, require more storage space than the actual data.

This website proves the point by letting you search the first 200 million digits of Pi for any string you want. My birthday begins at digit 196,469,286; my office phone number begins at digit 124,573,291. Because it is mathematically possible to calculate Pi from any arbitrary decimal place, it isn’t necessary to find all the prior digits to convert those back into the numbers they represent. That said, for large pieces of data (like the book and album mentioned), the Pi address would almost certainly be a lot longer than a data file containing the entire work. Pi addresses may not be a good way to refer to information, though they do provide a relatively dramatic perspective on the nature of infinity.

Author: Milan

In the spring of 2005, I graduated from the University of British Columbia with a degree in International Relations and a general focus in the area of environmental politics. In the fall of 2005, I began reading for an M.Phil in IR at Wadham College, Oxford. Outside school, I am very interested in photography, writing, and the outdoors. I am writing this blog to keep in touch with friends and family around the world, provide a more personal view of graduate student life in Oxford, and pass on some lessons I've learned here.

19 thoughts on “200 million pieces of Pi”

1. From the Wikipedia entry on Pi:

“The most pressing open question about π is whether it is a normal number — whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely “randomly”, and that this is true in every base, not just base 10. Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,…,9 occur infinitely often in the decimal expansion of π.”

2. Anon says:

Odds of finding any string in the first 100 million digits of Pi:

Number Length – Chance of Finding
1-5- 100%
6 – Nearly 100%
7 – 99.995%
8 – 63%
9 – 9.5%
10 – 0.995%%
11 – 0.09995%

3. R.K. says:

Converting Pi to binary: DON’T DO IT

If you compute it, you will be guilty of:

* Copyright infringement (of all books, all short stories, all
newspapers, all magazines, all web sites, all music, all movies,
and all software, including the complete Windows source code)
* Possession of child pornography
* Espionage (unauthorized possession of top secret information)
* Possession of DVD-cracking software
* Possession of threats to the President
* Possession of everyone’s SSN, everyone’s credit card numbers,
everyone’s PIN numbers, everyone’s unlisted phone numbers, and
* Defaming Islam. Not technically illegal, but you’ll have to go
into hiding along with Salman Rushdie.
* Defaming Scientology. Which IS illegal–just ask Keith Henson.

4. Tristan says:

as far as I understand it, the debate on the nature of infinity is between infinity as an unending succession, or as a totality. Does this help decide between those? It seems to point to the notion of infinity as totality, because all human knowledge would be contained within it. On the other hand, all new human knowledge is contained in it. All to-be-come-up-with human knowledge is contained in it.

It certainly does shift the emphasis away from succession.

5. Anon says:

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245890066063155881948815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298226733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277858813428588796091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638826537875937519577818577505321712268066130019278766111959092164201989

6. as far as I understand it, the debate on the nature of infinity is between infinity as an unending succession, or as a totality.

I am more interested in the debate about infinity among mathematicians and cosmologists, though it is entirely beyond me.

7. Pi is proven to go onto infinity, right?
Pi is also proven to be non-recurring.

From those two facts, does it follow that it must contain ANY number, of ANY non-infinite length, that you can come up with?

(For the record, my own telephone number wasn’t in the 20 million or whatever it searches. Waa.)

8. Pi is proven to go onto infinity, right?

This has been proven.

Pi is also proven to be non-recurring.

Something can be non-recurring without including all possible strings. For instance:

3.12123123412345123456….

Such a number is unending and non-recurring but does not contain every possible string.

9. I see, thank you for pointing that out. And reading your initial post closely, I see that you mention that pi is a normal number, and that’s why you can use it to store information, albeit with a bookmark possibly larger than datum you wish to store.

Ah!

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11. The other guy is 1 number off “you dont need to konk that”

12. solomon aminov says:

sorry i wanted to say u dont need to know that

13. solomon aminov says:

thats the name of the guy who is numbers off

14. God says:

@Milan-
Pi has NOT been proven to go to infinity, nor has it been proven nonrepeating. A number cannot be proven to be infinite, because by definition, an infinite number would have no end to its digits. By the same token, we are not aware of any pattern to Pi, but that does not mean that there isn’t a repeating pattern. Don’t post if you don’t know what you are talking about.

15. A number cannot be proven to be infinite, because by definition, an infinite number would have no end to its digits.

This isn’t a statement with any content. By definition, an infinitely long number has no end to its string of digits. That’s what infinite means. It doesn’t mean that mathematical proofs involving the number cannot be constructed.

By the same token, we are not aware of any pattern to Pi, but that does not mean that there isn’t a repeating pattern.

True. Hopefully, mathematicians will eventually figure this out. Either finding a pattern in Pi or proving that there is no pattern would be mathematically interesting.

As for the ‘God’ persona, it seems virtually certain that no such entity exits.

16. Matt says:

Don’t post if you don’t know what you are talking about.

More often than not, when you see a phrase like this in a web comment section, it is dripping with irony. No exception here.

17. The possibility that Pi is infinitely long and random in its contents is an interesting and amusing one, partly for the reason pointed out here.

If Pi is ‘normal’ and never-ending, it contains all possible finite strings. This would include all possible books, films, photos, etc.

18. Tristan says:

Any number which has no end to its string of digits, and yet about which some mathematical proof exists, is in at least one sense not infinite. “Finitude” means limit, shape, definition, idea, knowable. Lots of objects have unlimited distances on them, and yet are clearly finite objects (i.e. the distance around a ball is in a sense infinite – and this is at least tangentially analogous to how a number like 2.4241424142414241…. is infinite. You can travel along the number indefinitely, like the world, but you will nevertheless know where you are coming up to. In a sense, you’ve been there before (although never exactly there, because it is always a different time when you arrive the next time, just like the 3rd “1” in the above number is in some sense distinct from the 2nd “1” (counting from the beginning).

It seems odd to me that there could be any proof that a number repeated infinitely without coming to a pattern. How could you know in advance what pattern it might stabilize at? How do you define the lack of a pattern? What if we looked at the statistical correlations of the appearance of digits or sums of digits in the Pi string – if we could find statistically relevant data there would it prove that there is some finitude to Pi?

Finally, there is an ultimate proof that Pi is finite: Circles exist. They are as real as straight lines. Of course, circles don’t exist, and neither do straight lines – only representations of them exist, approximations, etc…

What is Pi? Pi is an ideal, an imagined form. A result of axioms, assumptions. The playing out of a pattern determined in advance by decisions made about what “geometry” is. About what “space” is (and how we can, at least for pretend, say it is flat).

It is thus not so interesting that Pi contains all possible finite strings. Pi is just a result of assumptions. Why is one set of assumptions more amazing than another?

Here’s a set of assumptions.
1) There is a never ending “normal” number, X.

And implications.
A) ‘X’ contains all possible books, films, photos, etc…