|
|
| Line 1: |
Line 1: |
| {{Refimprove|date=November 2011}}
| | My name is Ben Galvin. I life in Eindhoven (Netherlands).<br><br>Also visit my weblog; [http://hemorrhoidtreatmentfix.com/prolapsed-hemorrhoid prolapsed hemorrhoids] |
| {{cleanup-rewrite|date=March 2011}}
| |
| This article is about '''large numbers''' in the sense of [[number]]s that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive [[integer]]s, or more generally, large positive [[real number]]s, but it may also be used in other contexts.
| |
| | |
| Very large numbers often occur in fields such as [[mathematics]], [[physical cosmology|cosmology]], [[cryptography]], and [[statistical mechanics]]. Sometimes people refer to numbers as being "astronomically large". However, it is easy to mathematically define numbers that are much larger even than those used in astronomy.
| |
| | |
| == Using scientific notation to handle large and small numbers ==
| |
| {{See also|scientific notation|logarithmic scale|orders of magnitude}}
| |
| | |
| [[Scientific notation]] was created to handle the wide range of values that occur in scientific study. 1.0 × 10<sup>9</sup>, for example, means one [[1000000000 (number)|billion]], a 1 followed by nine zeros: 1 000 000 000, and 1.0 × 10<sup>−9</sup> means one billionth, or 0.000 000 001. Writing 10<sup>9</sup> instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is.
| |
| | |
| == Large numbers in the everyday world ==
| |
| | |
| Examples of large numbers describing everyday real-world objects are:
| |
| * The number of [[bit]]s on a computer [[hard disk]] ({{as of|2010|lc=true}}, typically about 10<sup>13</sup>, 500-1000 [[gigabyte|GB]])<!-- if you think this is wrong, read the numbers again, carefully. These are bits, not bytes. -->
| |
| * The estimated number of [[atom]]s in the observable Universe (10<sup>80</sup>)
| |
| * The number of [[Cell (biology)|cells]] in the human body (more than 10<sup>14</sup>)
| |
| * The number of [[Neuron|neuronal connections]] in the human brain (estimated at 10<sup>14</sup>)
| |
| * The [[Avogadro constant]], the number of "elementary entities" (usually atoms or molecules) in one [[Mole (unit)|mole]]; the number of atoms in 12 grams of [[carbon-12]]; (approximately 6.022 × 10<sup>23</sup> per mole)
| |
| | |
| == Astronomically large numbers ==
| |
| Other large numbers, as regards length and time, are found in [[astronomy]] and [[cosmology]]. For example, the current [[Big Bang model]] suggests that the Universe is 13.8 billion years (4.355 × 10<sup>17</sup> seconds) old, and that the [[observable universe]] is 93 billion [[light years]] across (8.8 × 10<sup>26</sup> metres), and contains about 5 × 10<sup>22</sup> stars, organized into around 125 billion (1.25 × 10<sup>11</sup>) galaxies, according to Hubble Space Telescope observations. There are about 10<sup>80</sup> atoms in the [[observable universe]], by rough estimation.<ref>[http://www.universetoday.com/36302/atoms-in-the-universe/#gsc.tab=0 Atoms in the Universe]. Universe Today. 30-07-2009. Retrieved 02-03-13.</ref>
| |
| | |
| According to [[Don Page (physicist)|Don Page]], physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
| |
| | |
| ::::<math>10^{10^{10^{10^{10^{1.1}}}}} \mbox{ years}</math>
| |
| | |
| which corresponds to the scale of an estimated [[Poincaré recurrence theorem|Poincaré recurrence time]] for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe, observable or not, assuming a certain [[inflation (cosmology)|inflationary]] model with an [[inflaton]] whose mass is 10<sup>−6</sup> [[Planck mass]]es.<ref name=page95>Information Loss in Black Holes and/or Conscious Beings?, Don N. Page, ''Heat Kernel Techniques and Quantum Gravity'' (1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics. {{arxiv|hep-th/9411193}}. ISBN 0-9630728-3-8.</ref><ref>[http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html How to Get A Googolplex]</ref> This time assumes a statistical model subject to [[Poincaré recurrence theorem|Poincaré recurrence]]. A much simplified way of thinking about this time is in a model where our universe's history [[Loschmidt's paradox|repeats itself]] arbitrarily many times due to [[Ergodic hypothesis|properties of statistical mechanics]]; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
| |
| | |
| [[Combinatorial]] processes rapidly generate even larger numbers. The [[factorial]] function, which defines the number of [[permutation]]s on a set of fixed objects, grows very rapidly with the number of objects. [[Stirling's formula]] gives a precise asymptotic expression for this rate of growth.
| |
| | |
| Combinatorial processes generate very large numbers in [[statistical mechanics]]. These numbers are so large that they are typically only referred to using their [[logarithm]]s.
| |
| | |
| [[Gödel number]]s, and similar numbers used to represent bit-strings in [[algorithmic information theory]], are very large, even for mathematical statements of reasonable length. However, some [[pathological (mathematics)|pathological]] numbers are even larger than the Gödel numbers of typical mathematical propositions.
| |
| | |
| Logician [[Harvey Friedman]] has done work related to very large numbers, such as with [[Kruskal's tree theorem]] and the [[Robertson–Seymour theorem]].
