# Orders of magnitude (numbers)

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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.

A number is a mathematical object used to count, measure, and also label. The original examples are the natural numbers 1, 2, 3, 4 and so forth. A notational symbol that represents a number is called a numeral. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, number may refer to a symbol, a word, or a mathematical abstraction.

In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned. Also known as a bare, pure, scalar or a quantity of dimension one.ed and the corresponding unit of measurement in the SI is one unit and it is not explicitly shown. Dimensionless quantities are widely used in many fields, such as mathematics, physics, chemistry, engineering, and economics. Examples of quantities to which dimensions are regularly assigned are length, time, and speed, which are measured in dimensional units, such as metre, second and metre per second. This is considered to aid intuitive understanding. However, especially in mathematical physics, it is often more convenient to drop the assignment of explicit dimensions and express the quantities without dimensions, e.g., addressing the speed of light simply by the dimensionless number 1.

Probability is the measure of the likelihood that an event will occur. See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking, 0 indicates impossibility and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.

## Smaller than 10−100 (one googolth)

• Mathematics – Numbers: The number zero is a natural, even number which quantifies a count or an amount of null size. [1]
• Mathematics – Writing: Approximately 10−183,800 is a rough first estimate of the probability that a monkey, placed in front of a typewriter, will perfectly type out William Shakespeare's play Hamlet on its first try. [2] However, taking punctuation, capitalization, and spacing into account, the actual probability is far lower: around 10−360,783. [3]
• 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.
• Computing: The number 4.9×10−324 is approximately equal to the smallest positive non-zero value that can be represented by a double-precision IEEE floating-point value.
• Computing: The number 1×10−101 is equal to the smallest positive non-zero value that can be represented by a single-precision IEEE decimal floating-point value.

Monkey is a common name that may refer to groups or species of mammals, in part, the simians of infraorder Simiiformes. The term is applied descriptively to groups of primates, such as families of new world monkeys and old world monkeys. Many monkey species are tree-dwelling (arboreal), although there are species that live primarily on the ground, such as baboons. Most species are also active during the day (diurnal). Monkeys are generally considered to be intelligent, especially the old world monkeys of Catarrhini.

The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would almost surely type every possible finite text an infinite number of times. However, the probability that monkeys filling the observable universe would type a complete work such as Shakespeare's Hamlet is so tiny that the chance of it occurring during a period of time hundreds of thousands of orders of magnitude longer than the age of the universe is extremely low.

The Tragedy of Hamlet, Prince of Denmark, often shortened to Hamlet, is a tragedy written by William Shakespeare sometime between 1599 and 1602. Set in Denmark, the play depicts Prince Hamlet and his revenge against his uncle, Claudius, who has murdered Hamlet's father in order to seize his throne and marry Hamlet's mother.

## 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 (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.

The standard 52-card deck of French playing cards is the most common deck of playing cards used today. It includes thirteen ranks in each of the four French suits: clubs, diamonds, hearts and spades, with reversible "court" or face cards. Each suit includes an ace, a king, queen and jack, each depicted with a symbol of its suit; and ranks two through ten, with each card depicting that many symbols (pips) of its suit. Anywhere from one to six jokers, often distinguishable with one being more colorful than the other, are added to commercial decks, as some card games require these extra cards. Modern playing cards carry index labels on opposite corners or in all four corners to facilitate identifying the cards when they overlap and so that they appear identical for players on opposite sides. The most popular standard pattern of the French deck is sometimes referred to as "English" or "Anglo-American" pattern.

In mathematics, the factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,

## 10−30

(0.000000000000000000000000000001; 1000−10; short scale: one nonillionth; long scale: one quintillionth)

The long and short scales are two of several large-number naming systems for integer powers of ten that use the same words with different meanings. The long scale is based on powers of one million, whereas the short scale is based on powers of one thousand.

• Mathematics: The probability in a game of bridge of all four players getting a complete suit each is approximately 4.47×10−28. [5]

Contract bridge, or simply bridge, is a trick-taking card game using a standard 52-card deck. In its basic format, it is played by four players in two competing partnerships, with partners sitting opposite each other around a table. Millions of people play bridge worldwide in clubs, tournaments, online and with friends at home, making it one of the world's most popular card games, particularly among seniors. The World Bridge Federation (WBF) is the governing body for international competitive bridge, with numerous other bodies governing bridge at the regional level.

