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Frequently Asked Questions (FAQS);faqs.093


    46 ( 16)  -  -  Tornado, Adventureland, IA
    47 ( 14)  -  -  Thriller, German Fairs
    47 ( 14) 26  -  Texas Tornado, Wonderland, TX
    47 ( 14) 39  -  Grand National, Blackpool, UK
    48 ( 13)  -  -  Jack Rabbit, Kennywood, PA
    49 ( 12) 39  -  Thunderhawk, Dorney Park, PA
    50 ( 11)  -  -  Big Dipper, Blackpool, UK
    50 ( 11)  -  -  Le Monstre, La Ronde, Canada
    50 ( 11)  -  -  Sea Serpent, Wildwood, NJ
    50 ( 11)  -  -  Excalibur, Valleyfair!, MN
     -   -   15  -  Revolution, Magic Mountain, CA
     -   -   33  -  Dragon Mountain, Marineland, Canada
     -   -   37  -  Wildcat, Lake Compounce, CT
     -   -   38  -  Gemini, Cedar Point, OH
     -   -   39  -  Space Center, Phantasialand, Germany
     -   -   40  -  Bandit, Yomiuriland, Tokyo, Japan
     -   -   40  -  Psyclone, Magic Mountain, CA
     -   -   40  -  Sidewinder, Hersheypark, PA

    G. List of Endangered Coasters in USA -- as of July 1992:

    Legend:

    DAMA - Damaged and non-operational
    DEMO - Demolished/Destroyed
    SBNO - Standing But Not Operating
    STOR - Dismantled and in storage

            ASSURED TO BE SAVED

    SBNO -Comet: 1946 Twister; Lincoln Park; N Dartmouth, MA
    SBNO -Leap The Dips: Side Friction; Lakemont Park; Altoona, PA

            COASTERS WITH A CHANCE

    STOR -Shooting Star: Out-and-Back (from Lakeside Park)
    STOR -Comet: Dbl Out-and-Back (from Crystal Beach)

            COASTERS IN DANGER

    OPER -Wildcat: 1926 Out-and-Back, Elitch Gardens; Denver, CO
    OPER -Coaster: Twister, PNE; Vancouver, BC
    SBNO -Blue Streak: Out-and-Back, Conneaut Lake, PA
    SBNO -Thunderbolt: 1925 Twister, Coney Island, NY
    SBNO -Mighty Lightnin: 1958 Wood, Rocky Glen; Moosic, PA
    SBNO -Jumper: 19?? Jr. Wood, West Point, PA
    SBNO -Red Streaker: 19?? Jr. Wood, Willow Mill; Mechanicsburg,PA
    SBNO -Jack Rabbit: 1910 Out/Back, Idora Park; Youngstown, OH
    DAMA -Wildcat: 1927 Twister, Idora Park; Youngstown, OH

            COASTERS WE'VE RECENTLY LOST FOREVER

    DEMO -CNE Flyer: 1956 Oval, CNE; Toronto, Canada
    DEMO -Speedway: 1937 Out/Back, Eldridge Park; Elmira, NY
    DEMO -Valley Volcano: 1956 Jr. Wood, Angela Park; Hazleton, PA
    DEMO -Tornado: 1968 Out/Back, Panama City, FL
    DEMO -Mountain Flyer: 1929  Out/Back, Mountain Park; Holyoke, MA
    DEMO -Coaster: 1931 Out/Back, Harvey's Lake, PA

*************************************************************************

Contributors:

     Mark Wyatt (Inside Track)
     [e-mail]
     [e-mail]
     [e-mail]
     [e-mail]
     [e-mail]

Editorial Assistance:

     Nora G.
     [e-mail]
     [e-mail]

Disclaimer:  I make no warranty on the information contained here-in.
Comments, corrections and questions are welcome via e-mail to
[e-mail]. You may redistribute this information freely as
long as it is distributed in its entirety. You may not charge, either
directly or indirectly, for this information.

