In today's world, Orders of magnitude (numbers) has become a topic of constant debate and discussion. From its origins to the present, Orders of magnitude (numbers) has greatly influenced different aspects of society, culture, technology and politics. Its impact has been so significant that it has given rise to a variety of opinions and perspectives, generating a constant exchange of ideas and arguments. In this article, we will explore in detail the importance of Orders of magnitude (numbers) and its influence in various areas, analyzing its implications over time and its relevance in the current context.
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.
Smaller than 10^{−100} (one googolth)
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.^{[1]} However, demanding correct punctuation, capitalization, and spacing, the probability falls to around 10^{−360,783}.^{[2]}
Computing: 2.2×10^{−78913} is approximately equal to the smallest non-zero value that can be represented by an octuple-precision IEEE floating-point value.
1×10^{−6176} is equal to the smallest non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value.
6.5×10^{−4966} is approximately equal to the smallest non-zero value that can be represented by a quadruple-precision IEEE floating-point value.
3.6×10^{−4951} is approximately equal to the smallest non-zero value that can be represented by an 80-bit x86 double-extended IEEE floating-point value.
1×10^{−398} is equal to the smallest non-zero value that can be represented by a double-precision IEEE decimal floating-point value.
1.5×10^{−157} is approximately equal to the probability that in a randomly selected group of 365 people, all of them will have different birthdays.^{[3]}
1×10^{−101} is equal to the smallest non-zero value that can be represented by a single-precision IEEE decimal floating-point value.
10^{−100} to 10^{−30}
Mathematics: The chances of shuffling a standard 52-card deck in any specific order is around 1.24×10^{−68} (or exactly 1⁄52!)^{[4]}
Computing: The number 1.4×10^{−45} is approximately equal to the smallest positive non-zero value that can be represented by a single-precision IEEE floating-point value.
10^{−30}
(0.000000000000000000000000000001; 1000^{−10}; short scale: one nonillionth; long scale: one quintillionth)
Mathematics: The Ramanujan constant, $e^{\pi {\sqrt {163}}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25\ldots ,$ is an almost integer, differing from the nearest integer by approximately 7.5×10^{−13}.
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^{}, 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^{}, 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^{}, are 13,983,815 to 1 against, for a probability of 7.151×10^{−8} (0.000007151%).
Mathematics – Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5×10^{−6} (0.00015%).^{[8]}
Mathematics – Poker: The odds of being dealt a straight flush (other than a royal flush) in poker are 72,192 to 1 against, for a probability of 1.4×10^{−5} (0.0014%).
Mathematics – Poker: The odds of being dealt a four of a kind in poker are 4,164 to 1 against, for a probability of 2.4×10^{−4} (0.024%).
Mathematics – Lottery: The odds of winning any prize in the UK National Lottery, with a single ticket, under the rules as of 2003, are 54 to 1 against, for a probability of about 0.018 (1.8%).
Mathematics – Poker: The odds of being dealt a three of a kind in poker are 46 to 1 against, for a probability of 0.021 (2.1%).
Mathematics – Lottery: The odds of winning any prize in the Powerball, with a single ticket, under the rules as of 2015, are 24.87 to 1 against, for a probability of 0.0402 (4.02%).
Mathematics – Poker: The odds of being dealt two pair in poker are 21 to 1 against, for a probability of 0.048 (4.8%).
Computing – Unicode: One character is assigned to the Lisu SupplementUnicode block, the fewest of any public-use Unicode block as of Unicode 15.0 (2022).
Mathematics:√2 ≈ 1.414213562373095049, the ratio of the diagonal of a square to its side length.
Mathematics: φ ≈ 1.618033988749894848, the golden ratio.
Mathematics:√3 ≈ 1.732050807568877293, the ratio of the diagonal of a unit cube.
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 silver ratio; the ratio of the 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:e ≈ 2.718281828459045087, the base of the natural logarithm.
Mathematics: the number system understood by ternary computers, the ternary system, uses 3 digits: 0, 1, and 2.
Religion: three manifestations of God in the Christian Trinity.
Mathematics:π ≈ 3.141592653589793238, the ratio of a circle's circumference to its diameter.
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.
Military history: 4,200 (Republic) or 5,200 (Empire) was the standard size of a Roman legion.
Linguistics: Estimates for the linguistic diversity of living human languages or dialects range between 5,000 and 10,000. (SIL Ethnologue in 2009 listed 6,909 known living languages.)
