| Notable Properties of Specific Numbers |
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Physicist Paul Dirac's estimate70 of the ratio of the universe's size to that of a proton. Using present-day values (see 1.73×10-15, 299792458 and 13.73×109) and a naive assumption that the universe grows at the speed of light, the number would be about 7×1039. Dirac noted that it was "close" to the ratio between the strength of gravity and electical attraction between a proton and an electron (about 4.4×1040). He hypothesized that it is more than just coincidence, and proposed that the strength of gravity diminishes with time such that the two numbers remain the same. That would mean that when gravity and electricity were of equal strength the universe was about the size of a proton. The choice of the proton's radius is a bit arbitrary; compare to 8.02×1060.
Ratio between the strength of the gravitational and electric attraction of a proton and an electron. This number was considered significant by Dirac, see 1040. Dirac put forth this hypothesis at a time when the proton was still considered fundamental (long before the quark model); note that for other similar pairs of particles (e.g. a proton and a muon, or a positron and an electron) you get different ratios.
1.15868...×1042 = 64! / (32!×8!2×2!4×24)
This is (a corrected value for) the number of possible chess positions, originally given by Shannon in the 1950 article "Programming a Computer for Playing Chess." (Phil. Mag. 41, 256-275). The formula is based on the idea that you can theoretically arrange all 32 pieces in any position whatsoever (giving 64!/32!) but that all pawns of a given color are equivalent (8! for each color), as is each pair of rooks (22) and each pair of knights (another 22); the bishops are not interchangable but each has only 32 squares to choose from (24). However, this is inaccurate for a number of reasons. First and most important, a pawn cannot switch columns (ranks), or move past the opposing pawn in its rank unless it captures. The more captures take place, the more flexibility the pawns have, but that decreases the number of pieces which decreases the number of board positions. Also, the possibility of pawn promotion increases the number of combinations somewhat. See also this estimate.
The number of possible chess games is much higher. See also 4.63×10170.
20988936657440586486151264256610222593863921 = (2148+1)/17 ~= 2.098893665744×1043
In July 1951 Ferrier found this 44-digit prime using a mechanical desk calculator. It became the largest-known prime, breaking the record set by Lucas in 1876. This record did not stand long; it was broken by Miller and Wheeler in the same month. 34
The value of the number called zài in Chinese. See also 104096.
824792557184288824246737061810550733633916929 = 3×(7×392-1)/2 ≅ 8.247925...×1044
This is a lower bound found by Milton Green for the value of BB(8), where BB(n) is the busy beaver function.
2054221614063184107682218077003539824552559296000 = 29×35×53×72×112×132×172×19×23×29×31×37×41×43×47×53×59×61×79×83×89×97×101 ≅ 2.054×1048
The smallest number that has at least 1010 distinct factors. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 457936×10917.
This is an upper bound on the number of possible chess positions, by my reckoning. It allows between 2 and 32 pieces in play, with no more than 16 of one color, including exactly one king of each color, and up to 8 pawns of each color (any of which might have been promoted to another piece). It is higher than Shannon's estimate because it allows pawn promotion.
The number of possible chess games is much higher. See also 4.63×10170.
808017424794512875886459904961710757005754368000000000 = 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19 × 23 × 29 × 31 × 41 × 47 × 59 × 71 ≅ 8.08...×1053
This is the "order" (number of elements) in the largest sporadic finite simple group, called the "Monster group" or the Fischer-Griess group.
(Some background: A "group" can be visualized as a set of transformations, e.g. rotations and reflections, that belong to an N-dimensional geometric structure such as a crystal lattice, or Rubik's Cube. A "simple" group has no "subgroups", which are subsets that themselves form a group; a "sporadic" group is one that does not fit into one of the infinite classes (cyclic, alternating, and Lie).)
See also 196883.
Another large number that appears in the Lotus sutra texts of Mahayana Buddhism, where it appears as the word A-so-gi (あそぎ). See also 1011.