| |
| | |
| == Computers and computational complexity ==
| |
| {{Original research|section|date=September 2009}}
| |
| | |
| Between 1980 and 2000, hard disk sizes increased from about 10 megabytes (10<sup>7</sup> bytes) to over 100 gigabytes (10<sup>11</sup> bytes). {{Citation needed|date=February 2013}} A 100 gigabyte disk could store the given names of all of Earth's seven billion inhabitants without using data compression. {{Citation needed|date=February 2013}} But what about a dictionary-on-disk storing all possible passwords containing up to 40 characters? Assuming each character equals one byte, there are about 2<sup>320</sup> such passwords, which is about 2 × 10<sup>96</sup>. In his paper ''Computational capacity of the universe'',<ref>{{cite journal | last = Lloyd | first = Seth | authorlink = Seth Lloyd | year = 2002 | title = Computational capacity of the universe | journal = Phys. Rev. Lett. | volume = 88 | pages = 237901 | arxiv = quant-ph/0110141 | doi = 10.1103/PhysRevLett.88.237901 | pmid = 12059399 | issue = 23 | bibcode=2002PhRvL..88w7901L}}</ref> [[Seth Lloyd]] points out that if every particle in the universe could be used as part of a huge computer, it could store only about 10<sup>90</sup> bits, less than one millionth of the size such a dictionary would require. However, storing information on hard disk and computing it are very different functions. On the one hand storage currently has limitations as stated, but computational speed is a different matter. It is quite conceivable {{By whom|date=February 2013}} that the stated limitations regarding storage have no bearing on the limitations of actual computational capacity, {{Citation needed|date=February 2013}} especially if the current research into quantum computers results in a "quantum leap" (but see ''[[holographic principle]]'').
| |
| | |
| Still, computers can easily be programmed to start creating and displaying all possible 40-character passwords one at a time. Such a program could be left to run indefinitely. Assuming a modern PC could output 1 billion strings per second, it would take one billionth of 2 × 10<sup>96</sup> seconds, or 2 × 10<sup>87</sup> seconds to complete its task, which is about 6 × 10<sup>79</sup> years. By contrast, the universe is estimated to be 13.8 billion (1.38 × 10<sup>10</sup>) years old. Computers will presumably continue to get faster, but the same paper mentioned before estimates that the entire universe functioning as a giant computer could have performed no more than 10<sup>120</sup> operations since the [[Big Bang]]. This is trillions of times more computation than is required for displaying all 40-character passwords, but computing all ''50'' character passwords would outstrip the estimated computational potential of the entire universe.
| |
| | |
| Problems like this [[exponential growth|grow exponentially]] in the number of computations they require, and they are one reason why exponentially difficult problems are called "intractable" in computer science: for even small numbers like the 40 or 50 characters described earlier, the number of computations required exceeds even theoretical limits {{Citation needed|date=February 2013}} on mankind's computing power. The [[P = NP problem|traditional division]] between "easy" and "hard" problems is thus drawn between programs that do and do not require exponentially increasing resources to execute.
| |
| | |
| Such limits are an advantage in [[cryptography]], since any [[cipher]]-breaking technique that requires more than, say, the 10<sup>120</sup> operations mentioned before will never be feasible. Such ciphers must be broken by finding efficient techniques unknown to the cipher's designer. Likewise, much of the research throughout all branches of [[computer science]] focuses on finding efficient solutions to problems that work with far fewer resources than are required by a [[naïve solution]]. For example, one way of finding the [[greatest common divisor]] between two 1000-digit numbers is to compute all their factors by trial division. This will take up to 2 × 10<sup>500</sup> division operations, far too large to contemplate. But the [[Euclidean algorithm]], using a much more efficient technique, takes only a fraction of a second to compute the GCD for even huge numbers such as these.
| |
| | |
| As a general rule, then, PCs in 2005 can perform 2<sup>40</sup> calculations in a few minutes. {{Citation needed|date=February 2013}} A few thousand PCs working for a few years could solve a problem requiring 2<sup>64</sup> calculations, but no amount of traditional computing power will solve a problem requiring 2<sup>128</sup> operations (which is about what would be required to brute-force the encryption keys in 128-bit [[Secure Sockets Layer|SSL]] commonly used in web browsers, assuming the underlying ciphers remain secure). Limits on computer storage are comparable. {{quantify|date=February 2013}} [[Quantum computing]] may allow certain problems {{vague|date=February 2013}} to become feasible, but it has practical and theoretical challenges that may never be overcome. {{examples|date=February 2013}}
| |
| | |
| {{See also|computation|computational complexity theory|algorithmic information theory|computability theory|big O notation}}
| |
| | |
| == Examples ==
| |
| {{See also|#Examples of numbers in numerical order|l1=Examples of numbers, in numerical order}}
| |
| | |
| *<math>10^{10}</math> <small>(10,000,000,000)</small>, called "ten billion" in the [[Long and short scales|short scale]] or "ten milliard" in the long scale.
| |
| *[[googol]] = <math>10^{100}</math> <small>(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)</small>.
| |
| | |
| *[[centillion]] = <math>10^{303}</math> or <math>10^{600}</math>, depending on number naming system
| |
| *[[googolplex]] = <math>10^{\text{googol}}=10^{10^{100}}</math>
| |
| *[[Skewes' number]]s: the first is approximately <math>10^{10^{10^{34}}}</math>, the second <math>10^{10^{10^{1000}}}</math>
| |
| *[[Graham's number]] = larger than can be represented here, even using power towers; however, it can be represented using [[Knuth's up-arrow notation]].
| |
| | |
| The total amount of printed material in the world is roughly 1.6 × 10<sup>18</sup> bits{{Citation needed|date=October 2009}}; therefore the contents can be represented by a number somewhere in the range 0 to roughly <math>2^{1.6 \times 10^{18}}\approx 10^{4.8 \times 10^{17}}</math>
| |
| | |
| Compare:
| |
| *<math>1.1^{1.1^{1.1^{1000}}} \approx 10^{10^{1.02\times10^{40}}}</math>
| |
| *<math>1000^{1000^{1000}}\approx 10^{10^{3000.48}}</math>
| |
| The first number is much larger than the second, due to the larger height of the power tower, and in spite of the small numbers 1.1. In comparing the magnitude of each successive exponent in the last number with <math>10^{10^{10}}</math>, we find a difference in the magnitude of effect on the final exponent.
