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.
The logarithmic scale can compactly represent the relationship among variously sized numbers.
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.
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%).
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.
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: 3,400,000,000 – the total number of speakers of Indo-European languages, of which 2,400,000,000 are native speakers; the other 1,000,000,000 speak Indo-European languages as a second language.
Mathematics and computing: 4,294,967,295 (232 − 1), the product of the five known Fermat primes and the maximum value for a 32-bit unsigned integer in computing.
Computing: 4,294,967,296 – the number of bytes in 4 gibibytes; in computation, 32-bit computers can directly access 232 units (bytes) of address space, which leads directly to the 4-gigabyte limit on main memory.
Mathematics – Known digits of π: As of 2013, the number of known digits of π is 12,100,000,000,000 (1.21×1013).
Biology – approximately 1014synapses in the human brain.
Biology – Cells in the human body: The human body consists of roughly 1014cells, of which only 1013 are human. The remaining 90% non-human cells (though much smaller and constituting much less mass) are bacteria, which mostly reside in the gastrointestinal tract, although the skin is also covered in bacteria.
Cryptography: 150,738,274,937,250 configuration of the plug-board of the Enigma machine used by the Germans in WW2 to encode and decode messages by cipher.
Biology-Insects: 1,000,000,000,000,000 to 10,000,000,000,000,000 (1015 to 1016) – The estimated total number of ants on Earth alive at any one time (their biomass is approximately equal to the total biomass of the human 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.
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 6×1018transistors were produced worldwide in 2008.
Computing – Computational limit of a 64-bit CPU: 9,223,372,036,854,775,807 (about 9.22×1018) is equal to 263−1, and as such is the largest number which can fit into a signed (two's complement) 64-bit integer on a computer.
Mathematics – Bases: 9,439,829,801,208,141,318 (≈9.44×1018) is the 10th and (by conjecture) largest number with more than one digit that can be written from base 2 to base 18 using only the digits 0 to 9.
Biology – Insects: It has been estimated that the insect population of the Earth is about 1019.
Mathematics – Answer to the wheat and chessboard problem: When doubling the grains of wheat on each successive square of a chessboard, beginning with one grain of wheat on the first square, the final number of grains of wheat on all 64 squares of the chessboard when added up is 264−1 = 18,446,744,073,709,551,615 (≈1.84×1019).
Mathematics – Legends: 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).
Mathematics – Rubik's Cube: There are 43,252,003,274,489,856,000 (≈4.33×1019) different positions of a 3×3×3 Rubik's Cube.
Password strength: Usage of the 95-character set found on standard computer keyboards for a 10-character password yields a computationally intractable 59,873,693,923,837,890,625 (9510, approximately 5.99×1019) permutations.
Chemistry – Physics: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.
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 (4×4×4 Rubik's Cube).
Chess: 4.52×1046 is a proven upper bound for the number of legal chess positions.
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).
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.
Mathematics: There are 282,870,942,277,741,856,536,180,333,107,150,328,293,127,731,985,672,134,721,536,000,000,000,000,000 (≈2.83×1074) possible permutations for the Professor's Cube (5×5×5 Rubik's Cube).
Cryptography: 2256 = 115,792,089,237,316,195,423,570,985,008,687,907,853,269,984,665,640,564,039,457,584,007,913,129,639,936 (≈1.15792089×1077), the total number of different possible keys in the AES 256-bit key space (symmetric cipher).
Computing: 69! (roughly 1.7112245×1098), is the highest factorial value that can be represented on a calculator with two digits for powers of ten without overflow.
Mathematics: One googol, 1×10100, 1 followed by one hundred zeros, or 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.
(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; 100033; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard)
Mathematics: 10googol (), a googolplex. A number 1 followed by 1 googol zeros. Carl Sagan has estimated that 1 googolplex, fully written out, would not fit in the observable universe because of its size, while also noting that one could also write the number as 1010100.
Mathematics:, a number in the googol family called a googolplexplex, googolplexian, or googolduplex. 1 followed by a googolplex zeros, or 10googolplex
Mathematics:, order of magnitude of another upper bound in a proof of Skewes.
Mathematics: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 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).
