Orders of magnitude (numbers)

Last updated
The logarithmic scale can compactly represent the relationship among variously sized numbers. Logarithmic scale.svg
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.

Contents

Smaller than 10100 (one googolth)

Chimpanzee probably not typing Hamlet Chimpanzee seated at typewriter.jpg
Chimpanzee probably not typing Hamlet

10−100 to 10−30

1/52! chance of a specific shuffle Card shuffle.jpg
1/52! chance of a specific shuffle

10−30

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

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)

10−18

Snake eyes Snake eyes dice.jpg
Snake eyes

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

ISO: atto- (a)

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)

10−6

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

ISO: micro- (μ)

Poker hands Poker Hands.png
Poker hands

10−3

(0.001; 1000−1; one thousandth)

ISO: milli- (m)

10−2

(0.01; one hundredth)

ISO: centi- (c)

10−1

(0.1; one tenth)

ISO: deci- (d)

100

Eight planets of the solar system Planets2013.svg
Eight planets of the solar system

(1; one)

101

Ten digits on two human hands Two hand, ten fingers.jpg
Ten digits on two human hands

(10; ten)

ISO: deca- (da)

102

128 ASCII characters ASCII-Table-wide.svg
128 ASCII characters

(100; hundred)

ISO: hecto- (h)

103

Roman legion (precise size varies) Legion Task ORG.png
Roman legion (precise size varies)

(1000; thousand)

ISO: kilo- (k)

104

(10000; ten thousand or a myriad)

105

100,000-150,000 strands of human hair Woman with long brown hair, close-up view.jpg
100,000–150,000 strands of human hair

(100000; one hundred thousand or a lakh).

106

3,674,160 Pocket Cube positions Pocket cube scrambled.jpg
3,674,160 Pocket Cube positions

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

ISO: mega- (M)

107

12,988,816 domino tilings of a checkerboard Pavage domino.svg
12,988,816 domino tilings of a checkerboard

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

108

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

109

World population estimates World population v3.svg
World population estimates

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

ISO: giga- (G)

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

10 stars in the Andromeda Galaxy Andromeda Galaxy (with h-alpha).jpg
10 stars in the Andromeda Galaxy

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

ISO: tera- (T)

1015

10 to 10 ants on Earth Ants eating01.jpg
10 to 10 ants on Earth

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

ISO: peta- (P)

1018

[?]4.33x10 Rubik's Cube positions Scrumbled Rubik's Cube.jpg
≈4.33×10 Rubik's Cube positions

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

ISO: exa- (E)

1021

[?]6.7x10 sudoku grids Sudoku Puzzle by L2G-20050714 solution standardized layout.svg
≈6.7×10 sudoku grids

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

ISO: zetta- (Z)

1024

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

ISO: yotta- (Y)

1027

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

1030

5 x 10 bacterial cells on Earth EscherichiaColi NIAID.jpg
5 × 10 bacterial cells on Earth

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

1033

(1000000000000000000000000000000000; 100011; short scale: one decillion; long scale: one thousand quintillion, or one quintilliard)

1036

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

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)

<4.52x10 legal chess positions ChessStartingPosition.jpg
<4.52×10 legal chess positions

10100 (one googol) to 101000

(10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000; short scale: ten duotrigintillion; long scale: ten thousand sexdecillion, or ten sexdecillard) [49]

[?]2.08x10 legal Go positions FloorGoban.JPG
≈2.08×10 legal Go positions

101000 to 1010100 (one googolplex)

Digit growth in the largest known prime Digits in largest prime found as a function of time.svg
Digit growth in the largest known prime

Larger than 1010100

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

See also

Related Research Articles

<span class="texhtml mvar" style="font-style:italic;">e</span> (mathematical constant) e ≈ 2.71828..., base of the natural logarithm

The number e, known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm. It 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

Floating-point arithmetic Computer format for representing real numbers

In computing, floating-point arithmetic (FP) is arithmetic using formulaic representation of real numbers as an approximation 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:

Hexadecimal Base 16 numerical system

In mathematics and computing, hexadecimal is a positional system that represents numbers using a base of 16. Unlike the common way of representing numbers with ten symbols, it uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values zero to nine, and "A"–"F" to represent values ten to fifteen.