| |
| | |
| ==Systematically creating ever faster increasing sequences==
| |
| {{Main|fast-growing hierarchy}}
| |
| Given a strictly increasing integer sequence/function <math>f_0(n)</math> (''n''≥1) we can produce a faster growing sequence <math>f_1(n) = f_0^n(n)</math> (where the superscript ''n'' denotes the ''n''<sup>th</sup> [[functional power]]). This can be repeated any number of times by letting <math>f_k(n) = f_{k-1}^n(n)</math>, each sequence growing much faster than the one before it. Then we could define <math>f_\omega(n) = f_n(n)</math>, which grows much faster than any <math>f_k</math> for finite ''k'' (here ω is the first infinite [[ordinal number]], representing the limit of all finite numbers k). This is the basis for the [[fast-growing hierarchy]] of functions, in which the indexing subscript is extended to ever-larger ordinals.
| |
| | |
| For example, starting with ''f''<sub>0</sub>(''n'') = ''n'' + 1:
| |
| | |
| * ''f''<sub>1</sub>(''n'') = ''f''<sub>0</sub><sup>''n''</sup>(''n'') = ''n'' + ''n'' = 2''n''
| |
| * ''f''<sub>2</sub>(''n'') = ''f''<sub>1</sub><sup>''n''</sup>(''n'') = 2<sup>''n''</sup>''n'' > (2 ↑) ''n''</sup> for n ≥ 2 (using [[Knuth up-arrow notation]])
| |
| * ''f''<sub>3</sub>(''n'') = ''f''<sub>2</sub><sup>''n''</sup>(''n'') > (2 ↑)<sup>''n''</sup> ''n'' ≥ 2 ↑<sup>2</sup> ''n'' for ''n'' ≥ 2.
| |
| * ''f''<sub>''k''+1</sub>(''n'') > 2 ↑<sup>''k''</sup> ''n'' for ''n'' ≥ 2, ''k'' < ω.
| |
| | |
| * ''f''<sub>ω</sub>(''n'') = ''f''<sub>''n''</sub>(''n'') > 2 ↑<sup>''n'' - 1</sup> ''n'' > 2 ↑<sup>''n'' − 2</sup> (''n'' + 3) − 3 = ''A''(''n'', ''n'') for ''n'' ≥ 2, where ''A'' is the [[Ackermann function]] (of which ''f''<sub>ω</sub> is a unary version).
| |
| * ''f''<sub>ω+1</sub>(64) > ''f''<sub>ω</sub><sup>64</sup>(6) > [[Graham's number#Definition|Graham's number]] (= ''g''<sub>64</sub> in the sequence defined by ''g''<sub>0</sub> = 4, ''g''<sub>''k''+1</sub> = 3 ↑<sup>''g''<sub>''k''</sub></sup> 3).
| |
| **This follows by noting ''f''<sub>ω</sub>(''n'') > 2 ↑<sup>''n'' - 1</sup> ''n'' > 3 ↑<sup>''n'' - 2</sup> 3 + 2, and hence ''f''<sub>ω</sub>(''g''<sub>''k''</sub> + 2) > ''g''<sub>''k''+1</sub> + 2.
| |
| | |
| * ''f''<sub>ω</sub>(''n'') > 2 ↑<sup>''n'' - 1</sup> ''n'' = (2 → ''n'' → ''n''-1) = (2 → ''n'' → ''n''-1 → 1) (using [[Conway chained arrow notation]])
| |
| * ''f''<sub>ω+1</sub>(''n'') = ''f''<sub>ω</sub><sup>''n''</sup>(''n'') > (2 → ''n'' → ''n''-1 → 2) (because if ''g''<sub>''k''</sub>(''n'') = X → ''n'' → ''k'' then X → ''n'' → ''k''+1 = ''g''<sub>''k''</sub><sup>''n''</sup>(1))
| |
| * ''f''<sub>ω+''k''</sub>(''n'') > (2 → ''n'' → ''n''-1 → ''k''+1) > (''n'' → ''n'' → ''k'')
| |
| * ''f''<sub>ω2</sub>(''n'') = ''f''<sub>ω+''n''</sub>(''n'') > (''n'' → ''n'' → ''n'') = (''n'' → ''n'' → ''n''→ 1)
| |
| * ''f''<sub>ω2+''k''</sub>(''n'') > (''n'' → ''n'' → ''n'' → ''k'')
| |
| * ''f''<sub>ω3</sub>(''n'') > (''n'' → ''n'' → ''n'' → ''n'')
| |
| * ''f''<sub>ω''k''</sub>(''n'') > (''n'' → ''n'' → ... → ''n'' → ''n'') (Chain of ''k''+1 ''n'''s)
| |
| * ''f''<sub>ω<sup>2</sup></sub>(''n'') = ''f''<sub>ω''n''</sub>(''n'') > (''n'' → ''n'' → ... → ''n'' → ''n'') (Chain of ''n''+1 ''n'''s)
| |
| | |
| == Standardized system of writing very large numbers ==
| |
| | |
| A standardized way of writing very large numbers allows them to be easily sorted in increasing order, and one can get a good idea of how much larger a number is than another one.
| |
| | |
| To compare numbers in scientific notation, say 5×10<sup>4</sup> and 2×10<sup>5</sup>, compare the exponents first, in this case 5 > 4, so 2×10<sup>5</sup> > 5×10<sup>4</sup>. If the exponents are equal, the ''mantissa'' (or coefficient) should be compared, thus 5×10<sup>4</sup> > 2×10<sup>4</sup> because 5 > 2.
| |
| | |
| [[Tetration]] with base 10 gives the sequence <math>10 \uparrow \uparrow n=10 \to n \to 2=(10\uparrow)^n 1</math>, the power towers of numbers 10, where <math>(10\uparrow)^n</math> denotes a [[functional power]] of the function <math>f(n)=10^n</math> (the function also expressed by the suffix "-plex" as in [[googolplex]], see [[Names of large numbers#The googol family|the Googol family]]).
| |
| | |
| These are very round numbers, each representing an [[order of magnitude]] in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.