War: 22,717 Union and Confederate soldiers were killed, wounded, or missing in the Battle of Antietam, the bloodiest single day of battle in American history.
Computing - Fonts: The maximum possible number of glyphs in a TrueType or OpenType font is 65,535 (2^{16}-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 (2^{16}) 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.
Biology – Plants: There are approximately 390,000 distinct plant species known, of which approximately 20% (or 78,000) are at risk of extinction.^{[14]}
Biology – Flowers: There are approximately 400,000 distinct flower species on Earth.^{[15]}
Demography: The population of Riga, Latvia was 1,003,949 in 2004, according to Eurostat.
Computing – UTF-8: There are 1,112,064 (2^{20} + 2^{16} - 2^{11}) 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 (2^{20} + 2^{16}) 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.^{[16]}
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.
Linguistics: The number of possible conjugations for each verb in the Archi language is 1,502,839.^{[17]}
Info: The freedb database of CD track listings has around 1,750,000 entries as of June 2005^{}.
Computing – UTF-8: 2,164,864 (2^{21} + 2^{16} + 2^{11} + 2^{7}) 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).
Geography/Computing – Geographic places: The NIMA GEOnet Names Server contains approximately 3.88 million named geographic features outside the United States, with 5.34 million names. The USGS Geographic Names Information System claims to have almost 2 million physical and cultural geographic features within the United States.
Computing - Supercomputer hardware: 4,981,760 processor cores in the final configuration of the Tianhe-2 supercomputer.
Genocide: Approximately 5,100,000–6,200,000 Jews were killed in the Holocaust.
Info – Web sites: As of November 12, 2024, the English Wikipedia contains approximately 6.9 million articles in the English language.
Genocide/Famine: 15 million is an estimated lower bound for the death toll of the 1959–1961 Great Chinese Famine, the deadliest known famine in human history.
War: 15 to 22 million casualties estimated as a result of World War I.
Computing: 16,777,216 different colors can be generated using the hex code system in HTML (note that the trichromaticcolor vision of the human eye can only distinguish between about an estimated 1,000,000 different colors).^{[18]}
Science Fiction: In Isaac Asimov's Galactic Empire, in 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.
Genocide/Famine: 55 million is an estimated upper bound for the death toll of the Great Chinese Famine.
Literature:Wikipedia contains a total of around 63 million articles in 352 languages as of November 2024.
War: 70 to 85 million casualties estimated as a result of World War II.
Video gaming: As of 2020^{}, approximately 200 million copies of Minecraft (the most-sold video game in history) have been sold.
Mathematics: More than 215,000,000 mathematical constants are collected on the Plouffe's Inverter as of 2010^{}.^{[20]}
Mathematics: 275,305,224 is the number of 5×5 normal magic squares, not counting rotations and reflections. This result was found in 1973 by Richard Schroeppel.
Demography: The population of the United States was 328,239,523 in 2019.
Transportation – Cars: As of 2018^{}, there are approximately 1.4 billion cars in the world, corresponding to around 18% of the human population.^{[21]}
Demographics – China: 1,409,670,000 – approximate population of the People's Republic of China in 2023.^{[22]}
Demographics – India 1,428,627,663 – approximate population of India in 2023.^{[23]}
Demographics – Africa: The population of Africa reached 1,430,000,000 sometime in 2023.
Internet – Google: There are more than 1,500,000,000 active Gmail users globally.^{[24]}
Internet: Approximately 1,500,000,000 active users were on Facebook as of October 2015.^{[25]}
Computing – Computational limit of a 32-bit CPU: 2,147,483,647 is equal to 2^{31}−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 (2^{31}) 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.
Biology – base pairs in the genome: approximately 3.3×10^{9}base pairs in the human genome.^{[11]}
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 (2^{32} − 1), the product of the five known Fermat primes and the maximum value for a 32-bit unsigned integer in computing.
Computing – IPv4: 4,294,967,296 (2^{32}) possible unique IP addresses.
Computing: 4,294,967,296 – the number of bytes in 4 gibibytes; in computation, 32-bit computers can directly access 2^{32} units (bytes) of address space, which leads directly to the 4-gigabyte limit on main memory.
Mathematics: 4,294,967,297 is a Fermat number and semiprime. It is the smallest number of the form $2^{2^{n}}+1$ which is not a prime number.