--
Geoff Allen          \  Please remain seated and keep your hands and arms
uunet!pmafire!geoff   \  above your head at all times.  Enjoy your ride.
[e-mail] \
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Archive-Name: sci-math-faq
Version: $Id: sci-math-faq,v 3.6 92/12/07 18:14:00 $

This is a list of frequently asked questions for sci.math (version 3.6).
Any contributions/suggestions/corrections are most welcome. Please use
* e-mail * on any comment concerning the FAQ list.

Changes of version will be important enough to deserve reading the FAQ
list again. Additions are marked with a # on the table of contents.
Still you may kill all versions of FAQ using the * wildcard. (Ask your
local unix guru for ways to do so). The FAQ is available via ftp in
rtfm.mit.edu (18.172.1.27).

The list of contributors to this FAQ list is to large to include here;
but thanks are due to all of them (you know who you are folks).

             Table of Contents
             -----------------

 1Q.- Fermat's Last Theorem, status of ..
 2Q.- Four Colour Theorem, proof of ..
 3Q.- Values of Record Numbers
 4Q.- General Netiquette
 5Q.- Computer Algebra Systems, application of ..
 6Q.- Computer Algebra Systems, references to ..
 7Q.- Fields Medal, general info ..
 8Q.- 0^0=1. A comprehensive approach ..
 9Q.- 0.999... = 1. Properties of the real numbers ..
10Q.- Digits of Pi, computation and references ..
11Q.- There are three doors, The Monty Hall problem, Master Mind and
      other games .. #
12Q.- Surface and Volume of the n-ball
13Q.- f(x)^f(x)=x, name of the function ..
14Q.- Projective plane of order 10 ..
15Q.- How to compute day of week of a given date ..
16Q.- Axiom of Choice and/or Continuum Hypothesis?
17Q.- Cutting a sphere into pieces of larger volume
18Q.- Pointers to Quaternions
19Q.- Erdos Number #


1Q:  What is the current status of Fermat's last theorem?
    (There are no positive integers x,y,z, and n > 2 such that
    x^n + y^n = z^n)
    I heard that <insert name here> claimed to have proved it but later
    on the proof was found to be wrong. ...
    (wlog we assume x,y,z to be relatively prime)

A:  The status of FLT has remained remarkably constant.  Every few
    years, someone claims to have a proof ... but oh, wait, not quite.
    Meanwhile, it is proved true for ever greater values of the exponent
    (but not all of them), and ties are shown between it and other
    conjectures (if only we could prove one of them), and ... so it has
    been for quite some time.  It has been proved that for each
    exponent, there are at most a finite number of counter-examples to
    FLT.

    Here is a brief survey of the status of FLT. It is not intended to
    be 'deep', but it is rather for non-specialists.

    The theorem is broken into 2 cases. The first case assumes
    (abc,n) = 1. The second case is the general case.

    What has been PROVED
    --------------------

    First Case.

    It has been proven true up to 7.568x10^17 by the work of Wagstaff &
    Tanner, Granville&Monagan, and Coppersmith. They all used extensions
    of the Wiefrich criteria and improved upon work performed by
    Gunderson and Shanks&Williams.

    The first case has been proven to be true for an infinite number of
    exponents by Adelman, Frey, et. al. using a generalization of the
    Sophie Germain criterion


    Second Case:

    It has been proven true up to n = 150,000 by Tanner & Wagstaff. The
    work used new techniques for computing Bernoulli numbers mod p and
    improved upon work of Vandiver. The work involved computing the
    irregular primes up to 150,000. FLT is true for all regular primes
    by a theorem of Kummer. In the case of irregular primes, some
    additional computations are needed.

    UPDATE :

    Fermat's Last Theorem has been proved true up to exponent 2,000,000
    in the general case. The method used was essentially that of Wagstaff:
    enumerating and eliminating irregular primes by Bernoulli number
    computations. The computations were performed on a set of NeXT
    computers by Richard Crandall.