## 10−27

(0.000000000000000000000000001; 1000−9; short scale: one octillionth; long scale: one quadrilliardth)

## 10−24

(0.000000000000000000000001; 1000−8; short scale: one septillionth; long scale: one quadrillionth)

ISO: yocto- (y)

## 10−21

(0.000000000000000000001; 1000−7; short scale: one sextillionth; long scale: one trilliardth)

ISO: zepto- (z)

• Mathematics: The probability of matching 20 numbers for 20 in a game of keno is approximately 2.83 × 10−19.

## 10−18

(0.000000000000000001; 1000−6; short scale: one quintillionth; long scale: one trillionth)

ISO: atto- (a)

• Mathematics: The probability of rolling snake eyes 10 times in a row on a pair of fair dice is about 2.74×10−16.

## 10−15

(0.000000000000001; 1000−5; short scale: one quadrillionth; long scale: one billiardth)

ISO: femto- (f)

## 10−12

(0.000000000001; 1000−4; short scale: one trillionth; long scale: one billionth)

ISO: pico- (p)

## 10−9

(0.000000001; 1000−3; short scale: one billionth; long scale: one milliardth)

ISO: nano- (n)

• 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 January 2014, are 175,223,510 to 1 against, for a probability of 5.707×10−9 (0.0000005707%).
• 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 March 2013, are 76,767,600 to 1 against, for a probability of 1.303×10−8 (0.000001303%).
• 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%).

## 10−6

(0.000001; 1000−2; long and short scales: one millionth)

ISO: micro- (μ)

• Mathematics Poker: The odds of being dealt a royal flush in poker are 649,739 to 1 against, for a probability of 1.5×106 (0.00015%). [7]
• 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 1.4×105 (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 ×104 (0.024%).

## 10−3

(0.001; 1000−1; one thousandth)

ISO: milli- (m)

• Mathematics – Poker: The odds of being dealt a full house in poker are 693 to 1 against, for a probability of 1.4 × 10−3 (0.14%).
• Mathematics – Poker: The odds of being dealt a flush in poker are 507.8 to 1 against, for a probability of 1.9 × 10−3 (0.19%).
• Mathematics – Poker: The odds of being dealt a straight in poker are 253.8 to 1 against, for a probability of 4 × 10−3 (0.39%).
• Physics:α = 0.007297352570(5), the fine-structure constant.

## 10−2

(0.01; one hundredth)

ISO: centi- (c)

• 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 2006, are 36.61 to 1 against, for a probability of 0.027 (2.7%).
• Mathematics – Poker: The odds of being dealt two pair in poker are 20 to 1 against, for a probability of 0.048 (4.8%).

## 10−1

(0.1; one tenth)

ISO: deci- (d)

• Legal history: 10% was widespread as the tax raised for income or produce in the ancient and medieval period; see tithe.
• Mathematics – Poker: The odds of being dealt only one pair in poker are about 5 to 2 against (2.37 to 1), for a probability of 0.42 (42%).
• Mathematics – Poker: The odds of being dealt no pair in poker are nearly 1 to 2, for a probability of about 0.5 (50%).

(1; one)

(10; ten)

ISO: deca- (da)

(100; hundred)

ISO: hecto- (h)

(1000; thousand)

ISO: kilo- (k)

## 104

(10000; ten thousand or a myriad)

## 105

(100000; one hundred thousand or a lakh).

## 106

(1000000; 10002; long and short scales: one million)

ISO: mega- (M)

• Demography: The population of Riga, Latvia was 1,003,949 in 2004, according to Eurostat.
• 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.
• Genocide: Approximately 800,000–1,500,000 (1.5 million) Armenians were killed in the Armenian Genocide.
• Info: The freedb database of CD track listings has around 1,750,000 entries as of June 2005.
• War: 1,857,619 casualties at the Battle of Stalingrad.
• 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).
• Info – Web sites: As of April9, 2019, Wikipedia contains approximately 5,837,000 articles in the English language.
• 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.
• Genocide: Approximately 5,100,000–6,200,000 Jews were killed in the Holocaust.

