However, I thought I'd post a preliminary version of the first three sections now. I'm hoping that some of you can double-check my computations and/or give me suggestions for topics that I could discuss in these sections. I would especially like to know of any interesting examples in which large numbers arise that I haven't already dealt with.
My plan is to carefully work up to the Howard-Bachmann ordinal level of the Grzegorczyk-Wainer Hierarchy, and then touch on some of the constructible ordinal notations that go beyond this. At present I have 10 sections (in various stages of completion), but there may be more than this by the time I'm done.
Dave L. Renfro
1. WARMING UP -- SOME ORDINARY LARGE NUMBERS
A. ANNOTATED LIST OF NUMBERS UP TO 10^^6 B. OTHER ANNOTATED LISTS OF LARGE NUMBERS C. REFERENCES FOR SECTION 1
10^12 -- One trillion. This many seconds is about 31,700 years. The thickness of five trillion sheets of typing paper equals the distance to the moon. If you magnify an atom by a factor of one trillion, its nucleus would be 1 mm in diameter. There are about one third of a trillion stars in our galaxy. In one trillionth of a second light travels 0.3 mm and sound (in air) travels about three times the diameter of an atom. Doubling something 40 times will make it about a trillion times larger. A 40-dimensional cube has about one trillion vertices. The sum of the first one trillion terms of the divergent series 1/1 + 1/2 + 1/3 + ... is 28.2082; the next trillion terms add to .693; the next TWO trillion terms add to 1.10; the next THREE trillion terms add to 1.39; the next TRILLION trillion terms add to 27.6.
4 x 10^14 -- This many bacteria weigh about a pound.
10^16 -- This many words have been spoken in all of human history.
1.6 x 10^18 -- This is the number of inches to Alpha Centauri, the nearest star.
6 x 10^23 -- Avogadro's number.
The aggregate impact of this many fleas falling 1 mm at the Earth's surface equals the energy released in a 600 kiloton atomic bomb (40 times the WW II Hiroshima bomb output). [Flea mass is m = 450 micrograms, kinetic energy from falling a distance h is mgh, where g is 9.8 m / sec^2, and 1 kiloton is approximately 4 x 10^12 Joules.] http://musr.physics.ubc.ca/~jess/p200/emc2/node4.html
10^(-200) -- This is the probability that an electron in a 1s orbital of a hydrogen atom is 5 nanometers from the nucleus.
10^(-1080) -- This is the probability of flipping a coin once a second for an hour and getting all heads.
10^(2576) -- In 1928 A. S. Besicovitch introduced and studied regular and irregular 1-sets in the plane (a fundamental notion in fractal geometry). He proved that the lower 1-density of a regular 1-set E in the plane is equal to 1 (the maximum value) at almost all points in E. To show how differently irregular 1-sets in the plane behave, Besicovitch proved that the lower 1-density of an irregular 1-set F in the plane is bounded below 1 at almost all points in F. The bound that Besicovitch obtained in 1928 was 1 - 10^(-2576). In 1934, Besicovitch managed to improve this to 3/4. [See Section 3.3 of Kenneth J. Falconer, THE GEOMETRY OF FRACTAL SETS, Cambridge University Press, 1985.] "Besicovitch's 1/2-problem" (presently unsolved) is to find the best almost everywhere upper bound for irregular sets. Besicovitch himself showed by a specific example that this bound is at least 1/2, and he conjectured that it is equal to 1/2. In 1992, David Preiss and Jaroslav Tiser proved the bound is at most [2 + sqrt(46)] / 12, which is approximately .73186. For a lot more about the Besicovitch 1/2-problem, see Hany M. Farag's papers at http://www.math.caltech.edu/people/farag.html
10^(-42,000) -- This is the probability of a monkey typing Hamlet by chance. [27,000 letters and spaces using 35 keys implies a one out of 35^(27,000) chance.]
10^(2,098,959) -- The 38'th Mersenne prime number is 2^(6,972,593)-1, which is approximately 4.37076 x 10^(2,098,959). All 2,098,960 digits of this number (a 2462 K file with 492 page print output) can be found at http://www.math.utah.edu/~alfeld/math/largeprime.html
10^(-10^8) -- This is the probability of flipping a coin once a second for 100 years and getting all heads.
Also, 10^(10^8) is the number of terms of the divergent series SUM(k=2 to infinity) [k*ln(k)]^(-1) that are needed for a partial sum to exceed 20. (See p. 244 of Boas .)
10^(3.7 x 10^8) -- The largest number that can be expressed using three digits and three selections of the operations +, *, ^ is 9^(9^9), which is approximately 10^(3.6969 x 10^8).