A very poor estimate of the radius of the "visible" universe in Planck units, based on its age and the idea that the objects currently at the "edge" have spent half of this time travelling to where they currently appear to be, and then their light spent an equal amount of time coming back to us. More adjustments would need to be made to account for changes in the rate of the universe's expansion, and the amount of its "curvature".
An approximate value for the age of the universe in Planck time units:
r = 13.73×109 × 365 × 24 × 3600 / 5.39 × 10-44
= 8.02 × 1060
For various reasons, this number is not equal to the "radius", nor is it exactly twice the radius of the visible universe. However, for rough calculations of things like the current volume and space-time volume, and particularly for larger derived values like the number of alternate universes, it is more than adequate.
See also 1040.
Archimedes, in his writing psammites (better known as The Sand Reckoner), estimated the size of the universe according to the heliocentric model of Artistarchus, and how many grains of sand would fit in it. He arrived at a value equivalent to one vigintillion, or 1063. Even more impressive, he described a system of numbers extending as high as 108×1016.
(Personal: For a while during 3rd grade this was the largest number I knew and on a few occasions I wrote it in the sand during recess: 1,000,000,000..., 21 sets of zeros. A mean kid would follow and wipe it out.)
5.2106440156792×1078 = 180×(2127-1)2+1
This is a prime, found by Miller and Wheeler in July 1951. This discovery has the distinction of being the first time the record for largest known prime was set by electronic computer. It broke the record set by Ferrier and was soon broken by Robinson. 34
31495448272550005155211307922363110936089435829054233418732462850152371262062592 = 2×136×2256 = 2×136×2223 = 17×2260 ≅ 3.149544...×1079
This is the Eddington number. According to Arthur Eddington in his book Mathematical Theory of Relativity (1923, London, Cambridge University press), it is the number of particles in the universe. It is notable for being the largest specific integer ever thought to have a unique and tangible relationship to the physical world. (All larger numbers in physics are estimates and approximations.)
Eddington was interested in showing that the various physical constants (the speed of light, the gravitational constant, the mass of the electron, etc.) were not accidental but were determined in some way that could be computed exactly. One of these constants was the fine-structure constant
In 1923 the fine-structure constant was known poorly enough that one could surmise that it is exactly 1/136. Eddington computed the number of particles in the universe from other measurements and observations and then found a simple mathematical formula based on integers that gave the same value. (When the fine-structure constant was later found to be closer to 1/137, Eddington repeated his work to make it fit that value!)
Eddington was shown to be wrong on other points. Many other estimates of the number of particles in the universe have been computed, all in the range from 1078 to 1080. Here is an example (which is way too simplistic for cosmologists but shows that the Eddington number was fairly close):
r = radius of visible universe
= age of universe × speed of light
= speed of light / Hubble constant
= 1.42 × 1026 meters
volume of universe = 4/3 π r3
= 1.2 × 1079 cubic meters
average density of universe
= 3 hydroden atoms per cubic meter
(from models that give the minimum mass of a "closed" universe)
1 hydrogen atom = 4 particles (proton + electron, a proton is 3 quarks)
number of particles = 4.8 × 1079
If you include the various massless particles (photons, gravitons, other gauge particles, perhaps neutrinos) and virtual particles, the estimates become much greater. The only estimate I have been able to find gives the density of neutrinos in the cosmic background radiation as being 107 per cubic meter, which gives a value of 1.2×1086 particles in the universe assuming that neutrinos comprise the vast majority.
2350988701644575015937473074444491355637331113544175043017503412556834518909454345703125 = 553 ≅ 2.3509887016443...×1087
In high school I wrote a special program (I think it was on a TRS-80) to calculate and display exact values of large exponents. Then I created tables of powerlogs, (which are integers of the form AAB where A and B are also integers), written by hand in a notebook. This is the largest of about 30 really big numbers in that table. See also 1.0621842147×104990856845.