Demographics – world population: 8,019,876,189 – Estimated population for the world as of 1 January 2024.^{[26]}
Astronomy – stars in our galaxy: of the order of 10^{11}stars in the Milky Way galaxy.^{[33]}
Mathematics: 608,981,813,029 is the smallest number for which there are more primes of the form 3k + 1 than of the form 3k + 2 up to the number.^{[34]}
Biology – Blood cells in the human body: The average human body has 2.5 × 10^{12} red blood cells.^{[medical citation needed]}
Biology: An estimate says there were 3.04 × 10^{12}trees on Earth in 2015.^{[36]}
Marine biology: 3,500,000,000,000 (3.5 × 10^{12}) – estimated population of fish in the ocean.^{[citation needed]}
Mathematics: 7,625,597,484,987 – a number that often appears when dealing with powers of 3. It can be expressed as $19683^{3}$, $27^{9}$, $3^{27}$, $3^{3^{3}}$ and ^{3}3 or when using Knuth's up-arrow notation it can be expressed as $3\uparrow \uparrow 3$ and $3\uparrow \uparrow \uparrow 2$.
Astronomy: A light-year, as defined by the International Astronomical Union (IAU), is the distance that light travels in a vacuum in one year, which is equivalent to about 9.46 trillionkilometers (9.46×10^{12}km).
Mathematics: 10^{13} – The approximate number of known non-trivial zeros of the Riemann zeta function as of 2004^{}.^{[37]}
Mathematics – Known digits of π: As of March 2019^{}, the number of known digits of π is 31,415,926,535,897 (the integer part of π×10^{13}).^{[38]}
Biology – approximately 10^{14}synapses in the human brain.^{[39]}
Biology – Cells in the human body: The human body consists of roughly 10^{14}cells, of which only 10^{13} are human.^{[40]}^{[41]} 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.
Mathematics: The first case of exactly 18 prime numbers between multiples of 100 is 122,853,771,370,900 + n,^{[42]} for n = 1, 3, 7, 19, 21, 27, 31, 33, 37, 49, 51, 61, 69, 73, 87, 91, 97, 99.
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 (10^{15} to 10^{16}) – 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).^{[43]}
Computing: 9,007,199,254,740,992 (2^{53}) – number until which all integer values can exactly be represented in IEEE double precision floating-point format.
Mathematics: 48,988,659,276,962,496 is the fifth taxicab number.
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 10^{17} sentient beings in the Star Wars galaxy.
Cryptography: There are 2^{56} = 72,057,594,037,927,936 different possible keys in the obsolete 56-bit DES symmetric cipher.
Mathematics: The first case of exactly 19 prime numbers between multiples of 100 is 1,468,867,005,116,420,800 + n,^{[42]} for n = 1, 3, 7, 9, 21, 31, 37, 39, 43, 49, 51, 63, 67, 69, 73, 79, 81, 87, 93.
Mathematics: 2^{61} − 1 = 2,305,843,009,213,693,951 (≈2.31×10^{18}) is the ninth Mersenne prime. It was determined to be prime in 1883 by Ivan Mikheevich Pervushin. This number is sometimes called Pervushin's number.
Mathematics:Goldbach's conjecture has been verified for all n ≤ 4×10^{18} by a project which computed all prime numbers up to that limit.^{[44]}
Computing – Manufacturing: An estimated 6×10^{18}transistors were produced worldwide in 2008.^{[45]}
Computing – Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22×10^{18}) is equal to 2^{63}−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 (≈9.44×10^{18}) 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 greater than 10.^{[46]}
Biology – Insects: It has been estimated that the insect population of the Earth is about 10^{19}.^{[47]}
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 2^{64}−1 = 18,446,744,073,709,551,615 (≈1.84×10^{19}).
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 2^{64}−1 = 18,446,744,073,709,551,615 (≈1.84×10^{19}) turns to complete the task (the same number as the wheat and chessboard problem above).^{[48]}
Computing – IPv6: 18,446,744,073,709,551,616 (2^{64}; ≈1.84×10^{19}) possible unique /64 subnetworks.
Mathematics – Rubik's Cube: There are 43,252,003,274,489,856,000 (≈4.33×10^{19}) 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 (95^{10}, approximately 5.99×10^{19}) permutations.
Economics:Hyperinflation in Zimbabwe estimated in February 2009 by some economists at 10 sextillion percent,^{[49]} or a factor of 10^{20}.