10^(10^9) -- This is Rudy Rucker's "gigaplex" (, p. 81), which he obtains as an upper bound on the number of possible thoughts a person can have. The is the number of ways to assign "on" or "off" to the 3 billion synapses in our brains is 2^(3 billion), which is approximately one gigaplex.
10^(-2.9 x 10^12) -- This is the Yukawa nuclear force between two nucleons that are 1 cm apart. [Evaluate the derivative of (K/r)*exp(-ar) for r = .01, k = 4.75 x 10^(-25), and a = 6.67 x 10^14.]
10^(-10^13) -- This is the probability that by randomly typing ASCII characters a monkey will type every post, in a pre-selected order, that appears at the Google Groups archive.
They claim over 700 million posts (so let's use 10^9). Taking the average post to be 50 lines at 70 characters per line (spaces count, so empty lines count as well), this is an average of 3500 characters per post (so let's use 5000). This totals to 5 x 10^12 characters in the Google Groups archive. Since there are 95 ASCII characters (so let's use 100), the number of (5 x 10^12)-length sequences using 100 characters is 100^(5 x 10^12).
10^(-5.2 x 10^18) -- Using non-relativistic quantum mechanics, this is the probability that an electron in a 1s orbital of a hydrogen atom is 240,000 miles from the nucleus (distance to the Moon).
10^(10^21) -- This is Rudy Rucker's "sextillionplex" (, p. 83), which he obtains as an upper bound on the number of possible lives a person can have. This is a count of the number of sequences of thoughts during a lifetime, which Rucker estimates as [10^(10^9)] ^ (10^12). [Assume 10^(10^9) possible thoughts to choose from with a choice of a thought occurring 500 times a second. Then a sequence of 10^12 of these .002 sec thought intervals is about 63 years.]
10^(-3 x 10^26) -- Gamow observes (see Chapter 8.4 of ) there is a nonzero probability that all of the air molecules in a room will collect together simultaneously on one side of the room, namely (1/2)^n where n = 10^27 is the number of air molecules in the room.
10^(-10^36) -- The probability given by life insurance calculations that someone will live 1000 years. [This appears at the beginning of Chapter 1 in Volume I of William Feller's text PROBABILITY THEORY AND ITS APPLICATIONS. Feller writes: "We hesitate to admit that man can grow 1000 years old, and yet current actuarial practice admits no such limit to life. According to formulas on which modern mortality tables are based, the proportion of men surviving 1000 years is of the order of magnitude of one in 10^(10^36) -- a number with 10^27 billions of digits."]
Using non-relativistic quantum mechanics, this is also the probability that an electron in a 1s orbital of a hydrogen atom is 10 billion light years away from the nucleus.
10^(-2.7 x 10^40) -- This is the Yukawa nuclear force between two nucleons that are 10 billion light years apart.
10^(2.5 x 10^70) -- This is an estimate by J. E. Littlewood (see pp. 106-107 of ) for an upper bound on the number of possible chess games. G. H. Hardy obtained a smaller estimate, 10^(10^50). See p. 17 of Hardy, RAMANUJAN: TWELVE LECTURES ON SUBJECTS SUGGESTED BY HIS LIFE AND WORK", 3'rd edition, Chelsea, 1999. http://mathworld.wolfram.com/Chess.html
10^(3.14 x 10^86) -- This is the number of terms of the convergent series SUM(k=2 to infinity) 1 / [k*ln(k)*(ln(ln k)^2] that are needed to obtain a sum accurate to within .005. (See p. 242 of Boas .)
10^(-10^93) -- Using non-relativistic quantum mechanics, the probability that every electron in all the hydrogen atoms in the sun are 10 billion light years away is more than this. [This number is [10^(-10^36)]^n, where n = 10^57 is the mass of the sun divided by the mass of a hydrogen atom. ("More than", because the sun is 91% hydrogen and I would imagine that very few of these electrons are in the lowest energy 1s orbital.)]
10^(10^100) -- One googolplex.
googolplex + 1 -- This number is NOT prime. One of its factors is 316,912,650,057,057,350,374,175,801,344,000,001. See Crandall .
10^(9.95657 x 10^101) -- This is the factorial of a googol.
10^(1.2 x 10^109) -- This is Asimov's T-(T-9). Asimov's T-n is a trillion trillion ... trillion (n-times), or 10^(12*n). From the last page of Asimov : "As for T-(T-9), that is far larger than a googolplex; in fact, it is far larger than a googol googolplexes."