Some of the larger estimates of the number of particles in the known visible universe are around this value, and result from including photons and other massless particles. The actual number of particles in the universe may be much larger — for example, it might be that most of the universe is beyond our event horizon (redshift horizon). See also 7×1022.
Most pocket calculators max out at 9.9999999...×1099, which is just below a googol (10100, see next entry). See also 9.9999999...×10999 and the computer overflow values starting with 3.4028236692093×1038.
10100, can be called "10 duotrigintillion" but it is better known by the name googol. It is perhaps the best-known example of a number that was "invented" just for the purpose of being large. Only slightly less well-known is the much larger googolplex.
Stories about the invention of these names differ. According to the most reliable sources, in 1938 mathematician Edward Kasner asked his nephew Milton Sirotta (who was 9 years old at the time) to invent a name for the number 1 with a hundred zeros written after it, and the nephew chose the name "googol". The name "googolplex" was also suggested by Milton Sirotta (or invented by Kasner himself, or by a colleague, depending on which story you believe) and the value 1010100 was assigned to that name by Kasner.
An estimate of the number of subatomic particles that it would take to fill all the space in the universe. See also 8.45×10184. (From Straight Dope)
Shannon's estimate of the number of chess games from his original 1950 paper on the topic. It is calculated by the approximation 100040, based on the idea that at each move by White there are about 30 choices, to which Black has about 30 responses, and typical games last 40 moves. See also 1.15x1042 and 101050.
In India's ancient writings there are many references to large numbers with names; some are hard to attach to a specific value because of multiple conflicting or ambiguous uses. One of the larger numbers given a name in India is asankhyeya, commonly said to be 10140. See also 10421 and 103.7218×1037.
6.8647976601306×10156 = 2521-1
This is the 13th Mersenne Prime and the first to be found by electronic computer. It was discovered in 1952 by Robinson and breaks the record set by Lucas in 1876, although that record was also broken by the non-Mersenne primes (2148+1)/17 and 180×(2127-1)2+1, which were found the year before. 34
This is (approximately) the number of possible positions in Go, played on a 19×19 board, as given by M. Beeler in HAKMEM72 item 96. See also 1.15x1042.
The number of years in the longest time-period in the cosmology of Jainism, a religion and philosophy from India in the 6th century B.C. (From an article by J J O'Connor and E F Robertson)
The current volume of the universe in Planck units. See also 8.02×1060, 1.69×10245, and 5.1843×1022652507173.
5.7324701932...×10207 = 75600000000000 × 840000028
Another number from measurement of time in Jainism. A purvi is 22×33×7×1011 = 7.56×1013 years, and a shirsha prahelika is 840000028 purvis, which works out to 5.7324701932...×10207 years.
1.267650600228...×10230 = 200100
Value of "googoc" from the Googology page. This page introduces a lot of novel number names and also summarizes a lot of other people's names for special large numbers.
The basic idea is to derive word endings, prefixes and infix parts by reverse etymology from existing names like "googol". According to that page, André Joyce noticed that googol is "googo" followed by the Roman numeral l (representing 50) and that googol=10100 is equal to 10050. He extrapolated this with the generalization that "googo" followed by any Roman numeral(s) z is (2z)z, which he chose to express as ack-h(2,ack-h(1,2,l),l) using the Herbert version of the Ackermann function. This means that "googoc" would be 200^100, "googod" is 1000500=101500, etc. In another similar generalization, anything followed by -ple- and Roman numeral(s) such as x is the result of raising x (or whatever) to the power of the thing to which -plex was added (perhaps he imagined "ple" stood for something like "placé dans l'exposant de"). Thus, for example, twoplex = two, 2, followed by -ple- and x representing 10, is 102; threepleiii is 33=27; and googodplem = 1000googod = 10001000500 = 103×101500.
The four-dimensional volume (in space + time) of the known universe using the formula for the volume of a hypercone (with spherical cross-section) and the universe's age, expressed in Planck units. The hypercone volume formula is
V = (1/4 h) (4/3 π r3)
where h is the height of the hypercone and r is the radius of the sphere that forms the hypercone's base. h is the age of the universe in Planck time units and r is its current radius in Planck length units. Due to complexities of relativity and the way the universe expands, h and r are not the same. This gives 6.2×10243 for the 4-D volume.