| |
| | |
| More accurately, numbers in between can be expressed in the form <math>(10\uparrow)^n a</math>, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g. <math>10^{10^{10^{10^{10^{4.829}}}}}</math>, a number between <math>10\uparrow\uparrow 5</math> and <math>10\uparrow\uparrow 6</math> (note that <math>10 \uparrow\uparrow n < (10\uparrow)^n a < 10 \uparrow\uparrow (n+1)</math> if <math> 1 < a < 10</math>). (See also [[Tetration#Extension to real heights|extension of tetration to real heights]].)
| |
| | |
| Thus googolplex is <math>10^{10^{100}} = (10\uparrow)^2 100 = (10\uparrow)^3 2</math>
| |
| | |
| Another example:
| |
| :<math>2 \uparrow\uparrow\uparrow 4 =
| |
| \begin{matrix}
| |
| \underbrace{2_{}^{2^{{}^{.\,^{.\,^{.\,^2}}}}}}\\
| |
| \qquad\quad\ \ \ 65,536\mbox{ copies of }2 \end{matrix}
| |
| \approx (10\uparrow)^{65,531}(6.0 \times 10^{19,728}) \approx (10\uparrow)^{65,533} 4.3
| |
| </math> (between <math>10\uparrow\uparrow 65,533</math> and <math>10\uparrow\uparrow 65,534</math>)
| |
| | |
| Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (''n'') one has to take the <math>log_{10}</math> to get a number between 1 and 10. Thus, the number is between <math>10\uparrow\uparrow n</math> and <math>10\uparrow\uparrow (n+1)</math>. As explained, a more accurate description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 10<sup>10</sup>, or the next, between 0 and 1.
| |
| | |
| Note that
| |
| :<math>10^{(10\uparrow)^{n}x}=(10\uparrow)^{n}10^x</math>
| |
| I.e., if a number ''x'' is too large for a representation <math>(10\uparrow)^{n}x</math> we can make the power tower one higher, replacing ''x'' by log<sub>10</sub>''x'', or find ''x'' from the lower-tower representation of the log<sub>10</sub> of the whole number. If the power tower would contain one or more numbers different from 10, the two approaches would lead to different results, corresponding to the fact that extending the power tower with a 10 at the bottom is then not the same as extending it with a 10 at the top (but, of course, similar remarks apply if the whole power tower consists of copies of the same number, different from 10).
| |
| | |
| If the height of the tower is large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so we can use the double-arrow notation, e.g. <math>10\uparrow\uparrow(7.21\times 10^8)</math>. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.
| |
| | |
| Examples:
| |
| :<math>10\uparrow\uparrow 10^{\,\!10^{10^{3.81\times 10^{17}}}}</math> (between <math>10\uparrow\uparrow\uparrow 2</math> and <math>10\uparrow\uparrow\uparrow 3</math>)
| |
| :<math>10\uparrow\uparrow 10\uparrow\uparrow (10\uparrow)^{497}(9.73\times 10^{32})=(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math> (between <math>10\uparrow\uparrow\uparrow 4</math> and <math>10\uparrow\uparrow\uparrow 5</math>)
| |
| | |
| Similarly to the above, if the exponent of <math>(10\uparrow)</math> is not exactly given then giving a value at the right does not make sense, and we can, instead of using the power notation of <math>(10\uparrow)</math>, add 1 to the exponent of <math>(10\uparrow\uparrow)</math>, so we get e.g. <math>(10\uparrow\uparrow)^{3} (2.8\times 10^{12})</math>.
| |
| | |
| If the exponent of <math>(10\uparrow \uparrow)</math> is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and we can, instead of using the power notation of <math>(10\uparrow \uparrow)</math>, use the triple arrow operator, e.g. <math>10\uparrow\uparrow\uparrow(7.3\times 10^{6})</math>.
| |
| | |
| If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. <math>10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math> (between <math>10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 4</math> and <math>10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 5</math>). This can be done recursively, so we can have a power of the triple arrow operator.
| |
| | |
| We can proceed with operators with higher numbers of arrows, written <math>\uparrow^n</math>.
| |
| | |
| Compare this notation with the [[hyper operator]] and the [[Conway chained arrow notation]]:
| |
| :<math>a\uparrow^n b</math> = ( ''a'' → ''b'' → ''n'' ) = hyper(''a'', ''n'' + 2, ''b'')
| |
| An advantage of the first is that when considered as function of ''b'', there is a natural notation for powers of this function (just like when writing out the ''n'' arrows): <math>(a\uparrow^n)^k b</math>. For example:
| |
| | |
| :<math>(10\uparrow^2)^3 b</math> = ( 10 → ( 10 → ( 10 → ''b'' → 2 ) → 2 ) → 2 )
| |
| and only in special cases the long nested chain notation is reduced; for ''b'' = 1 we get:
| |
| :<math>10\uparrow^3 3 = (10\uparrow^2)^3 1</math> = ( 10 → 3 → 3 )
| |
| | |
| Since the ''b'' can also be very large, in general we write a number with a sequence of powers <math>(10 \uparrow^n)^{k_n}</math> with decreasing values of ''n'' (with exactly given integer exponents <math>{k_n}</math>) with at the end a number in ordinary scientific notation. Whenever a <math>{k_n}</math> is too large to be given exactly, the value of <math>{k_{n+1}}</math> is increased by 1 and everything to the right of <math>({n+1})^{k_{n+1}}</math> is rewritten.
| |
| | |
| For describing numbers approximately, deviations from the decreasing order of values of ''n'' are not needed. For example, <math>10 \uparrow (10 \uparrow \uparrow)^5 a=(10 \uparrow \uparrow)^6 a</math>, and <math>10 \uparrow (10 \uparrow \uparrow \uparrow 3)=10 \uparrow \uparrow (10 \uparrow \uparrow 10 + 1)\approx 10 \uparrow \uparrow \uparrow 3</math>. Thus we have the somewhat counterintuitive result that a number ''x'' can be so large that, in a way, ''x'' and 10<sup>x</sup> are "almost equal" (for arithmetic of large numbers see also below).