10^(10^125) -- This is an upper bound on the number of known universes at any specific time. This number equals (10^80)^(10^123), the number of ways the 10^80 particles in our universe can be placed into the 10^123 particle-sized locations in our universe, with repetitions allowed (i.e. more than one particle can be put into a single location). This also assumes that each particle is distinguishable from every other particle.
10^(10^166) -- This is an upper bound on the number of 10 billion year long "universes". This is [10^(10^125)]^(10^41), the number of sequences of length 10^41 (the number of 10^(-24) second intervals in 10 billion years) each of whose terms can be any of the 10^(10^125) arrangements of particles in the known universe. This is an upper bound on the number of branches for the known universe, for 10 billion years, in the "many worlds interpretation of quantum mechanics".
This is also equivalent to the number of (10^41)-move "universal chess games" in which there are 10^80 *distinct* chess pieces and any number of these chess pieces can relocate to any of 10^123 locations during each move (and more than one chess piece can occupy each location).
10^(-10^166) is the probability of "randomly typing" the entire history of the universe by selecting the correct particle distribution (out of the 10^(10^125) total ways that particles can be distributed in the universe at any given time) during each of the 10^41 instants of time since the Big Bang ("instant" equals the time it takes light to travel the diameter of an atomic nucleus).
10^(10^7065) -- The largest Fermat number for which a factor is known as of 1984 is F[23,471] = 2^(2^23471) + 1, which is approximately 10^(8.98748 x 10^7064).
10^(10^(10^41)) -- This is the number of terms of the divergent series SUM(k=2 to infinity) 1 / [k*ln(k)*ln(ln k)] that are needed for a partial sum to exceed 100. (See p. 244 of Boas .)
10^(10^(10^100)) -- This is the factorial of a googolplex. A much better approximation is 10^(10^(100 + 10^100)). An even better approximation is 10^(10^(N + 10^100)), where N is log[ 10^100 - log(e) ] and "log" is base-10 logarithm.
10^(10^(10^(1000)) -- Assuming the Riemann hypothesis is FALSE, S. Skewes (1955) obtained this number as an upper bound for the first sign change of the difference between Integral(t=2 to t=x) of (ln t)^(-1) and the number of primes less than x. http://www.emis.de/cgi-bin/Zarchive?an=0068.26802
10^(10^(10^(4.3 x 10^5))) -- This is the number of terms of the divergent series SUM(k=2 to infinity) 1 / [k*ln(k)*ln(ln k)] that are needed for a partial sum to exceed 10^6. (See p. 244 of Boas .)
10^(10^(10^(10^(10^15)))) -- The number e^(e^(e^(e^(e^35)))), which is approximately 10^(10^(10^(10^(6.888 x 10^14)))), arises in the paper S. Knapowski, "On sign changes of the difference Pi(x) - Li(x)", Acta Arithmetica 7 (1962), 107-119. http://www.emis.de/cgi-bin/Zarchive?an=0126.07502
 George Gamow, ONE, TWO, THREE, ... INFINITY, Viking, 1947. [See Chapters 1 ("Big Numbers") and 8.4 ("The Law of Disorder", Section 4: "The 'Mysterious' Entropy"). But note the incorrect c = aleph_0 identification that slipped in during his discussion of cardinal numbers at the end of Chapter 1: .]
 Quinn Tyler Jackson, "Patterns of Randomness". [An essay on pseudorandom number generators and the super astronomically large" periods. For example, the Mersenne Twister has a period of 2^19937 = 4.3 x 10^6001.]
 John E. Littlewood, "Large Numbers", Mathematical Gazette 32 #300 (July 1948), 163-171. [Reprinted on pp. 100-113 of Bela Bollobas, LITTLEWOOD'S MISCELLANY, Cambridge Univ. Press, 1986. The largest number in Archimedes' "The Sand Reckoner" is given incorrectly as 10^(8 x 10^15) on p. 163 of the article and on p. 100 of the book reprint.] http://uk.cambridge.org/mathematics/catalogue/052133702X/default.htm
 Ilan Vardi, "Archimede face a l'innombrable", preprint, July 2000. [ABSTRACT: "Archimedes was the first person to invent a system for denoting very large numbers. He required these in his paper "The Sand Reckoner" in which he gave an upper bound on the number of sand grains that could fill the universe. My paper describes this system and also proposes that linguistic constraints of Ancient Greek were responsible for Archimedes stopping at the number 10^(8 x 10^16). An alternate system is given which would have allowed Archimedes to express much larger numbers."] http://www.ihes.fr/PREPRINTS/M00/Resu/resu-M00-76.html http://www.lix.polytechnique.fr/~ilan/publications.html