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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) buck@cavlry.enet.dec.com geoff@pmafire.inel.gov swain@aludra.usc.edu Tom_-_Obszanski@cup.portal.com betsyp@apollo.hp.com Editorial Assistance: Nora G. geoff@pmafire.inel.gov Tom_-_Obszanski@cup.portal.com Disclaimer: I make no warranty on the information contained here-in. Comments, corrections and questions are welcome via e-mail to geoff@pmafire.inel.gov. 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. geoff@pmafire.inel.gov \ Xref: bloom-picayune.mit.edu sci.math:37765 news.answers:4736 Path: bloom-picayune.mit.edu!enterpoop.mit.edu!hri.com!spool.mu.edu!olivea!sun-barr!cs.utexas.edu!torn!watserv2.uwaterloo.ca!watdragon.uwaterloo.ca!maytag.uwaterloo.ca!alopez-o From: alopez-o@maytag.uwaterloo.ca (Alex Lopez-Ortiz) Newsgroups: sci.math,news.answers Subject: sci.math: Frequently Asked Questions Summary: (version 3.4) Message-ID: <BzM4Gn.LCw@watdragon.uwaterloo.ca> Date: 21 Dec 92 14:05:10 GMT Sender: alopez-o@maytag.uwaterloo.ca Reply-To: alopez-o@maytag.uwaterloo.ca Followup-To: sci.math Organization: University of Waterloo Lines: 1147 Approved: news-answers-request@MIT.Edu Originator: alopez-o@maytag.uwaterloo.ca 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 bf04@UTMartn.bitnet (preferred) or kvax@utkvx.UTK.edu, 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 alopez-o@maytag.UWaterloo.ca) 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: (519) 747-2373 wmsi@daisy.uwaterloo.ca wmsi@daisy.waterloo.edu A: DOE-Macsyma Purpose: Symbolic and mathematical manipulations. Contact: National Energy Software Center Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60439 Phone: (708) 972-7250 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 pari@mizar.greco-prog.fr A: Mathematica Purpose: Mathematical computation and visualization, symbolic programming. Contact: Wolfram Research, Inc. 100 Trade Center Drive Champaign, IL 61820-7237 Phone: 1-800-441-MATH A: Macsyma Purpose: Symbolic and mathematical manipulations. Contact: Symbolics, Inc. 8 New England Executive Park East Burlington, Massachusetts 01803 United States of America (617) 221-1250 macsyma@Symbolics.COM 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 info@mathworks.com 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) (02) 692 3338 Fax: (61) (02) 692 4534 cayley@maths.su.oz.au 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 jk87377@cc.tut.fi) 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.