## 107

(10000000; a crore; long and short scales: ten million)

## 108

(100000000; long and short scales: one hundred million)

## 109

(1000000000; 10003; short scale: one billion; long scale: one thousand million, or one milliard)

ISO: giga- (G)

• Demography: The population of Africa reached 1,000,000,000 sometime in 2009.
• Demographics – India: 1,359,000,000 – approximate population of India in 2018.
• Demographics – China: 1,417,000,000 – approximate population of the People's Republic of China in 2018.
• Internet: Approximately 1,500,000,000 active users were on Facebook as of October 2015. [13]
• 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.
• Biology – base pairs in the genome: approximately 3×109 base pairs in the human genome. [9]
• 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 IPv4: 4,294,967,296 (232) possible unique IP addresses.
• 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.
• Mathematics: 4,294,967,297 is a Fermat number and semiprime. It is the smallest number of the form ${\displaystyle 2^{2^{n}}+1}$ which is not a prime number.
• Demographics world population: 7,650,000,000 – Estimated population for the world as of October 2018.

## 1010

(10000000000; short scale: ten billion; long scale: ten thousand million, or ten milliard)

## 1011

(100000000000; short scale: one hundred billion; long scale: hundred thousand million, or hundred milliard)

## 1012

(1000000000000; 10004; short scale: one trillion; long scale: one billion)

ISO: tera- (T)

• Astronomy: Andromeda Galaxy, which is part of the same Local Group as our galaxy, contains about 1012 stars.
• Biology – Bacteria on the human body: The surface of the human body houses roughly 1012 bacteria. [14]
• Wikipedia: 1.9786782 × 1012 is a rough estimate of the total number of links on Wikipedia.[ citation needed ]
• Astronomy – Galaxies: A 2016 estimate says there are 2 × 1012 galaxies in the observable universe. [18]
• Biology: An estimate says there were 3.04 × 1012 trees on Earth in 2015. [19]
• 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 ${\displaystyle 19683^{3}}$, ${\displaystyle 27^{9}}$, ${\displaystyle 3^{27}}$, ${\displaystyle 3^{3^{3}}}$ and 33 or when using Knuth's up-arrow notation it can be expressed as ${\displaystyle 3\uparrow \uparrow 3}$ and ${\displaystyle 3\uparrow \uparrow \uparrow 2}$.
• Mathematics: 1013 – The approximate number of known non-trivial zeros of the Riemann zeta function as of 2004. [20]
• Mathematics – Known digits of π:As of 2013, the number of known digits of π is 12,100,000,000,000 (1.21×1013). [21]
• Biology – approximately 1014 synapses in the human brain. [22]
• Biology – Cells in the human body: The human body consists of roughly 1014 cells, of which only 1013 are human. [23] [24] 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 configuration of the plug-board of the Enigma machine used by the Germans in WW2 to encode and decode messages by cipher.
• Computing MAC-48: 281,474,976,710,656 (248) possible unique physical addresses.
• Mathematics: 953,467,954,114,363 is the largest known Motzkin prime.

## 1015

(1000000000000000; 10005; short scale: one quadrillion; long scale: one thousand billion, or one billiard)

ISO: peta- (P)

• 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 race). [25]
• Computing: 9,007,199,254,740,992 (253) – 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.
• Cryptography: There are 7.205759×1016 different possible keys in the obsolete 56-bit DES symmetric cipher.

## 1018

(1000000000000000000; 10006; short scale: one quintillion; long scale: one trillion)

ISO: exa- (E)

• Mathematics: Goldbach's conjecture has been verified for all n ≤ 4×1018; that is, all prime numbers up to that value at least have been computed, but not necessarily stored.
• Computing – Manufacturing: An estimated 6×1018 transistors were produced worldwide in 2008. [26]
• 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 NCAA Basketball Tournament: There are 9,223,372,036,854,775,808 (263) possible ways to enter the bracket.
• 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. [27]
• Biology – Insects: It has been estimated that the insect population of the Earth is about 1019. [28]
• 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, moving only one at a time. 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). [29]
• 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.
• Economics: Hyperinflation in Zimbabwe estimated in February 2009 by some economists at 10 sextillion percent, [30] or a factor of 1020

## 1021

(1000000000000000000000; 10007; short scale: one sextillion; long scale: one thousand trillion, or one trilliard)

ISO: zetta- (Z)

