The logarithmic scale can compactly represent the relationship among variously sized numbers.
This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.
Mathematics – random selections: Approximately 10−183,800 is a rough first estimate of the probability that a typing "monkey", or an English-illiterate typing robot, when placed in front of a typewriter, will type out William Shakespeare's play Hamlet as its first set of inputs, on the precondition it typed the needed number of characters. However, demanding correct punctuation, capitalization, and spacing, the probability falls to around 10−360,783.
Computing: 2.2×10−78913 is approximately equal to the smallest positive non-zero value that can be represented by an octuple-precision IEEE floating-point value.
Mathematics – Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the US Powerball lottery, with a single ticket, under the rules as of October 2015[update], are 292,201,338 to 1 against, for a probability of 3.422×10−9 (0.0000003422%).
Mathematics – Lottery: The odds of winning the Grand Prize (matching all 6 numbers) in the Australian Powerball lottery, with a single ticket, under the rules as of April 2018[update], are 134,490,400 to 1 against, for a probability of 7.435×10−9 (0.0000007435%).
Mathematics – Lottery: The odds of winning the Jackpot (matching the 6 main numbers) in the UK National Lottery, with a single ticket, under the rules as of August 2009[update], are 13,983,815 to 1 against, for a probability of 7.151×10−8 (0.000007151%).
Mathematics: the number system understood by most computers, the binary system, uses 2 digits: 0 and 1.
Mathematics:√5 ≈ 2.236 067 9775, the correspondent to the diagonal of a rectangle whose side lengths are 1 and 2.
Mathematics:√2 + 1 ≈ 2.414213562373095049, The ratio of smaller of the two quantities to the larger quantity is the same as the ratio of the larger quantity to the sum of the smaller quantity and twice the larger quantity.
Mathematics: The hexadecimal system, a common number system used in computer programming, uses 16 digits where the last 6 are usually represented by letters: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
Computing – Unicode: The minimum possible size of a Unicode block is 16 contiguous code points (i.e., U+abcde0 - U+abcdeF).
Computing – UTF-16/Unicode: There are 17 addressable planes in UTF-16, and, thus, as Unicode is limited to the UTF-16 code space, 17 valid planes in Unicode.
Syllabic writing: There are 49 letters in each of the two kana syllabaries (hiragana and katakana) used to represent Japanese (not counting letters representing sound patterns that have never occurred in Japanese).
Chess: Either player in a chess game can claim a draw if 50 consecutive moves are made by each side without any captures or pawn moves.
Computing - Fonts: The maximum possible number of glyphs in a TrueType or OpenType font is 65,535 (216-1), the largest number representable by the 16-bit unsigned integer used to record the total number of glyphs in the font.
Computing – Unicode: A plane contains 65,536 (216) code points; this is also the maximum size of a Unicode block, and the total number of code points available in the obsolete UCS-2 encoding.
Computing – UTF-8: There are 1,112,064 (220 + 216 - 211) valid UTF-8 sequences (excluding overlong sequences and sequences corresponding to code points used for UTF-16 surrogates or code points beyond U+10FFFF).
Computing – UTF-16/Unicode: There are 1,114,112 (220 + 216) distinct values encodable in UTF-16, and, thus (as Unicode is currently limited to the UTF-16 code space), 1,114,112 valid code points in Unicode (1,112,064 scalar values and 2,048 surrogates).
Ludology – Number of games: Approximately 1,181,019 video games have been created as of 2019.
Biology – Species: The World Resources Institute claims that approximately 1.4 million species have been named, out of an unknown number of total species (estimates range between 2 and 100 million species). Some scientists give 8.8 million species as an exact figure.
Computing – UTF-8: 2,164,864 (221 + 216 + 211 + 27) possible one- to four-byte UTF-8 sequences, if the restrictions on overlong sequences, surrogate code points, and code points beyond U+10FFFF are not adhered to. (Note that not all of these correspond to unique code points.)
Mathematics – Playing cards: There are 2,598,960 different 5-card poker hands that can be dealt from a standard 52-card deck.
Mathematics: There are 3,149,280 possible positions for the Skewb.
Mathematics – Rubik's Cube: 3,674,160 is the number of combinations for the Pocket Cube (2×2×2 Rubik's Cube).
Internet – Google: There are more than 1,500,000,000 active Gmail users globally.
Internet: Approximately 1,500,000,000 active users were on Facebook as of October 2015.
Computing – Computational limit of a 32-bit CPU: 2,147,483,647 is equal to 231−1, and as such is the largest number which can fit into a signed (two's complement) 32-bit integer on a computer.
Computing – UTF-8: 2,147,483,648 (231) possible code points (U+0000 - U+7FFFFFFF) in the pre-2003 version of UTF-8 (including five- and six-byte sequences), before the UTF-8 code space was limited to the much smaller set of values encodable in UTF-16.