Logarithm Inverse of the exponential function, which maps products to sums

In mathematics, the logarithm is the inverse function to exponentiation. 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 the number of occurrences of the same factor in repeated multiplication; e.g., since 1000 = 10 × 10 × 10 = 103, the "logarithm base 10" of 1000 is 3, or log10(1000) = 3. The logarithm of x to baseb is denoted as logb(x), or without parentheses, logbx, or even without the explicit base, log x, when no confusion is possible, or when the base does not matter such as in big O notation.

The number π is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It 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, and is spelled out as "pi". It is also referred to as Archimedes' constant.

Prime number Positive integer with exactly two divisors, 1 and itself

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

Benfords law Observation about the frequency distribution of leading digits in many real-life sets of numerical data

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 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.

Rounding replacing numerical value by another approximately equal

Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For example, replacing $23.4476 with $23.45, the fraction 312/937 with 1/3, or the expression 2 with 1.414.

Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce the outcome to the number that don't. Odds are commonly used in gambling and statistics.

The significant figures of a number written in positional notation are digits that carry meaningful contributions to its measurement resolution. This includes all digits except:

The IEEE Standard for Floating-Point Arithmetic is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.

1,000,000,000 Natural number

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn.

Randomness lack of pattern or predictability in events, or, a measure of uncertainty of an outcome

In the common parlance, randomness is the apparent lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but since they often follow a probability distribution, the frequency of different outcomes over numerous 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 its haphazardness, and applies to concepts of chance, probability, and information entropy.

In computing, quadruple precision is a binary floating point–based computer number format that occupies 16 bytes with precision more than twice the 53-bit double precision.

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.

In computing, octuple precision is a binary floating-point-based computer number format that occupies 32 bytes in computer memory. This 256-bit octuple precision is for applications requiring results in higher than quadruple precision. This format is rarely used and very few environments support it.

A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol, or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.