• Geo – Grains of sand: All the world's beaches combined have been estimated to hold roughly 1021 grains of sand. [31]
• Computing – Manufacturing: Intel predicted that there would be 1.2×1021 transistors in the world by 2015 [32] and Forbes estimated that 2.9×1021 transistors had been shipped up to 2014. [33]
• Mathematics – Sudoku: There are 6,670,903,752,021,072,936,960 (≈6.7×1021) 9×9 sudoku grids. [34]
• Astronomy – Stars: 70 sextillion = 7×1022, the estimated number of stars within range of telescopes (as of 2003). [35]
• Astronomy – Stars: in the range of 1023 to 1024 stars in the observable universe. [36]
• Mathematics: 146,361,946,186,458,562,560,000 (≈1.5×1023) is the fifth unitary perfect number.
• Chemistry – Physics: Avogadro constant (≈6×1023) 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.

## 1024

(1000000000000000000000000; 10008; short scale: one septillion; long scale: one quadrillion)

ISO: yotta- (Y)

• Mathematics: 2,833,419,889,721,787,128,217,599 (≈2.8×1024) is a Woodall prime.
• Mathematics: 286 = 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. [37]

## 1027

(1000000000000000000000000000; 10009; short scale: one octillion; long scale: one thousand quadrillion, or one quadrilliard)

• Biology – Atoms in the human body: the average human body contains roughly 7×1027 atoms. [38]
• Mathematics – Poker: the number of unique combinations of hands and shared cards in a 10-player game of Texas Hold'em is approximately 2.117×1028; see Poker probability (Texas hold 'em).

## 1030

(1000000000000000000000000000000; 100010; short scale: one nonillion; long scale: one quintillion)

• Biology – Bacterial cells on Earth: The number of bacterial cells on Earth is estimated at around 5,000,000,000,000,000,000,000,000,000,000, or 5 × 1030. [39]
• Mathematics: The number of partitions of 1000 is 24,061,467,864,032,622,473,692,149,727,991. [40]
• Mathematics: 2108 = 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. [41]

## 1033

(1000000000000000000000000000000000; 100011; 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 7.24×1034) different positions of Alexander's Star.

## 1036

(1000000000000000000000000000000000000; 100012; short scale: one undecillion; long scale: one sextillion)

• Physics: ke e 2 / G m2, the ratio of the electromagnetic to the gravitational forces between two protons, is roughly 1036.
• Mathematics:${\displaystyle 2^{2^{7}-1}-1}$ = 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).

## 1039

(1000000000000000000000000000000000000000; 100013; short scale: one duodecillion; long scale: one thousand sextillion, or one sextilliard)

## 1042 to 10100

(1000000000000000000000000000000000000000000; 100014; short scale: one tredecillion; long scale: one septillion)

• 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 (4×4×4 Rubik's Cube).
• Chess: 4.52×1046 is a proven upper bound for the number of legal chess positions. [42]
• Geo: 1.33×1050 is the estimated number of atoms in the Earth.
• 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).
• Cosmology: 8×1060 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. [43]
• Cosmology: 1×1063 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 (≈8.07×1067) – 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. [44]
• 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).
• Cosmology: Various sources estimate the total number of fundamental particles in the observable universe to be within the range of 1080 to 1085. [45] [46] However, these estimates are generally regarded as guesswork. (Compare the Eddington number, the estimated total number of protons in the observable universe.)
• Computing: 9.999 999×1096 is equal to the largest value that can be represented in the IEEE decimal32 floating-point format.
• 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.

## 10100 (one googol) to 1010100 (one googolplex)

(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; 100033; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard) [47]