| |
| | |
| If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it acts. We can simply use a standard value at the right, say 10, and the expression reduces to <math>10 \uparrow^n 10=(10 \to 10 \to n)</math> with an approximate ''n''. For such numbers the advantage of using the upward arrow notation no longer applies, and we can also use the chain notation.
| |
| | |
| The above can be applied recursively for this ''n'', so we get the notation <math>\uparrow^n</math> in the superscript of the first arrow, etc., or we have a nested chain notation, e.g.:
| |
| | |
| :(10 → 10 → (10 → 10 → <math>3 \times 10^5</math>) ) = <math>10 \uparrow ^{10 \uparrow ^{3 \times 10^5} 10} 10 \!</math>
| |
| | |
| If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function <math>f(n)=10 \uparrow^{n} 10</math> = (10 → 10 → ''n''), these levels become functional powers of ''f'', allowing us to write a number in the form <math>f^m(n)</math> where ''m'' is given exactly and n is an integer which may or may not be given exactly (for the example: <math>f^2(3 \times 10^5)</math>. If ''n'' is large we can use any of the above for expressing it. The "roundest" of these numbers are those of the form ''f''<sup>''m''</sup>(1) = (10→10→''m''→2). For example, <math>(10 \to 10 \to 3\to 2) = 10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 \!</math>
| |
| | |
| Compare the definition of [[Graham's number]]: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus <math> G < 3\rightarrow 3\rightarrow 65\rightarrow 2 <(10 \to 10 \to 65\to 2)=f^{65}(1)</math>, but also <math> G < f^{64}(4)<f^{65}(1)</math>.
| |
| | |
| If ''m'' in <math>f^m(n)</math> is too large to give exactly we can use a fixed ''n'', e.g. ''n'' = 1, and apply the above recursively to ''m'', i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation of ''f'' this gives multiple levels of ''f''. Introducing a function <math>g(n)=f^{n}(1)</math> these levels become functional powers of ''g'', allowing us to write a number in the form <math>g^m(n)</math> where ''m'' is given exactly and n is an integer which may or may not be given exactly. We have (10→10→''m''→3) = ''g''<sup>''m''</sup>(1). If ''n'' is large we can use any of the above for expressing it. Similarly we can introduce a function ''h'', etc. If we need many such functions we can better number them instead of using a new letter every time, e.g. as a subscript, so we get numbers of the form <math>f_k^m(n)</math> where ''k'' and ''m'' are given exactly and n is an integer which may or may not be given exactly. Using ''k''=1 for the ''f'' above, ''k''=2 for ''g'', etc., we have (10→10→''n''→''k'') = <math>f_k(n)=f_{k-1}^n(1)</math>. If ''n'' is large we can use any of the above for expressing it. Thus we get a nesting of forms <math>{f_k}^{m_k}</math> where going inward the ''k'' decreases, and with as inner argument a sequence of powers <math>(10 \uparrow^n)^{p_n}</math> with decreasing values of ''n'' (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
| |
| | |
| When ''k'' is too large to be given exactly, the number concerned can be expressed as <math>{f_n}(10)</math>=(10→10→10→''n'') with an approximate ''n''. Note that the process of going from the sequence <math>10^{n}</math>=(10→''n'') to the sequence <math>10 \uparrow^n 10</math>=(10→10→''n'') is very similar to going from the latter to the sequence <math>{f_n}(10)</math>=(10→10→10→''n''): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). Numbering the subsequent versions of this function a number can be described using functions <math>{f_{qk}}^{m_{qk}}</math>, nested in [[lexicographical order]] with ''q'' the most significant number, but with decreasing order for ''q'' and for ''k''; as inner argument we have a sequence of powers <math>(10 \uparrow^n)^{p_n}</math> with decreasing values of ''n'' (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
| |
| | |
| For a number too large to write down in the Conway chained arrow notation we can describe how large it is by the length of that chain, for example only using elements 10 in the chain; in other words, we specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number we can apply the same techniques again for that.
| |
| | |
| ===Examples of numbers in numerical order===
| |
| Numbers expressible in decimal notation:
| |
| *2<sup>2</sup> = 4
| |
| *2<sup>2<sup>2</sup></sup> = 2 ↑↑ 3 = 16
| |
| *3<sup>3</sup> = 27
| |
| *4<sup>4</sup> = 256
| |
| *5<sup>5</sup> = 3125
| |
| *6<sup>6</sup> = 46,656
| |
| *<math>2^{2^{2^{2}}}</math> = 2 ↑↑ 4 = 2↑↑↑3 = 65,536
| |
| *7<sup>7</sup> = 823,543
| |
| *10<sup>6</sup> = 1,000,000 = 1 million
| |
| *8<sup>8</sup> = 16,777,216
| |
| *9<sup>9</sup> = 387,420,489
| |
| *10<sup>9</sup> = 1,000,000,000 = 1 billion
| |
| *10<sup>10</sup> = 10,000,000,000
| |
| *10<sup>12</sup> = 1,000,000,000,000 = 1 trillion
| |
| *3<sup>3<sup>3</sup></sup> = 3 ↑↑ 3 = 7,625,597,484,987 ≈ 7.63 × 10<sup>12</sup>
| |
| *10<sup>15</sup> = 1,000,000,000,000,000 = 1 million billion = 1 quadrillion
| |
| | |
| Numbers expressible in scientific notation:
| |
| *Approximate [[Observable_universe#Matter_content_-_number_of_atoms|number of atoms in the observable universe]] = 10<sup>80</sup> = 100,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
| |
| *[[googol]] = 10<sup>100</sup> = 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
| |
| *4<sup>4<sup>4</sup></sup> = 4 ↑↑ 3 ≈ 1.34 × 10<sup>154</sup> ≈ (10 ↑)<sup>2</sup> 2.2
| |
| *Approximate number of [[Planck length|Planck volumes]] composing the volume of the observable [[universe]] = 8.5 × 10<sup>184</sup>
| |
| *5<sup>5<sup>5</sup></sup> = 5 ↑↑ 3 ≈ 1.91 × 10<sup>2184</sup> ≈ (10 ↑)<sup>2</sup> 3.3
| |
| *<math>2^{2^{2^{2^2}}} = 2 \uparrow \uparrow 5 = 2^{65,536} \approx 2.0 \times 10^{19,728} \approx (10 \uparrow)^2 4.3</math>
| |
| *6<sup>6<sup>6</sup></sup> = 6 ↑↑ 3 ≈ 2.66 × 10<sup>36,305</sup> ≈ (10 ↑)<sup>2</sup> 4.6
| |
| *7<sup>7<sup>7</sup></sup> = 7 ↑↑ 3 ≈ 3.76 × 10<sup>695,974</sup> ≈ (10 ↑)<sup>2</sup> 5.8
| |
| *8<sup>8<sup>8</sup></sup> = 8 ↑↑ 3 ≈ 6.01 × 10<sup>15,151,335</sup> ≈ (10 ↑)<sup>2</sup> 7.2
| |
| *<math>M_{57,885,161} \approx 5.81 \times 10^{17,425,169} \approx 10^{10^{7.2}} = (10 \uparrow)^2 7.2</math>, the 48th and as of January 2013 the largest known [[Mersenne prime]].