Real models of the universe used by astrophysicists and cosmologists are much more complex and do not admit to such a simple calculation of volume, but most models would arrive at a figure close to this one.
See also 1.36×10403.
centillion, often said to be the largest number with a single-word name in English (and many other languages that use the Chuquet names).
This is (approximately) the maximum value that can be represented in the commonly-used IEEE 754 double-precision (1+11+52 bit) floating-point format.
See also 3.4028236692093×1038, 1.1897314953572318×104932, 4.26448742×102525222 and 1.4403971939817846×10323228010.
This is the first point at which the prime counting function pi(x) exceeds the logarithmic integral li(x). This is the quantity that was originally estimated by an upper bound of eee79 ≅ 10101034, the "Skewes number".
If you chose a random particle in the known universe, then chose a random moment in the universe's history and moved the particle to a random location somewhere else in the universe at that moment, you would have a total of 6.2×10340 distinct choices (as limited by the uncertainty principle). The formula for this is equivalent to the universe's 4-dimensional volume times the age times the number of particles. Moving a particle instantly by a large distance would usually violate Einstein's relativity (by exceeding the speed of light), but "particles" moving at faster than the speed of light must be considered when modeling physics by one of the gauge theories such as Quantum Electrodynamics.
6.2×10340 is the single-perturbation count. Its factorial gives the total number of complete shufflings of the entire known universe's history.
8.1847946207224960623437×10370
This is the revised, smaller value of the Skewes number, equal to ee27/4.
In the Lalitavistara, a biography of Gautama Buddha which was written mainly during the first couple centuries A.D., Gautama is asked to name the powers of ten starting with koti, which is 107. He gives names for powers of ten up to the tallaksana, 1053. He then describes successive "numerations", the dvajagravati=1099, the dvajagranisamani=10145, and several more culminating in 10421, which is given the name uttaraparamanurajahpravesa28. There are many other stories like this in the culture of India from that period, which shows the extent to which they were interested in large numbers, and also helps explain their need to use place-value notation with a symbol for zero.29
See also 10140 and 103.7218×1037.
A 6-state turing machine, found by Heiner Marxen and Juergen Buntrock in 2001 March, takes 3.00233×101730 steps before halting with 1.29149×10865 ones on the tape. It was a busy beaver record holder for a while. Using a very small set of state transition rules, it iterates X'=2K×X several times in a row, with a chaotic deterministic low-probability exit condition. Its record was broken by Terry and Shawn Ligocki, whose machine leaves 4.6×101439 marks.
457936...×10917 = 215×310×56×75×114×...×2039×2053×2063
The smallest number that has at least a googol distinct factors. Its exact value is 457936006084633875 260691932542213506579481395376 080192442872707759996212114957 373537195900697943283211344130 969977204683723647091975242566 556807073476262370119366712949 612051508874565615465951982148 103948322515169952026557331614 199239782652240565877185274882 891122589783986489974588207230 026310073238799349251084594897 863556829085566422093207975001 895285824382289647389848615424 710629561529529589935914349946 023950287863307022313442880758 800532983282085207377266536998 146723331964258315488766981883 904240306133944424567760471103 539279962416731476757145320641 439420037963516042879919957607 890943287019373144639492683640 803862704805497501551907216898 677744138585826270309663329962 841518933729157858558919253022 063551926057138672786596389094 200184031909805595086778342937 081605771699885426749776777391 919555685119629369584896777148 250878775274042686107865894781 763500774758450843791837394393 056896301600021929961984000000. The factorization of this number, along with the other record setters up to 103535, was found by Achim Flammenkamp7. See also 12, 840, 45360, 720720, 3603600, 245044800, 278914005382139703576000 and 2054221614063184107682218077003539824552559296000.
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