Wednesday, December 13, 2006



Want more? Click here to get one million and a quarter digits of Pi!

Yesterday, I read that the bible says Pi equals 3. Consequently I thought, gosh, somebody will insist to replace Pi with three in all schoolbooks, so they are in agreement with the bible. It didn't take me long to find out this was hardly a new concern, and has already status of an urban legend.

For the basics: Pi is the ratio of a circle's circumference to its diameter (in Euclidean geometry). It is named "π" because it is the first letter of the Greek words περιφέρεια 'periphery' and περίμετρος 'perimeter', i.e. 'circumference'. And it's not equal to three. In fact it's roughly equal to the long number shown above, the essential thing being the dots in the end.

But some more interesting info: Pi is a transcendental number, which means it can not be written as the solution (root) of a polynomial with coefficients in the integer numbers. This also implies that the number of digits after the point is infinite, they do never repeat, and every possible sequence appears at some point* (see here for the probability of finding some, and here for searching them).

Since this so far only explains what Pi is not, it seems some people are still concerned whether it actually exists. Well, this might sound somewhat philosophic, but I mean, you can't just write it down and say, there it is. The definition that I recall is that Pi/2 is the first zero of the sinus function. Which seems to me quite easy to prove that it exists (the function being smooth and having a sign change and all). If you don't want to use Euler's number for the sinus (another transcendental number), the sinus function can be defined as an infinite polynomial, which I would write down here, if latex could speak on my blog...

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  1. You mean this? (I would do an img tag but does not let me.

    I am using mimetex for atdotde which is not as beautiful as what Jacques does but readable I would say.

  2. Missing a factorial.

  3. Yasumasa Kanada

    6,442,450,000 decimal digits, March 1996.
    51,539,600,000 decimal digits, August 1997.
    206,158,430,000 decimal digits, September 1999.
    1,241,099,999,501 decimal digits, December 2002.

    Frequency distributions for (pi minus 3)

    0: 119,999,636,735
    1: 120,000,035,569
    2: 120,000,620,567
    3: 119,999,716,885
    4: 120,000,114,112
    5: 119,999,710,206
    6: 119,999,941,333
    7: 119,999,740,505
    8: 120,000,830,484
    9: 119,999,653,604

    Interesting sequences (digits after the decimal):

    271828182845 from 1,016,065,419,627-th
    314159265358 from 1,142,905,318,634-th

    012345678910 from 1,198,842,766,717-th

    01234567890 from 53,217,681,704-th
    01234567890 from 148,425,641,592-th
    01234567890 from 461,766,198,041-th
    01234567890 from 542,229,022,495-th
    01234567890 from 674,836,914,243-th
    01234567890 from 731,903,047,549-th
    01234567890 from 751,931,754,993-th
    01234567890 from 884,326,441,338-th
    01234567890 from 1,073,216,766,668-th

    98765432109 from 123,040,860,473-th
    98765432109 from 133,601,569,485-th
    98765432109 from 150,339,161,883-th
    98765432109 from 183,859,550,237-th
    98765432109 from 300,854,719,683-th
    98765432109 from 534,846,931,487-th
    98765432109 from 593,100,546,152-th
    98765432109 from 609,238,336,350-th
    98765432109 from 647,565,670,462-th
    98765432109 from 936,998,389,684-th
    98765432109 from 1,116,106,038,318-th

    09876543210 from 42,321,758,803-th
    09876543210 from 57,402,068,394-th
    09876543210 from 83,358,197,954-th
    09876543210 from 264,556,921,332-th
    09876543210 from 437,898,859,384-th
    09876543210 from 454,479,252,941-th
    09876543210 from 614,717,584,937-th
    09876543210 from 704,023,668,380-th
    09876543210 from 718,507,192,392-th
    09876543210 from 790,092,685,538-th
    09876543210 from 818,935,607,491-th
    09876543210 from 907,466,125,920-th
    09876543210 from 963,868,617,364-th
    09876543210 from 965,172,356,422-th
    09876543210 from 1,097,578,063,492-th

    10987654321 from 89,634,825,550-th
    10987654321 from 137,803,268,208-th
    10987654321 from 152,752,201,245-th
    10987654321 from 265,616,128,905-th
    10987654321 from 524,896,938,580-th
    10987654321 from 560,934,871,496-th
    10987654321 from 912,609,366,275-th
    10987654321 from 990,180,271,473-th
    10987654321 from 1,041,179,396,679-th
    10987654321 from 1,171,566,790,976-th

    777777777777 from 368,299,898,266-th
    999999999999 from 897,831,316,556-th
    111111111111 from 1,041,032,609,981-th
    888888888888 from 1,141,385,905,180-th
    666666666666 from 1,221,587,715,177-th

  4. Hi Uncle,

    that's nice :-) I once read a book which evolved around a sequence of 23 23-s in Pi, but I can't really remember the details (not the name of the book, will try to find out). You know where to find the sequence?