| |
| *9<sup>9<sup>9</sup></sup> = 9 ↑↑ 3 ≈ 4.28 × 10<sup>369,693,099</sup> ≈ (10 ↑)<sup>2</sup> 8.6
| |
| *10<sup>10<sup>10</sup></sup> =10 ↑↑ 3 = 10<sup>10,000,000,000</sup> = (10 ↑)<sup>3</sup> 1
| |
| *<math>3^{3^{3^{3}}} = 3 \uparrow \uparrow 4 \approx 1.26 \times 10^{3,638,334,640,024} \approx (10 \uparrow)^3 1.10</math>
| |
| | |
| Numbers expressible in (10 ↑)<sup>''n''</sup> ''k'' notation:
| |
| *[[googolplex]] = <math>10^{10^{100}} = (10 \uparrow)^3 2</math>
| |
| *<math>2^{2^{2^{2^{2^2}}}} = 2 \uparrow \uparrow 6 = 2^{2^{65,536}} \approx 2^{(10 \uparrow)^2 4.3} \approx 10^{(10 \uparrow)^2 4.3} = (10 \uparrow)^3 4.3</math>
| |
| *<math>10^{10^{10^{10}}}=10 \uparrow \uparrow 4=(10 \uparrow)^4 1</math>
| |
| *<math>3^{3^{3^{3^3}}} = 3 \uparrow \uparrow 5 \approx 3^{10^{3.6 \times 10^{12}}} \approx (10 \uparrow)^4 1.10</math>
| |
| *<math>2^{2^{2^{2^{2^{2^2}}}}} = 2 \uparrow \uparrow 7 = \approx (10 \uparrow)^4 4.3</math>
| |
| *10 ↑↑ 5 = (10 ↑)<sup>5</sup> 1
| |
| *3 ↑↑ 6 ≈ (10 ↑)<sup>5</sup> 1.10
| |
| *2 ↑↑ 8 ≈ (10 ↑)<sup>5</sup> 4.3
| |
| *10 ↑↑ 6 = (10 ↑)<sup>6</sup> 1
| |
| *10 ↑↑↑ 2 = 10 ↑↑ 10 = (10 ↑)<sup>10</sup> 1
| |
| *2 ↑↑↑↑ 3 = 2 ↑↑↑ 4 = 2 ↑↑ 65,536 ≈ (10 ↑)<sup>65,533</sup> 4.3 is between 10 ↑↑ 65,533 and 10 ↑↑ 65,534
| |
| | |
| Bigger numbers:
| |
| *3 ↑↑↑ 3 = 3 ↑↑ (3 ↑↑ 3) ≈ 3 ↑↑ 7.6 × 10<sup>12</sup> ≈ 10 ↑↑ 7.6 × 10<sup>12</sup> is between (10 ↑↑)<sup>2</sup> 2 and (10 ↑↑)<sup>2</sup> 3
| |
| *<math>10\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow)^3 1</math> = ( 10 → 3 → 3 )
| |
| *<math>(10\uparrow\uparrow)^2 11</math>
| |
| *<math>(10\uparrow\uparrow)^2 10^{\,\!10^{10^{3.81\times 10^{17}}}}</math>
| |
| *<math>10\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow)^4 1</math> = ( 10 → 4 → 3 )
| |
| *<math>(10\uparrow\uparrow)^{2} (10\uparrow)^{497}(9.73\times 10^{32})</math>
| |
| *<math>10\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow)^5 1</math> = ( 10 → 5 → 3 )
| |
| *<math>10\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow)^6 1</math> = ( 10 → 6 → 3 )
| |
| *<math>10\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow)^7 1</math> = ( 10 → 7 → 3 )
| |
| *<math>10\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow)^8 1</math> = ( 10 → 8 → 3 )
| |
| *<math>10\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow)^9 1</math> = ( 10 → 9 → 3 )
| |
| *<math>10 \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow)^10 1</math> = ( 10 → 2 → 4 ) = ( 10 → 10 → 3 )
| |
| *The first term in the definition of [[Graham's number]], ''g''<sub>1</sub> = 3 ↑↑↑↑ 3 = 3 ↑↑↑ (3 ↑↑↑ 3) ≈ 3 ↑↑↑ (10 ↑↑ 7.6 × 10<sup>12</sup>) ≈ 10 ↑↑↑ (10 ↑↑ 7.6 × 10<sup>12</sup>) is between (10 ↑↑↑)<sup>2</sup> 2 and (10 ↑↑↑)<sup>2</sup> 3 (See [[Graham's number#Magnitude]])
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 3=(10 \uparrow \uparrow\uparrow)^3 1</math> = (10 → 3 → 4)
| |
| *<math>4 \uparrow \uparrow \uparrow \uparrow 4</math> = ( 4 → 4 → 4 ) <math>\approx (10 \uparrow \uparrow \uparrow)^2 (10 \uparrow \uparrow)^3 154</math>
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 4=(10 \uparrow \uparrow\uparrow)^4 1</math> = ( 10 → 4 → 4 )
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 5=(10 \uparrow \uparrow\uparrow)^5 1</math> = ( 10 → 5 → 4 )
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 6=(10 \uparrow \uparrow\uparrow)^6 1</math> = ( 10 → 6 → 4 )
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 7=(10 \uparrow \uparrow\uparrow)^7 1=</math> = ( 10 → 7 → 4 )
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 8=(10 \uparrow \uparrow\uparrow)^8 1=</math> = ( 10 → 8 → 4 )
| |
| *<math>10\uparrow\uparrow\uparrow\uparrow 9=(10 \uparrow \uparrow\uparrow)^9 1=</math> = ( 10 → 9 → 4 )
| |
| *<math>10 \uparrow \uparrow \uparrow \uparrow \uparrow 2 = 10\uparrow\uparrow\uparrow\uparrow 10=(10 \uparrow \uparrow\uparrow)^{10} 1</math> = ( 10 → 2 → 5 ) = ( 10 → 10 → 4 )
| |
| *( 2 → 3 → 2 → 2 ) = ( 2 → 3 → 8 )
| |
| *( 3 → 2 → 2 → 2 ) = ( 3 → 2 → 9 ) = ( 3 → 3 → 8 )
| |
| *( 10 → 10 → 10 ) = ( 10 → 2 → 11 )