• Mathematics: There are 157 152 858 401 024 063 281 013 959 519 483 771 508 510 790 313 968 742 344 694 684 829 502 629 887 168 573 442 107 637 760 000 000 000 000 000 000 000 000 (≈1.57×10116) distinguishable permutations of the V-Cube 6 (6×6×6 Rubik's Cube).
• Chess: Shannon number, 10120, an estimation of the game-tree complexity of chess.
• Physics: 10120, the orders of magnitude of the vacuum catastrophe, the observed values of the quantum vacuum versus the values calculated by Quantum Field Theory.
• Physics: 8×10120, ratio of the mass-energy in the observable universe to the energy of a photon with a wavelength the size of the observable universe.
• History – Religion: Asaṃkhyeya is a Buddhist name for the number 10140. It is listed in the Avatamsaka Sutra and metaphorically means "innumerable" in the Sanskrit language of ancient India.
• Xiangqi: 10150, an estimation of the game-tree complexity of xiangqi.
• Mathematics: There are 19 500 551 183 731 307 835 329 126 754 019 748 794 904 992 692 043 434 567 152 132 912 323 232 706 135 469 180 065 278 712 755 853 360 682 328 551 719 137 311 299 993 600 000 000 000 000 000 000 000 000 000 000 000 (≈1.95×10160) distinguishable permutations of the V-Cube 7 (7×7×7 Rubik's Cube).
• 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 (≈2.08×10170) legal positions in the game of Go. See Go and mathematics.
• Board games: 3.457×10181, number of ways to arrange the tiles in English Scrabble on a standard 15-by-15 Scrabble board.
• Physics: 10186, approximate number of Planck volumes in the observable universe.
• Physics: 7×10245, approximate number of Planck units that have ever existed in the observable universe. [48]
• Computing: 1.797 693 134 862 315 807×10308 is approximately equal to the largest value that can be represented in the IEEE double precision floating-point format.
• Go: 10365, an estimation of the game-tree complexity in the game of Go. [ citation needed ]
• Computing: (10 – 10−15)×10384 is equal to the largest value that can be represented in the IEEE decimal64 floating-point format.
• Mathematics: There are approximately 1.869×104099 distinguishable permutations of the world's largest Rubik's cube (33×33×33).
• Computing: 1.189 731 495 357 231 765 05×104932 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 0×104932 is approximately equal to the largest value that can be represented in the IEEE quadruple precision floating-point format.
• Computing: (10 – 10−33)×106144 is equal to the largest value that can be represented in the IEEE decimal128 floating-point format.
• Computing: 1010,000 − 1 is equal to the largest value that can be represented in Windows Phone's calculator.
• Mathematics: 26384405 + 44052638 is a 15,071-digit Leyland prime; the largest which has been proven as of 2010. [49]
• Mathematics: 3,756,801,695,685 × 2666,669 ± 1 are 200,700-digit twin primes; the largest known as of December 2011. [50]
• Mathematics: 18,543,637,900,515 × 2666,667 − 1 is a 200,701-digit Sophie Germain prime; the largest known as of April 2012. [51]
• Mathematics: approximately 7.76 × 10206,544 cattle in the smallest herd which satisfies the conditions of Archimedes' cattle problem.
• Mathematics: 10290,253 – 2 × 10145,126 + 1 is a 290,253-digit palindromic prime, the largest known as of April 2012. [52]
• Mathematics: 1,098,133# – 1 is a 476,311-digit primorial prime; the largest known as of March 2012. [53]
• Mathematics: 150,209! + 1 is a 712,355-digit factorial prime; the largest known as of August 2013. [54]
• Mathematics – Literature: Jorge Luis Borges' Library of Babel contains at least 251,312,000 ≈ 1.956 × 101,834,097 books (this is a lower bound). [55]
• Mathematics: 475,856524,288 + 1 is a 2,976,633-digit Generalized Fermat prime, the largest known as of December 2012. [56]
• Mathematics: 19,249 × 213,018,586 + 1 is a 3,918,990-digit Proth prime, the largest known Proth prime [57] and non-Mersenne prime as of 2010. [58]
• Mathematics: 277,232,917 − 1 is a 23,249,425-digit Mersenne prime; the largest known prime of any kind as of 2018. [58]
• Mathematics: 277,232,916 × (277,232,917  1) is a 46,498,850-digit perfect number, the largest known as of 2018. [59]
• Mathematics – History: 1080,000,000,000,000,000, largest named number in Archimedes' Sand Reckoner .
• Mathematics: 10googol (${\displaystyle 10^{10^{100}}}$), 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 1010100. [60]

## Larger than 1010100

(One googolplex; 10googol; short scale: googolplex; long scale: googolplex)