Geo – Grains of sand: All the world's beaches combined have been estimated to hold roughly 10^{21} grains of sand.^{[50]}
Computing – Manufacturing: Intel predicted that there would be 1.2×10^{21}transistors in the world by 2015^{[51]} and Forbes estimated that 2.9×10^{21} transistors had been shipped up to 2014.^{[52]}
Mathematics – Sudoku: There are 6,670,903,752,021,072,936,960 (≈6.7×10^{21}) 9×9 sudoku grids.^{[53]}
Mathematics: The first case of exactly 20 prime numbers between multiples of 100 is 20,386,095,164,137,273,086,400 + n,^{[42]} for n = 1, 3, 7, 9, 13, 19, 21, 31, 33, 37, 49, 57, 63, 73, 79, 87, 91, 93, 97, 99.
Astronomy – Stars: 70 sextillion = 7×10^{22}, the estimated number of stars within range of telescopes (as of 2003).^{[54]}
Astronomy – Stars: in the range of 10^{23} to 10^{24} stars in the observable universe.^{[55]}
Mathematics: 146,361,946,186,458,562,560,000 (≈1.5×10^{23}) is the fifth unitary perfect number.
Mathematics: 357,686,312,646,216,567,629,137 (≈3.6×10^{23}) is the largest left-truncatable prime.
Chemistry – Physics: The Avogadro constant (6.02214076×10^{23}) 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: 2,833,419,889,721,787,128,217,599 (≈2.8×10^{24}) is the fifth Woodall prime.
Mathematics: 3,608,528,850,368,400,786,036,725 (≈3.6×10^{24}) is the largest polydivisible number.
Mathematics: 2^{86} = 77,371,252,455,336,267,181,195,264 is the largest known power of two not containing the digit '0' in its decimal representation.^{[56]}
10^{27}
(1000000000000000000000000000; 1000^{9}; short scale: one octillion; long scale: one thousand quadrillion, or one quadrilliard)
Biology – Bacterial cells on Earth: The number of bacterial cells on Earth is estimated at 5,000,000,000,000,000,000,000,000,000,000, or 5 × 10^{30}.^{[58]}
Mathematics: 5,000,000,000,000,000,000,000,000,000,027 is the largest quasi-minimal prime.
Mathematics: The number of partitions of 1000 is 24,061,467,864,032,622,473,692,149,727,991.^{[59]}
Mathematics: 3^{68} = 278,128,389,443,693,511,257,285,776,231,761 is the largest known power of three not containing the digit '0' in its decimal representation.
Mathematics: 2^{108} = 324,518,553,658,426,726,783,156,020,576,256 is the largest known power of two not containing the digit '9' in its decimal representation.^{[60]}
Mathematics: 7^{39} = 909,543,680,129,861,140,820,205,019,889,143 is the largest known power of 7 not containing the digit '7' in its decimal representation.
10^{33}
(1000000000000000000000000000000000; 1000^{11}; short scale: one decillion; long scale: one thousand quintillion, or one quintilliard)
Mathematics – Alexander's Star: There are 72,431,714,252,715,638,411,621,302,272,000,000 (about 7.24×10^{34}) different positions of Alexander's Star.
Mathematics: 2^{27−1} − 1 = 170,141,183,460,469,231,731,687,303,715,884,105,727 (≈1.7×10^{38}) is the largest known double Mersenne prime and the 12th Mersenne prime.
Computing: 2^{128} = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×10^{38}), 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: 2^{128} = 340,282,366,920,938,463,463,374,607,431,768,211,456 (≈3.40282367×10^{38}), the total number of different possible keys in the AES 128-bit key space (symmetric cipher).
10^{39}
(1000000000000000000000000000000000000000; 1000^{13}; short scale: one duodecillion; long scale: one thousand sextillion, or one sextilliard)
Mathematics:97# × 2^{5} × 3^{3} × 5 × 7 = 69,720,375,229,712,477,164,533,808,935,312,303,556,800 (≈6.97×10^{40}) is the least common multiple of every integer from 1 to 100.
Mathematics: 141×2^{141}+1 = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833 (≈3.93×10^{44}) is the second Cullen prime.
Mathematics: There are 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (≈7.4×10^{45}) possible permutations for the Rubik's Revenge (4×4×4 Rubik's Cube).
Chess: 4.52×10^{46} is a proven upper bound for the number of chess positions allowed according to the rules of chess.^{[61]}
Geo: 1.33×10^{50} is the estimated number of atoms on Earth.