    Since the genus of the curve a^n + b^n = 1, is greater than or equal
    to 2 for n > 3, it follows from Mordell's theorem [proved by
    Faltings], that for any given n, there are at most a finite number
    of solutions.


    Conjectures
    -----------

    There are many open conjectures that imply FLT. These conjectures
    come from different directions, but can be basically broken into
    several classes: (and there are interrelationships between the
    classes)

    (a) conjectures arising from Diophantine approximation theory such
    as the ABC conjecture, the Szpiro conjecture, the Hall conjecture,
    etc.

    For an excellent survey article on these subjects see the article
    by Serge Lang in the Bulletin of the AMS, July 1990 entitled
    "Old and new conjectured diophantine inequalities".

    Masser and Osterle formulated the following known as the ABC
    conjecture:

    Given epsilon > 0, there exists a number C(epsilon) such that for
    any set of non-zero, relatively prime integers a,b,c such that
    a+b = c we have

    max( |a|, |b|, |c|) <= C(epsilon) N(abc)^(1 + epsilon)

    where N(x) is the product of the distinct primes dividing x.

    It is easy to see that it implies FLT asymptotically. The conjecture
    was motivated by a theorem, due to Mason that essentially says the
    ABC conjecture IS true for polynomials.

    The ABC conjecture also implies Szpiro's conjecture [and vice-versa]
    and Hall's conjecture. These results are all generally believed to
    be true.

    There is a generalization of the ABC conjecture [by Vojta] which is
    too technical to discuss but involves heights of points on
    non-singular algebraic varieties . Vojta's conjecture also implies
    Mordell's theorem [already known to be true]. There are also a
    number of inter-twined conjectures involving heights on elliptic
    curves that are related to much of this stuff. For a more complete
    discussion, see Lang's article.

    (b) conjectures arising from the study of elliptic curves and
    modular forms. -- The Taniyama-Weil-Shmimura conjecture.

    There is a very important and well known conjecture known as the
    Taniyama-Weil-Shimura conjecture that concerns elliptic curves.
    This conjecture has been shown by the work of Frey, Serre, Ribet,
    et. al. to imply FLT uniformly, not just asymptotically as with the
    ABC conj.

    The conjecture basically states that all elliptic curves can be
    parameterized in terms of modular forms.

    There is new work on the arithmetic of elliptic curves. Sha, the
    Tate-Shafarevich group on elliptic curves of rank 0 or 1. By the way
    an interesting aspect of this work is that there is a close
    connection between Sha, and some of the classical work on FLT. For
    example, there is a classical proof that uses infinite descent to
    prove FLT for n = 4. It can be shown that there is an elliptic curve
    associated with FLT and that for n=4, Sha is trivial. It can also be
    shown that in the cases where Sha is non-trivial, that
    infinite-descent arguments do not work; that in some sense 'Sha
    blocks the descent'. Somewhat more technically, Sha is an
    obstruction to the local-global principle [e.g. the Hasse-Minkowski
    theorem].



    (c) Conjectures arising from some conjectured inequalities involving
    Chern classes and some other deep results/conjectures in arithmetic
    algebraic geometry.

    I can't describe these results since I don't know the math. Contact
    Barry Mazur [or Serre, or Faltings, or Ribet, or ...]. Actually the
    set of people who DO understand this stuff is fairly small.


    The diophantine and elliptic curve conjectures all involve deep
    properties of integers. Until these conjecture were tied to FLT,
    FLT had been regarded by most mathematicians as an isolated problem;
    a curiosity. Now it can be seen that it follows from some deep and
    fundamental properties of the integers. [not yet proven but
    generally believed].

    This synopsis is quite brief. A full survey would run to many pages.

    References:

    [1] J.P.Butler, R.E.Crandall, & R.W.Sompolski
    "Irregular Primes to One Million"
     Math. Comp. 59 (October 1992) pp. 717-722






2Q: Has the Four Colour Theorem been solved?
    (Every planar map with regions of simple borders can be coloured
    with 4 colours in such a way that no two regions sharing a non-zero
    length border have the same colour.)