^"there was, to our knowledge, no actual, direct estimate of numbers of cells or of neurons in the entire human brain to be cited until 2009. A reasonable approximation was provided by Williams and Herrup (1988), from the compilation of partial numbers in the literature. These authors estimated the number of neurons in the human brain at about 85 billion [...] With more recent estimates of 21–26 billion neurons in the cerebral cortex (Pelvig et al., 2008 ) and 101 billion neurons in the cerebellum (Andersen et al., 1992 ), however, the total number of neurons in the human brain would increase to over 120 billion neurons." Herculano-Houzel, Suzana (2009). "The human brain in numbers: a linearly scaled-up primate brain". Front. Hum. Neurosci. 3: 31. doi:10.3389/neuro.09.031.2009. PMC2776484. PMID19915731.
^Kapitsa, Sergei P (1996). "The phenomenological theory of world population growth". Physics-Uspekhi. 39 (1): 57–71. Bibcode:1996PhyU...39...57K. doi:10.1070/pu1996v039n01abeh000127. (citing the range of 80 to 150 billion, citing K. M. Weiss, Human Biology 56637, 1984, and N. Keyfitz, Applied Mathematical Demography, New York: Wiley, 1977). C. Haub, "How Many People Have Ever Lived on Earth?", Population Today 23.2), pp. 5–6, cited an estimate of 105 billion births since 50,000 BC, updated to 107 billion as of 2011 in Haub, Carl (October 2011). "How Many People Have Ever Lived on Earth?". Population Reference Bureau. Archived from the original on April 24, 2013. Retrieved April 29, 2013. (due to the high infant mortality in pre-modern times, close to half of this number would not have lived past infancy).
^From the third paragraph of the story: "Each book contains 410 pages; each page, 40 lines; each line, about 80 black letters." That makes 410 x 40 x 80 = 1,312,000 characters. The fifth paragraph tells us that "there are 25 orthographic symbols" including spaces and punctuation. The magnitude of the resulting number is found by taking logarithms. However, this calculation only gives a lower bound on the number of books as it does not take into account variations in the titles – the narrator does not specify a limit on the number of characters on the spine. For further discussion of this, see Bloch, William Goldbloom. The Unimaginable Mathematics of Borges' Library of Babel. Oxford University Press: Oxford, 2008.
1,000,000 (one million), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione (milione in modern Italian), from mille, "thousand", plus the augmentative suffix -one. It is commonly abbreviated as m (not to be confused with the metric prefix for 1×10−3) or M; further MM ("thousand thousands", from Latin "Mille"; not to be confused with the Roman numeral MM = 2,000), mm, or mn in financial contexts.In scientific notation, it is written as 1×106 or 106. Physical quantities can also be expressed using the SI prefix mega (M), when dealing with SI units; for example, 1 megawatt (1 MW) equals 1,000,000 watts.
The meaning of the word "million" is common to the short scale and long scale numbering systems, unlike the larger numbers, which have different names in the two systems.
The million is sometimes used in the English language as a metaphor for a very large number, as in "Not in a million years" and "You're one in a million", or a hyperbole, as in "I've walked a million miles" and "You've asked the million-dollar question".
Morris Muskat et al. developed the governing equations for multiphase flow (one vector equation for each fluid phase) in porous media as a generalisation of Darcy's equation (or Darcy's law) for water flow in porous media. The porous media are usually sedimentary rocks such as clastic rocks (mostly sandstone) or carbonate rocks.
where a = w, o, g
The present fluid phases are water, oil and gas, and they are represented by the subscript a = w,o,g respectively. The gravitational acceleration with direction is represented as or or . Notice that in petroleum engineering the spacial co-ordinate system is right-hand-oriented with z-axis pointing downward. The physical property that links the flow equations of the three fluid phases, is relative permeability of each fluid phase and pressure. This property of the fluid-rock system (i.e. water-oil-gas-rock system) is mainly a function of the fluid saturations, and it is linked to capillary pressure and the flowing process, implying that it is subject to hysteresis effect.
In 1940 M.C. Leverett pointed out that in order to include capillary pressure effects in the flow equation, the pressure must be phase dependent. The flow equation then becomes
where a = w, o, g
Leverett also pointed out that the capillary pressure shows significant hysteresis effects. This means that the capillary pressure for a drainage process is different from the capillary pressure of an imbibition process with the same fluid phases. Hysteresis does not change the shape of the governing flow equation, but it increases (usually doubles) the number of constitutive equations for properties involved in the hysteresis.