  5. Ah, found the book:

    The Visiting Professor by Robert Littel

    Here's the blurb:

    Forsaking his customary thriller territory, Littell ( The Revolutionist ) here finds fertile new ground in the farther reaches of mathematics, which prove a wellspring of rich and consistently surprising comedy. When Lemuel Falk, a Russian "theoretical chaoticist on the lam from terrestrial chaos," arrives to take up his visiting fellowship at Backwater University, he is immediately confronted by a blizzard of Americana: is it absolute confusion or, as Lemuel suspects, merely "fool's randomness"--the facade of disorder behind which lurks a pure meaning? Many turn to him for the answer: a dope-smoking Orthodox rabbi seeking "the chaos at the heart of the heart of the Torah," a libidinous female barber named Occasional Rain, and a multinational throng of spooks and spies all seeking to use Lemuel's mathematical genius for their encryption programs. A not-quite-innocent abroad fleeing Stalinist ghosts, the professor quests across the spiraling chaos of the American landscape, becoming in succession or in combination a lover, theologian, political protestor, media celebrity, homicide investigator and, finally, a refugee in the deceptively tranquil aisles of the local E-Z Mart. Littell's fast-paced satire is by turns bawdy, cerebral and touching.

    I quite liked the book, though I remember I found the plot somewhat vague.

  6. Pi is a transcendental number, which means [...] every possible sequence appears at some point.

    That's not strictly true. Pi is transcendental, of course, but it is not known that every transcendental number must be "normal" (have every possible finite sequence appearing with equal probability). Pretty much all that is known is that the set of non-normal numbers has measure zero. It is generally conjectured that pi is normal, but AFAIK nobody has shown normality for any number except special hand-crafted examples. For all that is known, the digit 9 might never appear in the decimal expansion of pi after the 10^(10^10)th digit.

  7. georg is right. It's easy to write down a transcendental number whose decimal digits are, for example, all 7s and 4s.

  8. Hi Georg, Hi Carl,

    Thanks for pointing this out! Now that you mention it, I recall there was a subtlety... I have corrected the sentence. So the status is, one thinks Pi is normal, but there's no prove?

    It's easy to write down a transcendental number whose decimal digits are, for example, all 7s and 4s.

    What exactly do you mean with 'write down'? I can imagine it is easy to show such a thing does exist (well, I guess you could take Pi and replace all 0,1,2,3,5,6,8,9-s with 4, and the result would still be transcendental, would that work?). Best,


  9. Without question, Pi's a symbol to be reckoned with. Admittedly, though, I hold a special fondness for e. Oh sure, Pi and e are equally two of the most celebrated transcendental numbers.

    In comparison to Pi, I'll remark, not only does e seem to allow one to perform more mathematical tricks, e also seems to have a stronger connection to Nature. Granted, this is nothing more than a hunch...

    Because e carries the weight of the hyperbola and Pi carries the weight of the circle, a great mystery is yet to unfold: Pi^e. If my memory serves me right, Euler was one the first to become fascinated by the relationship between e and Pi. Needless to say, anything Euler thought--not to mention--touched should never to taken lightly.

  10. Dear Cynthia,

    I admit, I also like Euler's number better, or the function e^z defined through it, respectively. Is there anything more beautiful than a function that reproduces itself under derivation?

    d/dz e^z = e^z



  11. To make a transcendental number with just 4s and 7s in the decimal expansion:

    Algebraic means that the number is the root of a rational polynomial. Transcendental means not algebraic.

    The algebraic numbers (which includes all the rationals) are countable. So one lists them in one order or another, as a countably infinite list {a_n}. Many such orderings are known.

    To write down the transcendental number made from 4s and 7s, one uses the nth digit of the a_n algebraic number to determine nth digit of the transcendental number.

    You use 4 or 7, either one, unless a_n's nth digit is 4 or 7, in which case you have to use 7 or 4.