A:  This theorem was proved with the aid of a computer in 1976.
    The proof shows that if aprox. 1,936  basic forms of maps
    can be coloured with four colours, then any given map can be
    coloured with four colours. A computer program coloured this
    basic forms. So far nobody has been able to prove it without
    using a computer. In principle it is possible to emulate the
    computer proof by hand computations.

    References:

    K. Appel and W. Haken, Every planar map is four colourable,
    Bulletin of the American Mathematical Society, vol. 82, 1976
    pp.711-712.

    K. Appel and W. Haken, Every planar map is four colourable,
    Illinois Journal of Mathematics, vol. 21, 1977, pp. 429-567.

    T. Saaty and Paul Kainen, The Four Colour Theorem: Assault and
    Conquest, McGraw-Hill, 1977. Reprinted by Dover Publications 1986.

    K. Appel and W. Haken, Every Planar Map is Four Colorable,
    Contemporary Mathematics, vol. 98, American Mathematical Society,
    1989, pp.741.

    F. Bernhart, Math Reviews. 91m:05007, Dec. 1991. (Review of Appel
    and Haken's book).




3Q:  What are the values of:

largest known Mersenne prime?

A:  It is 2^756839-1. It was discovered by a Cray-2 in England in 1992.
    It has 227,832 digits.


largest known prime?

A:  The largest known prime was 391581*2^216193 - 1.  See Brown, Noll,
    Parady, Smith, Smith, and Zarantonello, Letter to the editor,
    American Mathematical Monthly, vol. 97, 1990, p. 214.

    Now the largest known prime is the Mersenne prime described above.


largest known twin primes?

A:  The largest known twin primes are 1706595*2^11235 +- 1.
    See B. K. Parady and J. F. Smith and S. E. Zarantonello,
    Smith, Noll and Brown.
    Largest known twin primes, Mathematics of Computation,
    vol.55, 1990, pp. 381-382.


largest Fermat number with known factorization?

A:  F_11 = (2^(2^11)) + 1 which was  factored by Brent & Morain in
    1988. F9 = (2^(2^9)) + 1 = 2^512 + 1 was factored by
    A.K. Lenstra, H.W. Lenstra Jr., M.S. Manasse & J.M. Pollard
    in 1990. The factorization for F10 is NOT known.


Are there good algorithms to factor a given integer?

A:  There are several that have subexponential estimated
    running time, to mention just a few:

        Continued fraction algorithm,
        Class group method,
        Quadratic sieve algorithm,
        Elliptic curve algorithm,
        Number field sieve,
        Dixon's random squares algorithm,
        Valle's two-thirds algorithm,
        Seysen's class group algorithm,

    A.K. Lenstra, H.W. Lenstra Jr., "Algorithms in Number Theory",
    in: J. van Leeuwen (ed.), Handbook of Theoretical Computer
    Science, Volume A: Algorithms and Complexity, Elsevier, pp.
    673-715, 1990.


List of record numbers?

A:  Chris Caldwell maintains "THE LARGEST KNOWN PRIMES (ALL KNOWN
    PRIMES WITH 2000 OR MORE DIGITS)"-list. Send him mail to
    [e-mail] (preferred) or [e-mail], on any new
    gigantic primes (greater than 10,000 digits), titanic primes
    (greater than 1000 digits).


What is the current status on Mersenne primes?

A:  Mersenne primes are primes of the form 2^p-1. For 2^p-1 to be prime
    we must have that p is prime. The following Mersenne primes are
    known.