During 1951-1970 commercial computers entered the scene of scientific and engineering calculations and model simulations. Computer simulation of the dynamic behaviour of oil reservoirs soon became a target for the petroleum industry, but the computing power was very weak at that time.
With weak computing power, the reservoir models were correspondingly coarse, but upscaling of the static parameters were fairly simple and partly compensated for the coarseness. The question of upscaling relative permeability curves from the rock curves derived at core plug scale (which is often denoted the micro scale) to the coarse grids cells of the reservoir models (which is often called the macro scale) is much more difficult, and it became an important research field that is still ongoing. But the progress in upscaling was slow, and it was not until 1990-2000 that directional dependency of relative permeability and need for tensor representation was clearly demonstrated, even though at least one capable method was developed already in 1975. One such upscaling case is a slanted reservoir where the water (and gas) will segregate vertically relative to the oil in addition to the horizontal motion. The vertical size of a grid cell is also usually much smaller than the horizontal size of a grid cell, creating small and large flux areas respectively. All this requires different relative permeability curves for the x and z directions. Geological heterogeneities in the reservoirs like laminas or crossbedded permeability structures in the rock, also cause directional relative permeabilities. This tells us that relative permeability should, in the most general case, be represented by a tensor. The flow equations then become
where a = w, o, g
The above-mentioned case reflected downdip water injection (or updip gas injection) or production by pressure depletion. If you inject water updip (or gas downdip) for a period of time, it will give rise to different relative permeability curves in the x+ and x- directions. This is not a hysteresis process in the traditional sense, and it cannot be represented by a traditional tensor. It can be represented by an IF-statement in the software code, and it occurs
in some commercial reservoir simulators. The process (or rather sequence of processes) may be due to a backup plan for field recovery, or the injected fluid may flow to another reservoir rock formation due to an unexpected open part of a fault or a non-sealing cement behind casing of the injection well. The option for relative permeability is seldom used, and we just note that it does not change (the analytical shape of) the governing equation, but increases (usually doubles) the number of constitutive equations for the properties involved.
The above equation is a vector form of the most general equation for fluid flow in porous media, and it gives the reader a good overview of the terms and quantities involved. Before you go ahead and transform the differential equation into difference equations, to be used by the computers, you must write the flow equation in component form. The flow equation in component form (using summation convention) is
where a = w, o, g where = 1,2,3
The Darcy velocity is not the velocity of a fluid particle, but the volumetric flux (frequently represented by the symbol ) of the fluid stream. The fluid velocity in the pores (or short but inaccurately called pore velocity) is related to Darcy velocity by the relation
where a = w, o, g
The volumetric flux is an intensive quantity, so it is not good at describing how much fluid is coming per time. The preferred variable to understand this is the extensive quantity called volumetric flow rate which tells us how much fluid is coming out of (or going into) a given area per time, and it is related to Darcy velocity by the relation
where a = w, o, g
We notice that the volumetric flow rate is a scalar quantity and that the direction is taken care of by the normal vector of the surface (area) and the volumetric flux (Darcy velocity).
