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Topic: BIG NUMBERS #1
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Dave L. Renfro

Posts: 4,598
Registered: 12/3/04
BIG NUMBERS #1
Posted: Apr 8, 2002 6:47 PM
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I'm still working on an extensive revision of my sci.math
post "GRAHAM'S NUMBER AND RAPIDLY GROWING FUNCTIONS" at

http://groups.google.com/groups?selm=28ae5e5e.0203041003.2b10abad%40posting.google.com

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


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A. ANNOTATED LIST OF NUMBERS UP TO 10^^6

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

4.6 x 10^42 -- The number of possible chess positions according to
Claude E. Shannon, "Programming a computer for
playing chess", Philosophical Magazine (7) 41 (1950)
256-275. [Reprinted in D.N.L. Levy (editor), COMPUTER
CHESS COMPENDIUM, Springer-Verlag, 1988 and in CLAUDE
SHANNON: COLLECTED PAPERS (see 2'nd URL below).]
Littlewood [71] (p. 107) obtained 5 x 10^69, but
Littlewood's calculation includes illegal positions
(which he acknowledges).
http://mathworld.wolfram.com/Chess.html
http://www.research.att.com/~njas/doc/shannon.html
http://www.research.att.com/~njas/doc/shannonbib.html

5.53 x 10^(-67) -- The gravitational force (in Newtons) between
two electrons that are 1 cm apart.

8 x 10^67 -- The number of ways to shuffle a deck of cards.

10^80 -- The number of elementary particles in the known universe.

10^100 -- One googol. 69.9575744573535461362154966! is approximately
10^100. [It's about 1.18 x 10^75 less than 10^100.]

6.18 x 10^(-103) -- The gravitational force (in Newtons) between
two electrons that are 1 light year apart.

5.04 x 10^132 \_ _ The sum of the digits of 666^47 is 666.
9.94 x 10^143 / The sum of the digits of 666^51 is 666.
http://users.aol.com/s6sj7gt/mike666.htm

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

10^6001 -- The Mersenne Twister random number generator has a
period of 2^19937 = 4.3 x 10^6001. See Jackson [58] and
http://www.math.keio.ac.jp/~matumoto/emt.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^(4,053,946) -- The largest known prime number (found by Michael
Cameron on November 14, 2001), the 38'th Mersenne
prime number, is 2^(13,466,917) - 1. This number
is approximately 9.249477 x 10^(4,053,945).
http://www.mersenne.org/
http://www.utm.edu/research/primes/largest.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 [15].)

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" ([89], 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.

http://www.google.com/googlegroups/archive_announce_20.html

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^(8 x 10^16) -- In "The Sand Reckoner" Archimedes obtained a way
to name every number up to his "myriad-myriad
units of the myriad-myriad'th order of the
myriad-myriad'th period". See Rucker [88] (p. 98),
Vardi [103], and
http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml
http://web.fccj.org/~ethall/archmede/sandreck.htm

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" ([89], 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 [43]) 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 [71]) 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 [15].)

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 [25].

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 [7]:
"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^(10^115,127) -- The largest Fermat number for which a factor is
known as of 2001 is F[382,447] = 2^(2^382,447)+1,
which is approximately 10^(3.14312 x 10^115,127).
http://www.prothsearch.net/fermat.html
http://www.spd.dcu.ie/johnbcos/fermat.htm

10^(10^(2,000,000)) -- 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 20. (See p. 244 of Boas [15].)

10^(10^(10^34)) -- Skewes number. Assuming the Riemann hypothesis
is true, S. Skewes (1933) obtained an upper bound
of e^(e^(e^79))) = 10^(10^(8.852 x 10^33) 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://mathworld.wolfram.com/SkewesNumber.html
http://www.math.niu.edu/~rusin/known-math/97/skewes
http://www.math.niu.edu/~rusin/known-math/99/primedistr
http://www.emis.de/cgi-bin/Zarchive?an=0007.34003

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 [15].)

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 [15].)

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


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B. OTHER ANNOTATED LISTS OF LARGE NUMBERS


For other annotated lists of large numbers, see Determan [30],
Munafo [76], and Schneider [91].

The following web pages may also be of interest, although the
numbers involved are not nearly as large as what we've been
dealing with.

"Powers of Ten Tour of the Universe: A 26-step jaunt through
space and time" by Lisa Serio
http://cosmos2.phy.tufts.edu/~lserio/Astronomy_9/ast9lf.html

"Orders of Magnitude: Distance" by Erik Max Francis
http://www.alcyone.com/max/physics/orders/metre.html

"Data Powers of Ten" by Roy Williams Clickery
http://www.cacr.caltech.edu/~roy/dataquan/

COSMIC VIEW: The Universe in 40 Jumps by Kees Boeke
http://www.vendian.org/mncharity/cosmicview/
http://www.physics.rutgers.edu/~friedan/Boeke_Cosmic_View.html


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C. REFERENCES FOR SECTION 1


[7] Isaac Asimov, "T-Formation", Magazine of Fantasy and Science
Fact, August 1963. [Reprinted in ADDING A DIMENSION (1964)
and on pp. 45-56 of ASIMOV ON NUMBERS (1977). Both Skewes'
number and googolplex are one exponentiation by ten less than
they should be on the last page of the 1977 reprint.]
For some excerpts from Asimov's article, see
http://www.tfs.net/~bud/numbers2.html
http://www.angelfire.com/scifi/dreamweaver/quotes/qtwriters1.html

[15] Ralph P. Boas, "Partial sums of infinite series, and how they
grow", Amer. Math. Monthly 84 (1977), 237-258.

[25] Richard E. Crandall, "The challenge of large numbers",
Scientific American 276 (Feb. 1997), 74-79. This article can
be found at each of the following URL's:
http://www.fortunecity.com/emachines/e11/86/largeno.html
http://www.cryptosoft.com/snews/feb97/15029700.htm

[30] Scott Determan, "The Really Big Numbers Page".
http://varatek.com/scott/bnum.html

[43] 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: .]

[58] 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.]

http://members.shaw.ca/qjackson/writing_editing/articles/PatternsOfRandomness.html

[71] 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

[76] Robert P. Munafo, "Notable Properties of Specific Numbers"
http://home.earthlink.net/~mrob/pub/math/numbers-6.html
http://home.earthlink.net/~mrob/pub/math/numbers-7.html

[88] Rudy Rucker, INFINITY AND THE MIND, Princeton University
Press, 1995.
http://www.mathcs.sjsu.edu/faculty/rucker/ [Rucker's homepage]
http://pup.princeton.edu/TOCs/c5656.html
http://www.anselm.edu/homepage/dbanach/infin.htm

[89] Rudy Rucker, MIND TOOLS, Houghton Mifflin, 1987.

[91] Walter Schneider, "All Numbers Large and Beautifull".
[See "The List of Very Large Numbers" at the bottom.]
http://www.wschnei.de/number-theory/large-numbers.html

[103] 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


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