Linguistics: 3,400,000,000 – the total number of speakers of Indo-European languages, of which 2,400,000,000 are native speakers; the other 1,000,000,000 speak Indo-European languages as a second language.
Mathematics and computing: 4,294,967,295 (232 − 1), the product of the five known Fermat primes and the maximum value for a 32-bit unsigned integer in computing.
Computing: 4,294,967,296 – the number of bytes in 4 gibibytes; in computation, 32-bit computers can directly access 232 units (bytes) of address space, which leads directly to the 4-gigabyte limit on main memory.
Biology – Cells in the human body: The human body consists of roughly 1014cells, of which only 1013 are human. The remaining 90% non-human cells (though much smaller and constituting much less mass) are bacteria, which mostly reside in the gastrointestinal tract, although the skin is also covered in bacteria.
Cryptography: 150,738,274,937,250 configurations of the plug-board of the Enigma machine used by the Germans in WW2 to encode and decode messages by cipher.
Biology – Insects: 1,000,000,000,000,000 to 10,000,000,000,000,000 (1015 to 1016) – The estimated total number of ants on Earth alive at any one time (their biomass is approximately equal to the total biomass of the human species).
Science Fiction: In Isaac Asimov's Galactic Empire, in what we call 22,500 CE there are 25,000,000 different inhabited planets in the Galactic Empire, all inhabited by humans in Asimov's "human galaxy" scenario, each with an average population of 2,000,000,000, thus yielding a total Galactic Empire population of approximately 50,000,000,000,000,000.
Science Fiction: There are approximately 1017 sentient beings in the Star Wars galaxy.
Cryptography: There are 256 = 72,057,594,037,927,936 different possible keys in the obsolete 56-bit DES symmetric cipher.
Computing – Manufacturing: An estimated 6×1018transistors were produced worldwide in 2008.
Computing – Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22×1018) is equal to 263−1, and as such is the largest number which can fit into a signed (two's complement) 64-bit integer on a computer.
Mathematics – Bases: 9,439,829,801,208,141,318 (≈9.44×1018) is the 10th and (by conjecture) largest number with more than one digit that can be written from base 2 to base 18 using only the digits 0 to 9, meaning the digits for 10 to 17 are not needed in bases above 10.
Biology – Insects: It has been estimated that the insect population of the Earth is about 1019.
Mathematics – Answer to the wheat and chessboard problem: When doubling the grains of wheat on each successive square of a chessboard, beginning with one grain of wheat on the first square, the final number of grains of wheat on all 64 squares of the chessboard when added up is 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019).
Mathematics – Legends: The Tower of Brahmalegend tells about a Hindu temple containing a large room with three posts, on one of which are 64 golden discs, and the object of the mathematical game is for the Brahmins in this temple to move all of the discs to another pole so that they are in the same order, never placing a larger disc above a smaller disc, moving only one at a time. Using the simplest algorithm for moving the disks, it would take 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019) turns to complete the task (the same number as the wheat and chessboard problem above).
Computing – IPv6: 18,446,744,073,709,551,616 (264; ≈1.84×1019) possible unique /64 subnetworks.
Mathematics – Rubik's Cube: There are 43,252,003,274,489,856,000 (≈4.33×1019) different positions of a 3×3×3 Rubik's Cube.
Password strength: Usage of the 95-character set found on standard computer keyboards for a 10-character password yields a computationally intractable 59,873,693,923,837,890,625 (9510, approximately 5.99×1019) permutations.
Chemistry – Physics: The Avogadro constant (6.02214076×1023) is the number of constituents (e.g. atoms or molecules) in one mole of a substance, defined for convenience as expressing the order of magnitude separating the molecular from the macroscopic scale.
Mathematics: 227-1-1 = 170,141,183,460,469,231,731,687,303,715,884,105,727 (≈1.7×1038) is the largest known double Mersenne prime.
Computing: 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×1038), the theoretical maximum number of Internet addresses that can be allocated under the IPv6 addressing system, one more than the largest value that can be represented by a single-precision IEEE floating-point value, the total number of different Universally Unique Identifiers (UUIDs) that can be generated.
Cryptography: 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×1038), the total number of different possible keys in the AES 128-bit key space (symmetric cipher).
Geo: 1.33×1050 is the estimated number of atoms in Earth.
Mathematics: 2168 = 374,144,419,156,711,147,060,143,317,175,368,453,031,918,731,001,856 is the largest known power of two which is not pandigital: There is no digit '2' in its decimal representation.
Mathematics: 3106 = 375,710,212,613,636,260,325,580,163,599,137,907,799,836,383,538,729 is the largest known power of three which is not pandigital: There is no digit '4'.
Mathematics: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (≈8.08×1053) is the order of the monster group.
Cryptography: 2192 = 6,277,101,735,386,680,763,835,789,423,207,666,416,102,355,444,464,034,512,896 (6.27710174×1057), the total number of different possible keys in the AES 192-bit key space (symmetric cipher).