In a reservoir model the geometric volume is divided into grid cells, and the area of interest now is the intersectional area between two adjoining cells. If these are true neighboring cells, the area is the common side surface, and if a fault is dividing the two cells, the intersection area is usually less than the full side surface of both adjoining cells. A version of the multiphase flow equation, before it is discretized and used in reservoir simulators, is thus
where a = w, o, g
In expanded (component) form it becomes
where a = w, o, g
The (initial) hydrostatic pressure at a depth (or level) z above (or below) a reference depth z0 is calculated by
where a = w, o, g
When calculations of hydrostatic pressure are executed, one normally does not apply a phase subscript, but switch formula / quantity according to what phase is observed at the actual depth, but we have included the phase subscript here for clarity and consistency. However, when calculations of hydrostatic pressure are executed one may use an acceleration of gravity that varies with depth in order to increase accuracy. If such high accuracy is not needed, the acceleration of gravity is kept constant, and the calculated pressure is called overburden pressure. Such high accuracy is not needed in reservoir simulations so acceleration of gravity is treated as a constant in this discussion. The initial pressure in the reservoir model is calculated using the formula for (initial) overburden pressure which is
where a = w, o, g
In order to simplify the terms within the parenthesis of the flow equation, you can introduce the fluid potential (also called -potential) which is defined by
where a = w, o, g
It consists of two terms which are absolute pressure and gravity head. To save computing time the integral can be calculated initially and stored as a table to be used in the computationally cheaper table-lookup. Introduction of the fluid potential implies that
where a = w, o, g
The multiphase flow equation for porous media now becomes
where a = w, o, g
This multiphase flow equation has traditionally been the starting point for the software programmer when he/she starts transforming the equation from differential equation to difference equation in order to write a program code for a reservoir simulator to be used in the petroleum industry. The unknown dependent variables have traditionally been oil pressure (for oil fields) and volumetric quantities for the fluids involved, but one may rewrite the total set of model equations to be solved for oil pressure and mass or mole quantities for the fluid components involved.
The above equations are written in SI units and we are assuming that all material properties are also defined within the SI units. A result of this is that above versions of the equations do not need any unit conversion constants. The petroleum industry applies a variety of units, of which a least two have some prevalence. If you want to apply units other than SI units, you must establish correct unit conversion constants for the multiphase flow equations.
In dimensional analysis, a dimensionless quantity is a quantity to which no physical dimension is assigned, also known as a bare, pure, or scalar quantity or a quantity of dimension one, with a corresponding unit of measurement in the SI of one (or 1) unit that 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.
This is a list of articles about numbers. Due to the infinitude of many sets of numbers, this list will invariably be incomplete. Hence, only particularly notable numbers will be included.
This list focuses on numbers as mathematical objects and is not a list of numerals, which are linguistic devices: nouns, adjectives, or adverbs that designate numbers. The distinction is drawn between the number five, an abstract object equal to 2+3 and the numeral five, the noun referring to the number.
The definition of what is classed as a number is not concrete and is based on historical distinctions. For example the pair of numbers (3,4) is commonly regarded as a number when it is in the form of a complex number (3+4i), but not when it is in the form of a vector (3,4).
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.
For whole numbers less than a thousand million (< 109) the two scales are identical. From a thousand million up (≥ 109) the two scales diverge, using the same words for different numbers, which can cause misunderstanding.
This article lists and discusses the usage and derivation of names of large numbers, together with their possible extensions.
The following table lists those names of large numbers that are found in many English dictionaries and thus have a special claim to being "real words." The "Traditional British" values shown are unused in American English and are obsolete in British English, but their other-language variants are dominant in many non-English-speaking areas, including continental Europe and Spanish-speaking countries in Latin America; see Long and short scales.
Indian English does not use millions, but has its own system of large numbers including lakhs and crores.
English also has many words, such as "zillion", used informally to mean large but unspecified amounts; see indefinite and fictitious numbers.
An order of magnitude is an approximate measure of the number of digits that a number has in the commonly-used base-ten number system. It is equal to the whole number floor of logarithm (base 10). For example, the order of magnitude of 1500 is 3, because 1500 = 1.5 × 103.
Differences in order of magnitude can be measured on a base-10 logarithmic scale in “decades” (i.e., factors of ten). Examples of numbers of different magnitudes can be found at Orders of magnitude (numbers).
In mathematics, a power of 10 is any of the integer powers of the number ten; in other words, ten multiplied by itself a certain number of times (when the power is a positive integer). By definition, the number one is a power (the zeroth power) of ten. The first few non-negative powers of ten are:
1, 10, 100, 1,000, 10,000, 100,000, 1,000,000, 10,000,000. ... (sequence A011557 in the OEIS)
The Westinghouse Sign was the first computer-controlled sign in the United States. Located in Pittsburgh, Pennsylvania, the large animated display advertised the Westinghouse Electric Company, and was best known for the seemingly endless number of combinations in which its individual elements could be illuminated. The sign was removed in 1998 when the building on which it was mounted was demolished to make way for the construction of PNC Park.
This page is based on a Wikipedia article written by authors
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.