    nr            p                                 year  by
    -----------------------------------------------------------------
     1-5   2,3,5,7,13                    in or before the middle ages
     6-7       17,19                     1588  Cataldi
     8          31                       1750  Euler
     9          61                       1883  Pervouchine
    10          89                       1911  Powers
    11          107                      1914  Powers
    12          127                      1876  Lucas
    13-14       521,607                  1952  Robinson
    15-17       1279,2203,2281           1952  Lehmer
    18          3217                     1957  Riesel
    19-20       4253,4423                1961  Hurwitz & Selfridge
    21-23       9689,9941,11213          1963  Gillies
    24          19937                    1971  Tuckerman
    25          21701                    1978  Noll & Nickel
    26          23209                    1979  Noll
    27          44497                    1979  Slowinski & Nelson
    28          86243                    1982  Slowinski
    29          110503                   1988  Colquitt & Welsh jr.
    30          132049                   1983  Slowinski
    31          216091                   1985  Slowinski
    32?         756839                   1992  Slowinski & Gage

    The way to determine if 2^p-1 is prime is to use the Lucas-Lehmer
    test:
      Lucas_Lehmer_Test(p):
         u := 4
         for i from 3 to p do
            u := u^2-2 mod 2^p-1
         od
         if u == 0 then
            2^p-1 is prime
         else
            2^p-1 is composite
         fi

   The following ranges have been checked completely:
    2 - 355K and  430K - 520K

   More on Mersenne primes and the Lucas-Lehmer test can be found in:
      G.H. Hardy, E.M. Wright, An introduction to the theory of numbers,
      fifth edition, 1979, pp. 16, 223-225.


(Please send updates to [e-mail])




4Q:  I think I proved <insert big conjecture>.    OR
    I think I have a bright new idea.

    What should I do?

A:  Are you an expert in the area? If not, please ask first local
    gurus for pointers to related work (the "distribution" field
    may serve well for this purposes). If after reading them you still
    think your *proof is correct*/*idea is new* then send it to the net.


5Q:  I have this complicated symbolic problem (most likely
    a symbolic integral or a DE system) that I can't solve.
    What should I do?

A:  Find a friend with access to a computer algebra system
    like MAPLE, MACSYMA or MATHEMATICA and ask her/him to solve it.
    If packages cannot solve it, then (and only then) ask the net.


6Q:  Where can I get <Symbolic Computation Package>?
    This is not a comprehensive list. There are other Computer Algebra
    packages available that may better suit your needs. There is also
    a FAQ list in the group sci.math.symbolics which may have the
    info your looking for.

A: Maple
        Purpose: Symbolic and numeric computation, mathematical
        programming, and mathematical visualization.
        Contact: Waterloo Maple Software,
        160 Columbia Street West,
        Waterloo, Ontario, Canada     N2L 3L3
        Phone: [phone]
        [e-mail] [e-mail]

A: DOE-Macsyma
        Purpose: Symbolic and mathematical manipulations.
        Contact: National Energy Software Center
        Argonne National Laboratory 9700 South Cass Avenue
        Argonne, Illinois 60439
        Phone: [phone]

A: Pari
        Purpose: Number-theoretic computations and simple numerical
        analysis.
        Available for Sun 3, Sun 4, generic 32-bit Unix, and
        Macintosh II. This is a free package, available by ftp from
        math.ucla.edu (128.97.64.16).
        Contact: questions about pari can be sent to [e-mail]

A: Mathematica
        Purpose: Mathematical computation and visualization,
        symbolic programming.
        Contact: Wolfram Research, Inc.
        100 Trade Center Drive Champaign,
        IL 61820-7237
        Phone: [phone]-MATH

A: Macsyma
        Purpose: Symbolic and mathematical manipulations.
    Contact: Symbolics, Inc.
    8 New England Executive Park East
    Burlington, Massachusetts 01803
    United States of America
    [phone]
    [e-mail]

A: Matlab
        Purpose: `matrix laboratory' for tasks involving
    matrices, graphics and general numerical computation.
    Contact: The MathWorks, Inc.
         21 Eliot Street
         South Natick, MA 01760
         508-653-1415
         [e-mail]

A: Cayley
        Purpose: Computation in algebraic and combinatorial structures
        such as groups, rings, fields, modules and graphs.
        Available for: SUN 3, SUN 4, IBM running AIX or VM, DEC VMS, others
        Contact: Computational Algebra Group
        University of Sydney
        NSW 2006
        Australia
        Phone:  (61) [phone]
        Fax: (61) [phone]
        [e-mail]



7Q:  Let P be a property about the Fields Medal. Is P(x) true?