| |
| *( 10 → 2 → 2 → 2 ) = ( 10 → 2 → 100 )
| |
| *( 10 → 10 → 2 → 2 ) = ( 10 → 2 → <math>10^{10}</math> ) = <math>10 \uparrow ^{10^{10}} 10 \!</math>
| |
| *The second term in the definition of Graham's number, ''g''<sub>2</sub> = 3 ↑<sup>''g''<sub>1</sub></sup> 3 > 10 ↑<sup>''g''<sub>1</sub> - 1</sup> 10.
| |
| *( 10 → 10 → 3 → 2 ) = (10 → 10 → (10 → 10 → <math>10^{10}</math>) ) = <math>10 \uparrow ^{10 \uparrow ^{10^{10}} 10} 10 \!</math>
| |
| *''g''<sub>3</sub> = (3 → 3 → ''g''<sub>2</sub>) > (10 → 10 → ''g''<sub>2</sub> - 1) > (10 → 10 → 3 → 2)
| |
| *''g''<sub>4</sub> = (3 → 3 → ''g''<sub>3</sub>) > (10 → 10 → ''g''<sub>3</sub> - 1) > (10 → 10 → 4 → 2)
| |
| *...
| |
| *''g''<sub>9</sub> = (3 → 3 → ''g''<sub>8</sub>) is between (10 → 10 → 9 → 2) and (10 → 10 → 10 → 2)
| |
| *( 10 → 10 → 10 → 2 )
| |
| *''g''<sub>10</sub> = (3 → 3 → ''g''<sub>9</sub>) is between (10 → 10 → 10 → 2) and (10 → 10 → 11 → 2)
| |
| *...
| |
| *''g''<sub>63</sub> = (3 → 3 → ''g''<sub>62</sub>) is between (10 → 10 → 63 → 2) and (10 → 10 → 64 → 2)
| |
| *( 10 → 10 → 64 → 2 )
| |
| *[[Graham's number]], ''g''<sub>64</sub><ref>Regarding the comparison with the previous value: <math>10\uparrow ^n 10 < 3 \uparrow ^{n+1} 3</math>, so starting the 64 steps with 1 instead of 4 more than compensates for replacing the numbers 3 by 10</ref>
| |
| *( 10 → 10 → 65 → 2 )
| |
| *( 10 → 10 → 10 → 3 )
| |
| *( 10 → 10 → 10 → 4 )
| |
| | |
| ==Comparison of base values==
| |
| The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers, and the arithmetic.
| |
| | |
| <math>100^{12}=10^{24}</math>, with base 10 the exponent is doubled.
| |
| | |
| <math>100^{100^{12}}=10^{2*10^{24}}</math>, ditto.
| |
| | |
| <math>100^{100^{100^{12}}}=10^{10^{2*10^{24}+0.3}}</math>, the highest exponent is very little more than doubled.
| |
| | |
| *<math>100\uparrow\uparrow 2=10^ {200} </math>
| |
| *<math>100\uparrow\uparrow 3=10^ {2 \times 10^ {200}}</math>
| |
| *<math>100\uparrow\uparrow 4=(10\uparrow)^2 (2 \times 10^ {200}+0.3)=(10\uparrow)^2 (2\times 10^ {200})=(10\uparrow)^3 200.3=(10\uparrow)^4 2.3</math>
| |
| *<math>100\uparrow\uparrow n=(10\uparrow)^{n-2} (2 \times 10^ {200})=(10\uparrow)^{n-1} 200.3=(10\uparrow)^{n}2.3<10\uparrow\uparrow (n+1)</math> (thus if ''n'' is large it seems fair to say that <math>100\uparrow\uparrow n</math> is "approximately equal to" <math>10\uparrow\uparrow n</math>)
| |
| *<math>100\uparrow\uparrow\uparrow 2=(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow)^{100} 2.3</math>
| |
| *<math>100\uparrow\uparrow\uparrow 3=10\uparrow\uparrow(10\uparrow)^{98} (2 \times 10^ {200})=10\uparrow\uparrow(10\uparrow)^{100} 2.3</math>
| |
| *<math>100\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{98} (2 \times 10^ {200})=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{100} 2.3<10\uparrow\uparrow\uparrow (n+1)</math> (compare <math>10\uparrow\uparrow\uparrow n=(10\uparrow\uparrow)^{n-2}(10\uparrow)^{10}1<10\uparrow\uparrow\uparrow (n+1)</math>; thus if ''n'' is large it seems fair to say that <math>100\uparrow\uparrow\uparrow n</math> is "approximately equal to" <math>10\uparrow\uparrow\uparrow n</math>)
| |
| *<math>100\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow 2=(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>)
| |
| *<math>100\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow 3=10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>)
| |
| *<math>100\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{98}(10\uparrow)^{100} 2.3</math> (compare <math>10\uparrow\uparrow\uparrow\uparrow n=(10\uparrow\uparrow\uparrow)^{n-2}(10\uparrow\uparrow)^{8}(10\uparrow)^{10}1</math>; if ''n'' is large this is "approximately" equal)
| |
| | |
| == Accuracy ==
| |
| | |
| Note that for a number <math>10^n</math>, one unit change in ''n'' changes the result by a factor 10. In a number like <math>10^{\,\!6.2 \times 10^3}</math>, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor <math>10^{50}</math> too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).