• Cosmology: The highest estimated time for the Big Freeze to occur is about in 2×1010120 years.
• Mathematics–Literature: The number of different ways in which the books in Jorge Luis Borges' Library of Babel can be arranged is ${\displaystyle 10^{10^{1,834,102}}}$, 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 ${\displaystyle 10^{10^{10,000,000}}}$. [61]
• Mathematics:${\displaystyle 10^{\,\!10^{10^{34}}}}$, order of magnitude of an upper bound that occurred in a proof of Skewes (this was later estimated to be closer to 1.397 × 10316).
• Cosmology: The estimated number of years for quantum fluctuations and tunnelling to generate a new Big Bang is estimated to be ${\displaystyle 10^{10^{10^{56}}}}$.
• Mathematics:${\displaystyle 10^{\,\!10^{10^{100}}}}$, a number in the googol family called a googolplexplex, googolplexian, or googolduplex. 1 followed by a googolplex zeros, or 10googolplex
• Mathematics:${\displaystyle 10^{\,\!10^{10^{963}}}}$, 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 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 ${\displaystyle 10^{\,\!10^{10^{...}}}}$ 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 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 TREE(TREE(…TREE(3)…)) (the TREE function nested TREE(3) times with TREE(3) at the bottom).

## Related Research Articles

The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation so as to support a trade-off between range and precision. For this reason, floating-point computation is often found in systems which include very small and very large real numbers, which require fast processing times. A number is, in general, represented approximately to a fixed number of significant digits and scaled using an exponent in some fixed base; the base for the scaling is normally two, ten, or sixteen. A number that can be represented exactly is of the following form:

In mathematics, the logarithm is the inverse function to exponentiation (it is an example of a concave function). That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. In the simplest case, the logarithm counts repeated multiplication of the same factor; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm to base 10" of 1000 is 3. The logarithm of x to baseb is denoted as logb (x) (or, without parentheses, as logbx, or even without explicit base as log x, when no confusion is possible). More generally, exponentiation allows any positive real number to be raised to any real power, always producing a positive result, so the logarithm for any two positive real numbers b and x where b is not equal to 1, is always a unique real number y. More explicitly, the defining relation between exponentiation and logarithm is:

In poker, pot odds are the ratio of the current size of the pot to the cost of a contemplated call. Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value.

The number π is a mathematical constant. Originally defined as the ratio of a circle's circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, though it is also sometimes spelled out as "pi". It is also called Archimedes' constant.

Benford's law, also called the Newcomb–Benford law, the law of anomalous numbers, or the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. For example, in sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. If the digits were distributed uniformly, they would each occur about 11.1% of the time. Benford's law also makes predictions about the distribution of second digits, third digits, digit combinations, and so on.

In poker, the probability of each type of 5-card hand can be computed by calculating the proportion of hands of that type among all possible hands.

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Odds are a numerical expression, usually expressed as a pair of numbers, used in both gambling and statistics. In statistics, the odds for or odds of some event reflect the likelihood that the event will take place, while odds against reflect the likelihood that it will not. In gambling, the odds are the ratio of payoff to stake, and do not necessarily reflect exactly the probabilities. Odds are expressed in several ways, and sometimes the term is used incorrectly to mean simply the probability of an event. Conventionally, gambling odds are expressed in the form "X to Y", where X and Y are numbers, and it is implied that the odds are odds against the event on which the gambler is considering wagering. In both gambling and statistics, the 'odds' are a numerical expression of the likelihood of some possible event.

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An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently, the ratio of the odds of B in the presence of A and the odds of B in the absence of A. Two events are independent if and only if the OR equals 1: the odds of one event are the same in either the presence or absence of the other event. If the OR is greater than 1, then A and B are associated (correlated) in the sense that, compared to the absence of B, the presence of B raises the odds of A, and symmetrically the presence of A raises the odds of B. Conversely, if the OR is less than 1, then A and B are negatively correlated, and the presence of one event reduces the odds of the other event. The OR plays an important role in the logistic model, which generalizes beyond two events.

The mathematics of gambling are a collection of probability applications encountered in games of chance and can be included in game theory. From a mathematical point of view, the games of chance are experiments generating various types of aleatory events, the probability of which can be calculated by using the properties of probability on a finite space of events.

Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.

The hartley, also called a ban, or a dit, is a logarithmic unit which measures information or entropy, based on base 10 logarithms and powers of 10, rather than the powers of 2 and base 2 logarithms which define the bit, or shannon. One ban or hartley is the information content of an event if the probability of that event occurring is ​110. It is therefore equal to the information contained in one decimal digit, assuming a priori equiprobability of each possible value.