A:  There are a few gaps in the list. If you know any of the
    missing information (or if you notice any mistakes),
    please send me e-mail.

Year Name               Birthplace              Age Institution
---- ----               ----------              --- -----------
1936 Ahlfors, Lars      Helsinki       Finland   29 Harvard U         USA
1936 Douglas, Jesse     New York NY    USA       39 MIT               USA
1950 Schwartz, Laurent  Paris          France    35 U of Nancy        France
1950 Selberg, Atle      Langesund      Norway    33 Adv.Std.Princeton USA
1954 Kodaira, Kunihiko  Tokyo          Japan     39 Princeton U       USA
1954 Serre, Jean-Pierre Bages          France    27 College de France France
1958 Roth, Klaus        Breslau        Germany   32 U of London       UK
1958 Thom, Rene         Montbeliard    France    35 U of Strasbourg   France
1962 Hormander, Lars    Mjallby        Sweden    31 U of Stockholm    Sweden
1962 Milnor, John       Orange NJ      USA       31 Princeton U       USA
1966 Atiyah, Michael    London         UK        37 Oxford U          UK
1966 Cohen, Paul        Long Branch NJ USA       32 Stanford U        USA
1966 Grothendieck, Alexander Berlin    Germany   38 U of Paris        France
1966 Smale, Stephen     Flint MI       USA       36 UC Berkeley       USA
1970 Baker, Alan        London         UK        31 Cambridge U       UK
1970 Hironaka, Heisuke  Yamaguchi-ken  Japan     39 Harvard U         USA
1970 Novikov, Serge     Gorki          USSR      32 Moscow U          USSR
1970 Thompson, John     Ottawa KA      USA       37 U of Chicago      USA
1974 Bombieri, Enrico   Milan          Italy     33 U of Pisa         Italy
1974 Mumford, David     Worth, Sussex  UK        37 Harvard U         USA
1978 Deligne, Pierre    Brussels       Belgium   33 IHES              France
1978 Fefferman, Charles Washington DC  USA       29 Princeton U       USA
1978 Margulis, Gregori  Moscow         USSR      32 InstPrblmInfTrans USSR
1978 Quillen, Daniel    Orange NJ      USA       38 MIT               USA
1982 Connes, Alain      Draguignan     France    35 IHES              France
1982 Thurston, William  Washington DC  USA       35 Princeton U       USA
1982 Yau, Shing-Tung    Kwuntung       China     33 IAS               USA
1986 Donaldson, Simon   Cambridge      UK        27 Oxford U          UK
1986 Faltings, Gerd     1954           Germany   32 Princeton U       USA
1986 Freedman, Michael  Los Angeles CA USA       35 UC San Diego      USA
1990 Drinfeld, Vladimir Kharkov        USSR      36 Phys.Inst.Kharkov USSR
1990 Jones, Vaughan     Auckland       N Zealand 38 UC Berkeley       USA
1990 Mori, Shigefumi    Nagoya         Japan     39 U of Kyoto?       Japan
1990 Witten, Edward     ?              USA       38 Princeton U/IAS   USA

References :

International Mathematical Congresses, An Illustrated History 1893-1986,
Revised Edition, Including 1986, by Donald J.Alberts, G. L. Alexanderson
and Constance Reid, Springer Verlag, 1987.

Tropp, Henry S., ``The origins and history of the Fields Medal,''
Historia Mathematica, 3(1976), 167-181.


8Q:  What is 0^0 ?

A:  According to some Calculus textbooks, 0^0 is an "indeterminate
    form". When evaluating a limit of the form 0^0, then you need
    to know that limits of that form are called "indeterminate forms",
    and that you need to use a special technique such as L'Hopital's
    rule to evaluate them. Otherwise, 0^0=1 seems to be the most
    useful choice for 0^0. This convention allows us to extend
    definitions in different areas of mathematics that otherwise would
    require treating 0 as a special case. Notice that 0^0 is a
    discontinuity of the function x^y.