    The resulting decimal expansion is different from every algebraic number and is therefore transcendental, and made of 4s and 7s.

    After you select an ordering of the algebraic numbers, you can compute the decimal expansions of the first few algebraic numbers, and from that determine the first few digits of your transcendental number. It's finding all the digits that is difficult.

  12. You should also link to the rather well-known blog that has the PI favicon. ;-)

  13. carlbrannen--actually, it shouldn't do too difficult to find transcendental numbers on a line segment. After all, Georg Cantor discovered that there are more irrational numbers than rational numbers, and more transcendental numbers than algebraic ones. Restated, most real numbers are irrational; and among irrational numbers, most are transcendental.

    Hence, transcendentals are--by far--the most common set of numbers on a line segment. Oddly enough, though, the task of locating transcendentals is fairly easy. However, the task of devising a rigorous proof to determine--without a shadow of a doubt--that they are definitively transcendentals is incredibly grueling, to say the least.;)

  14. You can pretty easily write down some transcendental numbers with only zeros and ones. Liouville's constant:

    \sum 10^{-(k!)}

    does the trick.

  15. " Aaron Bergman said...

    You can pretty easily write down some transcendental numbers with only zeros and ones. Liouville's constant:

    x= \sum 10^{-(k!)}

    is trascendental"

    Hence, 4/9 + 3x is a trascendental number with only 4 and 7 in its decimal expansion :-). It's nice to be able to avoid the axiom of choice.

  16. Just a quick note on adding math support to Blogger blogs (repeat of post in Clifford's blog).

    Have a look here. Peter Jipsen of Chapman University has written ASCIIMathML a great javascript program that converts ASCII notation (more or less the one used in math and physics newsgroups, and of course in blogs) to MathML. It also recognizes latex notation. Firefox rendering is great, although you may need to download some fonts. IE needs a plugin and rendering is not so good. You are supposed to upload the javascript file to your server, but I suspect that it will work also if you put the content of the file in the head of your Blogger template. (Be warned though that the file is 42K, and that I have not tried this, I just assume it will work.)

    If you do not want to risk this, there are two more solutions mentioned in the page given above. The first one are the ASCIIMath Image Fallback Scripts which use a public mimetex server (meaning you do not have to have mimetex in your server).

    The second is LaTeXMathML, which as you may have guessed translates latex notation to mathml using again a public server if you cannot upload the file in your own. To use this, you just need to add one line in your template head, so that the file can be read from a public server.

  17. Well if I were a mathematician, I could tell if this paper by Bailey and Crandall includes pi in it's proven class of "normal" constants or not...
    *scratches head*

  18. nice to be able to avoid the axiom of choice.

    You don't need the axiom of choice to order a countable set, though it will do the job.

  19. Hi Gebar,

    Thanks so much! I'll give it a try if I find the time - hopefully soon :-)


  20. I had a laymen question: Why are they are looking for higher and higher digits of pi and whether they are used in anyway? I understand it could be fun in itself. Thank You!

  21. Now why didn't God communicate the first million digit via a prophet, so that it would have appeared in the Bible? :)

  22. Maybe he did and Pi is just a code for the bible. After all, if Pi is normal then it should contain the bible in whatever code you could think of...

  23. Hi Chinmaya,

    Honestly, I don't know. But mathematicians are funny people. They can spend years on a single number, or a function that isn't even named after them. Maybe it's good to easily come up with some coding mechanism? Say, I send you a scrambled text and a ten digit number indicating a starting point in Pi. Then we only need to have agreed on a recipe like: from the starting point on you take two numbers for each letter in the text, add this amount of steps on the letter, and get a new letter.

    But this can't be a good reason for a mathematician. Maybe they are indeed looking for God's name ;-) Or trying to find out whether Pi suddenly consists only of 4s and 7s or so.



  24. I did a lesson on computing pi for fifth graders, starting with inscribed and circumscribe squares, moving to Archimedes method, and Euler's formula in terms of alternating series. We finished up with a more modern algorithm with which we could compute the first 1500 or so digits, letting Maple draw all the pictures and do the aithmetic.

    I'm not sure how much they understood, but they seemed to like it, and got a glimpse of algebra, the Pythagorean Theorem, series and limits.

  25. A question: Suppose you write a trancendental number in binary, but then interpret the number thus written as a decimal: Is the result always transcendental? I feel sure that it it, but can't immediately prove it.


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