Mathematics: 2^{168} = 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.^{[62]}
Mathematics: 3^{106} = 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'.^{[62]}
Mathematics: 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 (≈8.08×10^{53}) is the order of the monster group.
Cryptography: 2^{192} = 6,277,101,735,386,680,763,835,789,423,207,666,416,102,355,444,464,034,512,896 (6.27710174×10^{57}), the total number of different possible keys in the Advanced Encryption Standard (AES) 192-bit key space (symmetric cipher).
Cosmology: 8×10^{60} is roughly the number of Planck time intervals since the universe is theorised to have been created in the Big Bang 13.799 ± 0.021 billion years ago.^{[63]}
Cosmology: 1×10^{63} is Archimedes' estimate in The Sand Reckoner of the total number of grains of sand that could fit into the entire cosmos, the diameter of which he estimated in stadia to be what we call 2 light-years.
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 (≈8.07×10^{67}) – the number of ways to order the cards in a 52-card deck.
Mathematics: There are ≈1.01×10^{68} 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 (≈1.81×10^{72}) – The largest known prime factor found by Lenstra elliptic-curve factorization (LECF) as of 2010^{}.^{[64]}
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 (≈2.83×10^{74}) possible permutations for the Professor's Cube (5×5×5 Rubik's Cube).
Cryptography: 2^{256} = 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 (≈1.15792089×10^{77}), the total number of different possible keys in the Advanced Encryption Standard (AES) 256-bit key space (symmetric cipher).
Cosmology: Various sources estimate the total number of fundamental particles in the observable universe to be within the range of 10^{80} to 10^{85}.^{[65]}^{[66]} However, these estimates are generally regarded as guesswork. (Compare the Eddington number, the estimated total number of protons in the observable universe.)
Computing: 9.999 999×10^{96} is equal to the largest value that can be represented in the IEEE decimal32 floating-point format.
Computing: 69! (roughly 1.7112245×10^{98}), is the largest factorial value that can be represented on a calculator with two digits for powers of ten without overflow.
Mathematics: One googol, 1×10^{100}, 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)^{[67]}
Physics: 8×10^{120}, ratio of the mass-energy in the observable universe to the energy of a photon with a wavelength the size of the observable universe.
Go: There are 208 168 199 381 979 984 699 478 633 344 862 770 286 522 453 884 530 548 425 639 456 820 927 419 612 738 015 378 525 648 451 698 519 643 907 259 916 015 628 128 546 089 888 314 427 129 715 319 317 557 736 620 397 247 064 840 935 (≈2.08×10^{170}) legal positions in the game of Go. See Go and mathematics.
Economics: The annualized rate of the hyperinflation in Hungary in 1946 was estimated to be 2.9×10^{177}%.^{[68]} It was the most extreme case of hyperinflation ever recorded.
Board games: 3.457×10^{181}, number of ways to arrange the tiles in English Scrabble on a standard 15-by-15 Scrabble board.
Shogi: 10^{226}, an estimation of the game-tree complexity of shogi.
Physics: 7×10^{245}, approximate spacetime volume of the history of the observable universe in Planck units.^{[69]}
Computing: 1.797 693 134 862 315 807×10^{308} is approximately equal to the largest value that can be represented in the IEEE double precision floating-point format.
Computing: (10 – 10^{−15})×10^{384} is equal to the largest value that can be represented in the IEEE decimal64 floating-point format.
Mathematics: There are approximately 1.869×10^{4099} distinguishable permutations of the world's largest Rubik's Cube (33×33×33).
Computing: 1.189 731 495 357 231 765 05×10^{4932} is approximately equal to the largest value that can be represented in the IEEE 80-bit x86 extended precision floating-point format.
Computing: 1.189 731 495 357 231 765 085 759 326 628 007 0×10^{4932} is approximately equal to the largest value that can be represented in the IEEE quadruple-precision floating-point format.
Computing: (10 – 10^{−33})×10^{6144} is equal to the largest value that can be represented in the IEEE decimal128 floating-point format.
Computing: 10^{10,000} − 1 is equal to the largest value that can be represented in Windows Phone's calculator.
Mathematics: 104,824^{5} + 5^{104,824} is the largest proven Leyland prime; with 73,269 digits as of April 2023^{}.^{[70]}
Mathematics: approximately 7.76 × 10^{206,544} cattle in the smallest herd which satisfies the conditions of Archimedes's cattle problem.
