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Large numbers

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Large numbers are numbers that are significantly larger than those typically used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts. The study of nomenclature and properties of large numbers is sometimes called googology.One Million Things: A Visual Encyclopedia «The study of large numbers is called googology»
Very large numbers often occur in fields such as mathematics, 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.

In the everyday world

Scientific notation was created to handle the wide range of values that occur in scientific study. 1.0 × 109, for example, means one billion, a 1 followed by nine zeros: 1 000 000 000, and 1.0 × 10−9 means one billionth, or 0.000 000 001. Writing 109 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is.
Examples of large numbers describing everyday real-world objects include:
  • The number of cells in the human body (estimated at 3.72 × 1013)
  • The number of bits on a computer hard disk ( , typically about 1013, 1–2 TB)
  • The number of neuronal connections in the human brain (estimated at 1014)
  • The Avogadro constant is the number of “elementary entities” (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 approximately .
  • The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3±3.6)×1037
  • The mass of Earth consists of about 4x1051 nucleons
  • The estimated number of atoms in the observable universe (1080)
  • The lower bound on the game-tree complexity of chess, also known as the “Shannon number” (estimated at around 10120)


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 × 1017 seconds) old, and that the observable universe is 93 billion light years across (8.8 × 1026 metres), and contains about 5 × 1022 stars, organized into around 125 billion (1.25 × 1011) galaxies, according to Hubble Space Telescope observations. There are about 1080 atoms in the observable universe, by rough estimation.Atoms in the Universe. Universe Today. 30-07-2009. Retrieved 02-03-13.
According to Don Page, physicist at the University of Alberta, Canada, the longest finite time that has so far been explicitly calculated by any physicist is
::10^{10^{10^{10^{10^{1.1 \mbox{ years
which corresponds to the scale of an estimated 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 inflationary model with an inflaton whose mass is 10−6 Planck masses.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. . .How to Get A Googolplex This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history repeats itself arbitrarily many times due to Ergodic hypothesisproperties 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 permutations 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 logarithms.
Gödel numbers, 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 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.

"Billions and billions"

To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b". Sagan never did, however, say "billions and billions". The public's association of the phrase and Sagan came from a Tonight Show skit. Parodying Sagan's affect, Johnny Carson quipped "billions and billions".Carl Sagan takes questions more from his 'Wonder and Skepticism' CSICOP 1994 keynote, Skeptical Inquirer The phrase has, however, now become a humorous fictitious number—the Sagan. Cf., Carl Sagan#Sagan unitsSagan Unit.


  • googol = 10^{100
  • centillion = 10^{303 or 10^{600, depending on number naming system
  • millinillion = 10^{3003 or 10^{6000, depending on number naming system
  • millinillinillion = 10^{3000003 or 10^{6000000, depending on number naming system
  • The largest known Smith number = (101031−1) × (104594 + 3 + 1)1476
  • The largest known Mersenne prime = 2^{82,589,933-1 (as of December 21, 2018)
  • googolplex = 10^{\text{googol=10^{10^{100
  • Skewes' numbers: the first is approximately 10^{10^{10^{34, the second 10^{10^{10^{964
  • Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using Knuth's up-arrow notation
  • Rayo's number is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at MIT on 26 January 2007

Standardized system of writing

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 5×104 and 2×105, compare the exponents first, in this case 5 > 4, so 2×105 > 5×104. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×104 > 2×104 because 5 > 2.
Tetration with base 10 gives the sequence 10 \uparrow \uparrow n=10 \to n \to 2=(10\uparrow)^n 1, the power towers of numbers 10, where (10\uparrow)^n denotes a functional power of the function f(n)=10^n (the function also expressed by the suffix "-plex" as in googolplex, see 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 (10\uparrow)^n a, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g. 10^{10^{10^{10^{10^{4.829 = (10\uparrow)^5 4.829, a number between 10\uparrow\uparrow 5 and 10\uparrow\uparrow 6 (note that 10 \uparrow\uparrow n < (10\uparrow)^n a < 10 \uparrow\uparrow (n+1) if 1 < a < 10). (See also extension of tetration to real heights.)
Thus googolplex is 10^{10^{100 = (10\uparrow)^2 100 = (10\uparrow)^3 2
Another example:
2 \uparrow\uparrow\uparrow 4 =
\qquad\quad\ \ \ 65,536\mbox{ copies of 2 \end{matrix
\approx (10\uparrow)^{65,531(6 \times 10^{19,728) \approx (10\uparrow)^{65,533 4.3
(between 10\uparrow\uparrow 65,533 and 10\uparrow\uparrow 65,534)
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 log_{10 to get a number between 1 and 10. Thus, the number is between 10\uparrow\uparrow n and 10\uparrow\uparrow (n+1). 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 1010, or the next, between 0 and 1.
Note that
I.e., if a number x is too large for a representation (10\uparrow)^{nx we can make the power tower one higher, replacing x by log10x, or find x from the lower-tower representation of the log10 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. 10\uparrow\uparrow(7.21\times 10^8). If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.
10\uparrow\uparrow 10^{\,\!10^{10^{3.81\times 10^{17 (between 10\uparrow\uparrow\uparrow 2 and 10\uparrow\uparrow\uparrow 3)
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) (between 10\uparrow\uparrow\uparrow 4 and 10\uparrow\uparrow\uparrow 5)
Similarly to the above, if the exponent of (10\uparrow) 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 (10\uparrow), add 1 to the exponent of (10\uparrow\uparrow), so we get e.g. (10\uparrow\uparrow)^{3 (2.8\times 10^{12).
If the exponent of (10\uparrow \uparrow) 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 (10\uparrow \uparrow), use the triple arrow operator, e.g. 10\uparrow\uparrow\uparrow(7.3\times 10^{6).
If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. 10\uparrow\uparrow\uparrow(10\uparrow\uparrow)^{2 (10\uparrow)^{497(9.73\times 10^{32) (between 10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 4 and 10\uparrow\uparrow\uparrow 10\uparrow\uparrow\uparrow 5). 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 \uparrow^n.
Compare this notation with the hyper operator and the Conway chained arrow notation:
a\uparrow^n b = ( abn ) = hyper(an + 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): (a\uparrow^n)^k b. For example:
(10\uparrow^2)^3 b = ( 10 → ( 10 → ( 10 → b → 2 ) → 2 ) → 2 )