    Rotando & Korn show that if f and g are real functions that vanish
    at the origin and are _analytic_ at 0 (infinitely differentiable is
    not sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from
    the right.

    From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):

    "Some textbooks leave the quantity 0^0 undefined, because the
    functions x^0 and 0^x have different limiting values when x
    decreases to 0. But this is a mistake. We must define

       x^0 = 1 for all x,

    if the binomial theorem is to be valid when x=0, y=0, and/or x=-y.
    The theorem is too important to be arbitrarily restricted! By
    contrast, the function 0^x is quite unimportant."

    Published by Addison-Wesley, 2nd printing Dec, 1988.

    Another reference is:

    H. E. Vaughan, The expression '0^0', Mathematics
    Teacher 63 (1970), pp.111-112.


    Louis M. Rotando & Henry Korn, "The Indeterminate Form 0^0",
    Mathematics Magazine, Vol. 50, No. 1 (January 1977),
    pp. 41-42.

    L. J. Paige, A note on indeterminate forms, American
    Mathematical Monthly, 61 (1954), 189-190; reprinted
    in the Mathematical Association of America's 1969
    volume, Selected Papers on Calculus, pp. 210-211.



9Q:  Why is 0.9999... = 1?

A:  In modern mathematics, the string of symbols "0.9999..." is
    understood to be a shorthand for "the infinite sum  9/10 + 9/100
    + 9/1000 + ...." This in turn is shorthand for "the limit of the
    sequence of real numbers 9/10, 9/10 + 9/100, 9/10 + 9/100 + 9/1000,
    ..."  Using the well-known epsilon-delta definition of limit, one
    can easily show that this limit is 1.  The statement that
    0.9999...  = 1 is simply an abbreviation of this fact.

                    oo              m
                   ---   9         ---   9
        0.999... = >   ---- = lim  >   ----
                   --- 10^n  m->oo --- 10^n
                   n=1             n=1
        Choose epsilon > 0. Suppose delta = 1/-log_10 epsilon, thus
        epsilon = 10^(-1/delta). For every m>1/delta we have that

        |  m           |
        | ---   9      |     1          1
        | >   ---- - 1 | = ---- < ------------ = epsilon
        | --- 10^n     |   10^m   10^(1/delta)
        | n=1          |

        So by the (epsilon-delta) definition of the limit we have
               m
              ---   9
         lim  >   ---- = 1
        m->oo --- 10^n
              n=1


    An *informal* argument could be given by noticing that the following
    sequence of "natural" operations has as a consequence 1 = 0.9999....
    Therefore it's "natural" to assume 1 = 0.9999.....

             x = 0.99999....
           10x = 9.99999....
       10x - x = 9
            9x = 9
             x = 1
    Thus
             1 = 0.99999....

    References:

    E. Hewitt & K. Stromberg, Real and Abstract Analysis,
    Springer-Verlag, Berlin, 1965.

    W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, 1976.



10Q:  Where I can get pi up to a few hundred thousand digits of pi?
    Does anyone have an algorithm to compute pi to those zillion
    decimal places?


A:  MAPLE or MATHEMATICA can give you 10,000 digits of Pi in a blink,
    and they can compute another 20,000-500,000 overnight (range depends
    on hardware platform).

    It is possible to retrieve 1.25+ million digits of pi via anonymous
    ftp from the site wuarchive.wustl.edu, in the files pi.doc.Z and
    pi.dat.Z which reside in subdirectory doc/misc/pi.

    References :
    (This is a short version for a more comprhensive list contact
    Juhana Kouhia at [e-mail])

    J. M. Borwein, P. B. Borwein, and D. H. Bailey, "Ramanujan,
    Modular Equations, and Approximations to Pi", American Mathematical
    Monthly, vol. 96, no. 3 (March 1989), p. 201 - 220.