References

  1. Kittel, Charles and Herbert Kroemer (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. p. 53. ISBN   978-0-7167-1088-2.
  2. 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.
  3. Robert Matthews. "What are the odds of shuffling a deck of cards into the right order?". Science Focus. Retrieved December 10, 2018.
  4. www.BridgeHands.com, Sales. "Probabilities Miscellaneous: Bridge Odds". Archived from the original on 2009-10-03.
  5. 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. PMID   13998626.
  6. Courtney Taylor. "The Probability of Being Dealt a Royal Flush in Poker". ThoughtCo. Retrieved December 10, 2018.
  7. 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. doi:10.1128/JVI.36.3.829-836.1980. ISSN   0022-538X. PMC   353710 . PMID   7463557.
  8. 1 2 "Homo sapiens – Ensembl genome browser 87". www.ensembl.org. Archived from the original on 2017-05-25. Retrieved 2017-01-28.
  9. "Pi World Ranking List". Archived from the original on 2017-06-29.
  10. Sloane, N. J. A. (ed.). "SequenceA283670". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2017-03-15.
  11. Kibrik, A. E. (2001). "Archi (Caucasian—Daghestanian)", The Handbook of Morphology, Blackwell, pg. 468
  12. Funk, Matthias (31 January 2020). "How Many YouTube Channels Are There?". tubics. Retrieved 2020-05-19.
  13. Plouffe's Inverter Archived 2005-08-12 at the Wayback Machine
  14. "How many cars are there in the world?". carsguide. 6 August 2018. Retrieved 18 May 2020.
  15. Christof Baron (2015). "Facebook users worldwide 2016 | Statista". Statista. statista.com. Archived from the original on 2016-09-09.
  16. 1 2 "Earth microbes on the moon". Science@Nasa. 1 September 1998. Archived from the original on 23 March 2010. Retrieved 2 November 2010.
  17. "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   2776484 . PMID   19915731.
  18. 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).
  19. 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).
  20. Hollis, Morgan (13 October 2016). "A universe of two trillion galaxies". The Royal Astronomical Society. Retrieved 9 November 2017.
  21. Jonathan Amos (3 September 2015). "Earth's trees number 'three trillion'". BBC. Archived from the original on 6 June 2017.
  22. Xavier Gourdon (October 2004). "Computation of zeros of the Zeta function". Archived from the original on 15 January 2011. Retrieved 2 November 2010.
  23. Haruka Iwao, Emma (14 March 2019). "Pi in the sky: Calculating a record-breaking 31.4 trillion digits of Archimedes' constant on Google Cloud". Archived from the original on 19 October 2019. Retrieved 12 April 2019.
  24. Koch, Christof. Biophysics of computation: information processing in single neurons. Oxford university press, 2004.
  25. 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.
  26. Berg, R. (1996). "The indigenous gastrointestinal microflora". Trends in Microbiology. 4 (11): 430–5. doi:10.1016/0966-842X(96)10057-3. PMID   8950812.
  27. Bert Holldobler and E.O. Wilson The Superorganism: The Beauty, Elegance, and Strangeness of Insect Societies New York:2009 W.W. Norton Page 5
  28. "60th Birthday of Microelectronics Industry". Semiconductor Industry Association. 13 December 2007. Archived from the original on 13 October 2008. Retrieved 2 November 2010.
  29. Sequence A131646 Archived 2011-09-01 at the Wayback Machine in The On-Line Encyclopedia of Integer Sequences
  30. "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).
  31. Ivan Moscovich, 1000 playthinks: puzzles, paradoxes, illusions & games, Workman Pub., 2001 ISBN   0-7611-1826-8 .
  32. "Scores of Zimbabwe farms 'seized'". BBC. 23 February 2009. Archived from the original on 1 March 2009. Retrieved 14 March 2009.
  33. "To see the Universe in a Grain of Taranaki Sand". Archived from the original on 2012-06-30.
  34. "Intel predicts 1,200 quintillion transistors in the world by 2015". Archived from the original on 2013-04-05.
  35. "How Many Transistors Have Ever Shipped? – Forbes". Archived from the original on 30 June 2015. Retrieved 1 September 2015.
  36. "Sudoku enumeration". Archived from the original on 2006-10-06.
  37. "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.
  38. "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: 1009.5992 . 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
  39. (sequence A007377 in the OEIS )
  40. "Questions and Answers – How many atoms are in the human body?". Archived from the original on 2003-10-06.
  41. 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   33863 . PMID   9618454.
  42. (sequence A070177 in the OEIS )
  43. (sequence A035064 in the OEIS )
  44. John Tromp (2010). "John's Chess Playground". Archived from the original on 2014-06-01.
  45. Planck Collaboration (2016). "Planck 2015 results. XIII. Cosmological parameters (See Table 4 on page 31 of pfd)". Astronomy & Astrophysics. 594: A13. arXiv: 1502.01589 . Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830.
  46. Paul Zimmermann, "50 largest factors found by ECM Archived 2009-02-20 at the Wayback Machine ".
  47. Matthew Champion, "Re: How many atoms make up the universe?" Archived 2012-05-11 at the Wayback Machine , 1998
  48. 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.
  49. "Names of large and small numbers". bmanolov.free.fr. Miscellaneous pages by Borislav Manolov. Archived from the original on 2016-09-30.
  50. "Richard Eldridge".
  51. Chris Caldwell, The Top Twenty: Elliptic Curve Primality Proof at The Prime Pages.
  52. Chris Caldwell, The Top Twenty: Twin Primes Archived 2013-01-27 at the Wayback Machine at The Prime Pages.
  53. Chris Caldwell, The Top Twenty: Sophie Germain (p) at The Prime Pages.
  54. Chris Caldwell, The Top Twenty: Palindrome at The Prime Pages.
  55. PrimeGrid's Primorial Prime Search Archived 2013-11-26 at the Wayback Machine
  56. Chris Caldwell, The Top Twenty: Factorial primes Archived 2013-04-10 at the Wayback Machine at The Prime Pages.
  57. 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.
  58. Chris Caldwell, The Top Twenty: Generalized Fermat Archived 2014-12-23 at the Wayback Machine at The Prime Pages.
  59. Chris Caldwell, The Top Twenty: Proth at The Prime Pages.
  60. 1 2 Chris Caldwell, The Top Twenty: Largest Known Primes at The Prime Pages.
  61. Chris Caldwell, Mersenne Primes: History, Theorems and Lists at The Prime Pages.
  62. asantos (15 December 2007). "Googol and Googolplex by Carl Sagan" via YouTube.
  63. Zyga, Lisa "Physicists Calculate Number of Parallel Universes" Archived 2011-06-06 at the Wayback Machine , PhysOrg , 16 October 2009.