and only in special cases the long nested chain notation is reduced; for b = 1 we get:
10\uparrow^3 3 = (10\uparrow^2)^3 1 = ( 10 → 3 → 3 )
Since the b can also be very large, in general we write a number with a sequence of powers (10 \uparrow^n)^{k_n with decreasing values of n (with exactly given integer exponents {k_n) with at the end a number in ordinary scientific notation. Whenever a {k_n is too large to be given exactly, the value of {k_{n+1 is increased by 1 and everything to the right of ({n+1)^{k_{n+1 is rewritten.
For describing numbers approximately, deviations from the decreasing order of values of n are not needed. For example, 10 \uparrow (10 \uparrow \uparrow)^5 a=(10 \uparrow \uparrow)^6 a, and 10 \uparrow (10 \uparrow \uparrow \uparrow 3)=10 \uparrow \uparrow (10 \uparrow \uparrow 10 + 1)\approx 10 \uparrow \uparrow \uparrow 3. Thus we have the somewhat counterintuitive result that a number x can be so large that, in a way, x and 10x 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 10 \uparrow^n 10=(10 \to 10 \to n) 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 \uparrow^n in the superscript of the first arrow, etc., or we have a nested chain notation, e.g.:
(10 → 10 → (10 → 10 → 3 \times 10^5) ) = 10 \uparrow ^{10 \uparrow ^{3 \times 10^5 10 10
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 f(n)=10 \uparrow^{n 10 = (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form f^m(n) where m is given exactly and n is an integer which may or may not be given exactly (for example: f^2(3 \times 10^5)). If n is large we can use any of the above for expressing it. The "roundest" of these numbers are those of the form fm(1) = (10→10→m→2). For example, (10 \to 10 \to 3\to 2) = 10 \uparrow ^{10 \uparrow ^{10^{10 10 10
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 G < 3\rightarrow 3\rightarrow 65\rightarrow 2 2 ↑n – 1 n > 3 ↑n – 2 3 + 2, and hence fω(gk + 2) > gk+1 + 2
  • fω(n) > 2 ↑n – 1 n = (2 → nn-1) = (2 → nn-1 → 1) (using Conway chained arrow notation)
  • fω+1(n) = fωn(n) > (2 → nn-1 → 2) (because if gk(n) = X → nk then X → nk+1 = gkn(1))
  • fω+k(n) > (2 → nn-1 → k+1) > (nnk)
  • fω2(n) = fω+n(n) > (nnn) = (nnn→ 1)
  • fω2+k(n) > (nnnk)
  • fω3(n) > (nnnn)
  • fωk(n) > (nn → ... → nn) (Chain of k+1 ns)
  • fω2(n) = fωn(n) > (nn → ... → nn) (Chain of n+1 ns)

  • In some noncomputable sequences

    The busy beaver function Σ is an example of a function which grows faster than any 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 . Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 3.5×1018267.

    Infinite numbers

    Although all the numbers discussed above are very large, they are all still decidedly finite. Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite set of natural numbers, and aleph-one is the next greatest cardinal number. \mathfrak{c is the cardinality of the reals. The proposition that \mathfrak{c = \aleph_1 is known as the continuum hypothesis.

    See also

    • Arbitrary-precision arithmetic
    • List of arbitrary-precision arithmetic software
    • Dirac large numbers hypothesis
    • Exponential growth
    • Fast-growing hierarchy of functions
    • Googol
    • Googolplex
    • Graham's number
    • History of large numbers
    • Human scale
    • Largest number
    • Law of large numbers
    • Myriads (10,000) in East Asia
    • Names of large numbers
    • Power of two
    • Power of 10
    • Tetration

    Category:Mathematical notation

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