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## References

1. Eric W. Weisstein. "Zero". MathWorld. Retrieved December 10, 2018.
2. Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 53. ISBN   978-0-7167-1088-2.
3. There are around 130,000 letters and 199,749 total characters in Hamlet; 26 letters ×2 for capitalization, 12 for punctuation characters = 64, 64199749 10360,783.
4. Robert Matthews. "What are the odds of shuffling a deck of cards into the right order?". Science Focus. Retrieved December 10, 2018.
5. www.BridgeHands.com, Sales. "Probabilities Miscellaneous: Bridge Odds". Archived from the original on 2009-10-03.
6. Walraven, P. L.; Lebeek, H. J. (1963). "Foveal Sensitivity of the Human Eye in the Near Infrared". J. Opt. Soc. Am. 53 (6): 765–766. doi:10.1364/josa.53.000765.
7. Courtney Taylor. "The Probability of Being Dealt a Royal Flush in Poker". ThoughtCo. Retrieved December 10, 2018.
8. Mason, W S; Seal, G; Summers, J (1980-12-01). "Virus of Pekin ducks with structural and biological relatedness to human hepatitis B virus". Journal of Virology. 36 (3): 829–836. ISSN   0022-538X. PMC  . PMID   7463557.
9. "Homo sapiens – Ensembl genome browser 87". www.ensembl.org. Archived from the original on 2017-05-25. Retrieved 2017-01-28.
10. "Pi World Ranking List". Archived from the original on 2017-06-29.
11. Sloane, N. J. A. (ed.). "SequenceA283670". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2017-03-15.
12. Plouffe's Inverter Archived 2005-08-12 at the Wayback Machine
13. Christof Baron (2015). "Facebook users worldwide 2016 | Statista". Statista. statista.com. Archived from the original on 2016-09-09.
14. "Earth microbes on the moon". Science@Nasa. 1 September 1998. Archived from the original on 23 March 2010. Retrieved 2 November 2010.
15. "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. PMC  . PMID   19915731.
16. 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).
17. Elizabeth Howell, How Many Stars Are in the Milky Way? Archived 2016-05-28 at the Wayback Machine , Space.com, 21 May 2014 (citing estimates from 100 to 400 billion).
18. Hollis, Morgan (13 October 2016). "A universe of two trillion galaxies". The Royal Astronomical Society. Retrieved 9 November 2017.
19. Jonathan Amos (3 September 2015). "Earth's trees number 'three trillion'". BBC. Archived from the original on 6 June 2017.
20. Xavier Gourdon (October 2004). "Computation of zeros of the Zeta function". Archived from the original on 15 January 2011. Retrieved 2 November 2010.
21. Alexander J. Yee & Shigeru Kondo (28 Dec 2013). "12.1 Trillion Digits of Pi". Archived from the original on 2014-02-21. Retrieved 17 Feb 2014.
22. Koch, Christof. Biophysics of computation: information processing in single neurons. Oxford university press, 2004.
23. Savage, D. C. (1977). "Microbial Ecology of the Gastrointestinal Tract". Annual Review of Microbiology. 31: 107–33. doi:10.1146/annurev.mi.31.100177.000543. PMID   334036.
24. Berg, R. (1996). "The indigenous gastrointestinal microflora". Trends in Microbiology. 4 (11): 430–5. doi:10.1016/0966-842X(96)10057-3. PMID   8950812.
25. Bert Holldobler and E.O. Wilson The Superorganism: The Beauty, Elegance, and Strangeness of Insect Societies New York:2009 W.W. Norton Page 5
26. "60th Birthday of Microelectronics Industry". Semiconductor Industry Association. 13 December 2007. Archived from the original on 13 October 2008. Retrieved 2 November 2010.
27. Sequence A131646 Archived 2011-09-01 at the Wayback Machine in The On-Line Encyclopedia of Integer Sequences
28. "Smithsonian Encyclopedia: Number of Insects Archived 2016-12-28 at the Wayback Machine ". Prepared by the Department of Systematic Biology, Entomology Section, National Museum of Natural History, in cooperation with Public Inquiry Services, Smithsonian Institution. Accessed 27 December 2016. Facts about numbers of insects. Puts the number of individual insects on Earth at about 10 quintillion (1019).
