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 quantity 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 – Writing: Approximately 10−183,800 is a rough first estimate of the probability that a monkey, placed in front of a typewriter and given adequate food, breaks, and sleep while doing so, will type all the letters of Hamlet on the first try. This is the same as the average number of letters needed to be typed for Hamlet to be produced. However, taking punctuation, capitalization, and spacing into account, the actual probability is far less: around 10−360,783.
Computing: The number 1×10−6176 is equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE decimal floating-point value.
Computing: The number 6.5×10−4966 is approximately equal to the smallest positive non-zero value that can be represented by a quadruple-precision IEEE floating-point value.
Computing: The number 3.6×10−4951 is approximately equal to the smallest positive non-zero value that can be represented by a 80-bit x86 double-extended IEEE floating-point value.
Computing: The number 1×10−398 is equal to the smallest positive non-zero value that can be represented by a double-precision IEEE decimal 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 August 2009[update], are 195,249,053 to 1 against, for a probability of 5.12×10−9 (0.000000512%).
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 7.15×10−8 (0.00000715%).
BioMed – 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.
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.
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.
BioMed – base pairs in the genome: approximately 3×109base pairs in the human genome
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
Computing: 4,294,967,296 – the number of bytes in 4 gibibytes; in computation, the 32-bit computers can directly access 232 pieces of address space, this leads directly to the 4 gigabyte limit on main memory.
Astronomy – stars in our galaxy: approximately 4×1011 stars in the Milky Way galaxy (Often the figure 1×1011 is quoted in error; that is the total mass of the galaxy in solar masses. However, the total number of stars in the galaxy is 4×1011 because 73% of the stars in the galaxy are red dwarves, which have a much smaller mass than the Sun.)
BioMed – Bacteria on the human body: The surface of the human body houses roughly 1012bacteria.
Marine biology: 3,500,000,000,000 (3.5 × 1012) – estimated population of fish in the ocean.
Mathematics: 7,625,597,484,987 – a number that often appears when dealing with powers of 3. It can be expressed as , , , and 33 or when using Knuth's up-arrow notation it can be expressed as and .
Mathematics: 1013 – The approximate number of known non-trivial zeros of the Riemann zeta function as of 2004[update].
Mathematics – Known digits of π: As of 2013[update], the number of known digits of π is 12,100,000,000,000 (1.21×1013).
BioMed – 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.
BioMed-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 race).
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.
Cryptography: There are 7.205759×1016 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 largest number with more than one digit that can be written from base 2 to base 18 using only the digits 0 to 9.
BioMed – 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: In the legend called the Tower of Brahma about a Hindu temple which contains a large room with three posts on one of which is 64 golden discs, the object of the mathematical game is for the Brahmins in the 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. It would take 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019) turns to complete the task (same number as the wheat and chessboard problem above).
Mathematics – Rubik's Cube: There are 43,252,003,274,489,856,000 (≈4.33×1019) different positions of a 3x3x3 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.
Mathematics: = 170,141,183,460,469,231,731,687,303,715,884,105,727 (≈1.7×1038) is a 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).
Mathematics: 141×2141+1 = 393,050,634,124,102,232,869,567,034,555,427,371,542,904,833 (≈3.93×1044) is the second Cullen prime
Mathematics: There are 7,401,196,841,564,901,869,874,093,974,498,574,336,000,000,000 (≈7.4×1045) possible permutations for the Rubik's Revenge (4x4x4 Rubik's Cube).
Chess: 4.52×1046 is a proven upper bound for the number of legal chess positions.
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: 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 (5x5x5 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).
Mathematics: , order of magnitude of an upper bound that occurred in a proof of Skewes (this was later estimated to be closer to 1.397 × 10316).
Mathematics: , order of magnitude of another upper bound in a proof of Skewes.
Mathematics:Moser's number "2 in a mega-gon" is approximately equal to 10↑↑↑...↑↑↑10, where there are 10↑↑257 arrows, the last two digits are ...56.
Mathematics:Graham's number, the last ten digits of which are ...24641 95387. Arises as an upper bound solution to a problem in Ramsey theory. Representation in powers of 10 would be impractical (the number of digits in the exponent far exceeds the number of particles in the observable universe).
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.
^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.