Mathematics – Cards: 52! = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 (≈8.07×1067) – the number of ways to order the cards in a 52-card deck.
Mathematics: There are ≈1.01×1068 possible combinations for the Megaminx.
Mathematics: 1,808,422,353,177,349,564,546,512,035,512,530,001,279,481,259,854,248,860,454,348,989,451,026,887 (≈1.81×1072) – The largest known prime factor found by ECM factorization as of 2010[update].
Mathematics: There are 282,870,942,277,741,856,536,180,333,107,150,328,293,127,731,985,672,134,721,536,000,000,000,000,000 (≈2.83×1074) possible permutations for the Professor's Cube (5×5×5 Rubik's Cube).
Cryptography: 2256 = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936 (≈1.15792089×1077), the total number of different possible keys in the AES 256-bit key space (symmetric cipher).
Computing: 69! (roughly 1.7112245×1098), is the highest factorial value that can be represented on a calculator with two digits for powers of ten without overflow.
Mathematics: One googol, 1×10100, 1 followed by one hundred zeros, or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard)
Mathematics: 10googol (), a googolplex. A number 1 followed by 1 googol zeros. Carl Sagan has estimated that 1 googolplex, fully written out, would not fit in the observable universe because of its size, while also noting that one could also write the number as 1010100.
Mathematics:, a number in the googol family called a googolplexplex, googolplexian, or googolduplex. 1 followed by a googolplex zeros, or 10googolplex
Mathematics:, order of magnitude of another upper bound in a proof of Skewes.
Mathematics:, a number in the googol family called a googolplexplexplex, googolplexianth, or googoltriplex. 1 followed by a googolduplex zeros, or 10googolduplex
Mathematics: Steinhaus' mega lies between 10257 and 10258 (where a[n]b is hyperoperation).
Mathematics: Moser's number, "2 in a mega-gon" in Steinhaus–Moser notation, is approximately equal to 10[10257]10, the last four digits are ...1056.
Mathematics:Graham's number, the last ten digits of which are ...2464195387, equals 3[3[3[...3[3[33+2]3+2]3...]3+2]3+2]3 with 64 levels of brackets. Arises as an upper bound solution to a problem in Ramsey theory. Representation in powers of 10 would be impractical (the number of 10s in the power tower would be virtually indistinguishable from the number itself).
Mathematics:TREE(3): appears in relation to a theorem on trees in graph theory. Representation of the number is difficult, but one weak lower bound is AA(187196)(1), where A(n) is a version of the Ackermann function.
Mathematics:SSCG(3): appears in relation to the Robertson–Seymour theorem. Known to be greater than both TREE(3) and the TREE function nested inside itself TREE(3) times with TREE(3) at the bottom.
^"there was, to our knowledge, no actual, direct estimate of numbers of cells or of neurons in the entire human brain to be cited until 2009. A reasonable approximation was provided by Williams and Herrup (1988), from the compilation of partial numbers in the literature. These authors estimated the number of neurons in the human brain at about 85 billion [...] With more recent estimates of 21–26 billion neurons in the cerebral cortex (Pelvig et al., 2008 ) and 101 billion neurons in the cerebellum (Andersen et al., 1992 ), however, the total number of neurons in the human brain would increase to over 120 billion neurons." Herculano-Houzel, Suzana (2009). "The human brain in numbers: a linearly scaled-up primate brain". Front. Hum. Neurosci. 3: 31. doi:10.3389/neuro.09.031.2009. PMC2776484. PMID19915731.
^Kapitsa, Sergei P (1996). "The phenomenological theory of world population growth". Physics-Uspekhi. 39 (1): 57–71. Bibcode:1996PhyU...39...57K. doi:10.1070/pu1996v039n01abeh000127. (citing the range of 80 to 150 billion, citing K. M. Weiss, Human Biology 56637, 1984, and N. Keyfitz, Applied Mathematical Demography, New York: Wiley, 1977). C. Haub, "How Many People Have Ever Lived on Earth?", Population Today 23.2), pp. 5–6, cited an estimate of 105 billion births since 50,000 BC, updated to 107 billion as of 2011 in Haub, Carl (October 2011). "How Many People Have Ever Lived on Earth?". Population Reference Bureau. Archived from the original on April 24, 2013. Retrieved April 29, 2013. (due to the high infant mortality in pre-modern times, close to half of this number would not have lived past infancy).
^From the third paragraph of the story: "Each book contains 410 pages; each page, 40 lines; each line, about 80 black letters." That makes 410 x 40 x 80 = 1,312,000 characters. The fifth paragraph tells us that "there are 25 orthographic symbols" including spaces and punctuation. The magnitude of the resulting number is found by taking logarithms. However, this calculation only gives a lower bound on the number of books as it does not take into account variations in the titles – the narrator does not specify a limit on the number of characters on the spine. For further discussion of this, see Bloch, William Goldbloom. The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press: Oxford, 2008.