29. Ivan Moscovich, 1000 playthinks: puzzles, paradoxes, illusions & games, Workman Pub., 2001 ISBN   0-7611-1826-8 .
30. "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. Archived from the original on 1 March 2009. Retrieved 14 March 2009.
31. "To see the Universe in a Grain of Taranaki Sand". Archived from the original on 2012-06-30.
32. "Intel predicts 1,200 quintillion transistors in the world by 2015". Archived from the original on 2013-04-05.
33. "How Many Transistors Have Ever Shipped? – Forbes". Archived from the original on 30 June 2015. Retrieved 1 September 2015.
34. "Sudoku enumeration". Archived from the original on 2006-10-06.
35. "Star count: ANU astronomer makes best yet". The Australian National University. 17 July 2003. Archived from the original on July 24, 2005. Retrieved 2 November 2010.
36. "Astronomers count the stars". BBC News. July 22, 2003. Archived from the original on August 13, 2006. Retrieved 2006-07-18. "trillions-of-earths-could-be-orbiting-300-sextillion-stars" van Dokkum, Pieter G.; Charlie Conroy (2010). "A substantial population of low-mass stars in luminous elliptical galaxies". Nature. 468 (7326): 940–942. arXiv:. Bibcode:2010Natur.468..940V. doi:10.1038/nature09578. PMID   21124316. "How many stars?" Archived 2013-01-22 at the Wayback Machine ; see mass of the observable universe
37. (sequence in the OEIS )
38. "Questions and Answers – How many atoms are in the human body?". Archived from the original on 2003-10-06.
39. William B. Whitman; David C. Coleman; William J. Wiebe (1998). "Prokaryotes: The unseen majority". Proceedings of the National Academy of Sciences of the United States of America. 95 (12): 6578–6583. Bibcode:1998PNAS...95.6578W. doi:10.1073/pnas.95.12.6578. PMC  . PMID   9618454.
40. (sequence in the OEIS )
41. (sequence in the OEIS )
42. John Tromp (2010). "John's Chess Playground". Archived from the original on 2014-06-01.
43. Planck Collaboration (2016). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". Astronomy & Astrophysics. 594: A13. arXiv:. Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830.
44. Paul Zimmermann, "50 largest factors found by ECM Archived 2009-02-20 at the Wayback Machine ".
45. Matthew Champion, "Re: How many atoms make up the universe?" Archived 2012-05-11 at the Wayback Machine , 1998
46. WMAP- Content of the Universe Archived 2016-07-26 at the Wayback Machine . Map.gsfc.nasa.gov (2010-04-16). Retrieved on 2011-05-01.
47. "Names of large and small numbers". bmanolov.free.fr. Miscellaneous pages by Borislav Manolov. Archived from the original on 2016-09-30.
48. Chris Caldwell, The Top Twenty: Elliptic Curve Primality Proof at The Prime Pages.
49. Chris Caldwell, The Top Twenty: Twin Primes Archived 2013-01-27 at the Wayback Machine at The Prime Pages.
50. Chris Caldwell, The Top Twenty: Sophie Germain (p) at The Prime Pages.
51. Chris Caldwell, The Top Twenty: Palindrome at The Prime Pages.
52. PrimeGrid's Primorial Prime Search Archived 2013-11-26 at the Wayback Machine
53. Chris Caldwell, The Top Twenty: Factorial primes Archived 2013-04-10 at the Wayback Machine at The Prime Pages.
54. 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.
55. Chris Caldwell, The Top Twenty: Generalized Fermat Archived 2014-12-23 at the Wayback Machine at The Prime Pages.
56. Chris Caldwell, The Top Twenty: Proth at The Prime Pages.
57. Chris Caldwell, The Top Twenty: Largest Known Primes at The Prime Pages.
58. Chris Caldwell, Mersenne Primes: History, Theorems and Lists at The Prime Pages.
59. asantos (15 December 2007). "Googol and Googolplex by Carl Sagan" via YouTube.
60. Zyga, Lisa "Physicists Calculate Number of Parallel Universes" Archived 2011-06-06 at the Wayback Machine , PhysOrg , 16 October 2009.