Mathematics: 2,618,163,402,417 × 2^{1,290,000} − 1 is a 388,342-digit Sophie Germain prime; the largest known as of April 2023^{}.^{[71]}
Mathematics: 2,996,863,034,895 × 2^{1,290,000} ± 1 are 388,342-digit twin primes; the largest known as of April 2023^{}.^{[72]}
Mathematics: 3,267,113# – 1 is a 1,418,398-digit primorial prime; the largest known as of April 2023^{}.^{[73]}
Mathematics – Literature:Jorge Luis Borges' Library of Babel contains at least 25^{1,312,000} ≈ 1.956 × 10^{1,834,097} books (this is a lower bound).^{[74]}
Mathematics: 10^{1,888,529} − 10^{944,264} – 1 is a 1,888,529-digit palindromic prime, the largest known as of April 2023^{}.^{[75]}
Mathematics: 4 × 72^{1,119,849} − 1 is the smallest prime of the form 4 × 72^{n} − 1.^{[76]}
Mathematics: 422,429! + 1 is a 2,193,027-digit factorial prime; the largest known as of April 2023^{}.^{[77]}
Mathematics: (2^{15,135,397} + 1)/3 is a 4,556,209-digit Wagstaff probable prime, the largest known as of June 2021^{}.
Mathematics: 1,963,736^{1,048,576} + 1 is a 6,598,776-digit Generalized Fermat prime, the largest known as of April 2023^{}.^{[78]}
Mathematics: (10^{8,177,207} − 1)/9 is a 8,177,207-digit probable prime, the largest known as of 8 May 2021^{}.^{[79]}
Mathematics: 10,223 × 2^{31,172,165} + 1 is a 9,383,761-digit Proth prime, the largest known Proth prime^{[80]} and non-Mersenne prime as of 2021^{}.^{[81]}
Mathematics: 10^{googol} ($10^{10^{100}}$), 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 10^{10100}.^{[83]}
Mathematics – Literature: The number of different ways in which the books in Jorge Luis Borges' Library of Babel can be arranged is approximately $10^{10^{1,834,102}}$, the factorial of the number of books in the Library of Babel.
Cosmology: In chaotic inflation theory, proposed by physicist Andrei Linde, our universe is one of many other universes with different physical constants that originated as part of our local section of the multiverse, owing to a vacuum that had not decayed to its ground state. According to Linde and Vanchurin, the total number of these universes is about $10^{10^{10,000,000}}$.^{[84]}
Mathematics:$10^{\,\!10^{10^{34}}}$, order of magnitude of an upper bound that occurred in a proof of Skewes (this was later estimated to be closer to 1.397 × 10^{316}).
Mathematics:$10^{\,\!10^{10^{100}}}$, a number in the googol family called a googolplexplex, googolplexian, or googolduplex. 1 followed by a googolplex zeros, or 10^{googolplex}
Cosmology: The uppermost estimate to the size of the entire universe is approximately $10^{10^{10^{122}}}$ times that of the observable universe.^{[85]}
Mathematics:$10^{\,\!10^{10^{963}}}$, order of magnitude of another upper bound in a proof of Skewes.
Mathematics: Steinhaus' mega lies between 10257 and 10258 (where ab is hyperoperation).
Mathematics: Moser's number, "2 in a mega-gon" in Steinhaus–Moser notation, is approximately equal to 10257]10, the last four digits are ...1056.
Mathematics:Graham's number, the last ten digits of which are ...2464195387. 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 $10^{\,\!10^{10^{...}}}$ 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 A^{A(187196)}(1), where A(n) is a version of the Ackermann function.
Mathematics:Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest number to have ever been named.^{[87]} It was originally defined in a "big number duel" at MIT on 26 January 2007.^{[88]}
^There are around 130,000 letters and 199,749 total characters in Hamlet; 26 letters ×2 for capitalization, 12 for punctuation characters = 64, 64^{199749} ≈ 10^{360,783}.
^Kibrik, A. E. (2001). "Archi (Caucasian—Daghestanian)", The Handbook of Morphology, Blackwell, pg. 468
^Judd DB, Wyszecki G (1975). Color in Business, Science and Industry. Wiley Series in Pure and Applied Optics (third ed.). New York: Wiley-Interscience. p. 388. ISBN978-0-471-45212-6.
^"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. S2CID250877833. (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.