| |
| | |
| === Accuracy for very large numbers ===
| |
| | |
| In the case of an approximation of an extremely large number, the relative error may be large, yet there may still be a sense in which we want to consider the numbers as "close in magnitude". For example, consider
| |
| | |
| :<math>10^{10}</math> and <math>10^9</math>
| |
| | |
| The relative error is
| |
| | |
| :<math>1 - \frac{10^9}{10^{10}} = 1 - \frac{1}{10} = 90\%</math>
| |
| | |
| a large relative error. However, we can also consider the relative error in the [[logarithms]]; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%.
| |
| | |
| The point is that [[exponential function]]s magnify relative errors greatly – if ''a'' and ''b'' have a small relative error,
| |
| | |
| :<math>10^a</math> and <math>10^b</math>
| |
| | |
| the relative error is larger, and
| |
| | |
| :<math>10^{10^a}</math> and <math>10^{10^b}</math>
| |
| | |
| will have even larger relative error. The question then becomes: on which level of iterated logarithms do we wish to compare two numbers? There is a sense in which we may want to consider
| |
| | |
| :<math>10^{10^{10}}</math> and <math>10^{10^9}</math>
| |
| | |
| to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:
| |
| | |
| :<math>\log_{10}(\log_{10}(10^{10^{10}})) = 10</math> and <math>\log_{10}(\log_{10}(10^{10^9})) = 9</math>
| |
| | |
| Such comparisons of iterated logarithms are common, e.g., in [[analytic number theory]].
| |
| | |
| ===Approximate arithmetic for very large numbers===
| |
| | |
| There are some general rules relating to the usual arithmetic operations performed on very large numbers:
| |
| | |
| *The sum and the product of two very large numbers are both "approximately" equal to the larger one.
| |
| *<math>(10^a)^{\,\!10^b}=10^{a 10^b}=10^{10^{b+\log _{10} a}}</math>
| |
| Hence:
| |
| *A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large n we have <math>n^n\approx 10^n</math> (see e.g. [[Steinhaus-Moser notation#Mega|the computation of mega]]) and also <math>2^n\approx 10^n</math>. Thus <math>2\uparrow\uparrow 65536 \approx 10\uparrow\uparrow 65533</math>, see [[Knuth's up-arrow notation#Tables of values|table]].
| |
| | |
| == Large numbers in some noncomputable sequences ==
| |
| | |
| The [[busy beaver]] function Σ is an example of a function which grows faster than any [[Computability theory (computer science)|computable]] function. Its value for even relatively small input is huge. The values of Σ(''n'') for ''n'' = 1, 2, 3, 4 are 1, 4, 6, 13 {{OEIS|id=A028444}}. Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 3.5×10<sup>18267</sup>.
| |
| | |
| == Infinite numbers ==
| |
| {{Main|cardinal number}}
| |
| {{See also|large cardinal|Mahlo cardinal|totally indescribable cardinal}}
| |
| Although all these numbers above are very large, they are all still decidedly [[finite set|finite]]. Certain fields of mathematics define [[Infinity|infinite]] and [[transfinite number]]s. For example, [[aleph-null]] is the [[cardinality]] of the [[infinite set]] of [[natural number]]s, and [[aleph-one]] is the next greatest cardinal number. <math>\mathfrak{c}</math> is the [[cardinality of the continuum|cardinality of the reals]]. The proposition that <math>\mathfrak{c} = \aleph_1</math> is known as the [[continuum hypothesis]].
| |
| | |
| == Notations ==
| |
| Some notations for extremely large numbers:
| |
| *[[Knuth's up-arrow notation]] / [[hyper operator]]s / [[Ackermann function]], including [[tetration]]
| |
| *[[Conway chained arrow notation]]
| |
| *[[Steinhaus-Moser notation]]; apart from the method of construction of large numbers, this also involves a graphical notation with [[polygons]]; alternative notations, like a more conventional function notation, can also be used with the same functions.
| |
| These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever faster increasing functions can easily be constructed recursively by applying these functions with large integers as argument.
| |
| | |
| Note that a function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.
| |
| | |
| == See also ==
| |
| <div style="-moz-column-count:2; column-count:2;">
| |
| | |
| * [[Arbitrary-precision arithmetic]]
| |
| * [[Dirac large numbers hypothesis]]
| |
| * [[Exponential growth]]
| |
| * [[Fast-growing hierarchy]] of functions
| |
| * [[Graham's Number]]
| |
| * [[History of large numbers]]
| |
| * [[Human scale]]
| |
| * [[Myriad#Sinosphere|Myriads (10,000) in Sinosphere]]
| |
| * [[Law of large numbers]]
| |
| * [[Names of large numbers]]
| |
| * [[Small number]]
| |
| * [[Tetration]]
| |
| | |
| </div>
| |
| | |
| ==Notes and references==
| |
| {{Reflist}}
| |
| | |
| {{Large numbers}}
| |
| | |
| {{DEFAULTSORT:Large Numbers}}
| |
| [[Category:Mathematical notation]]
| |
| [[Category:Large numbers|*]]
| |