Large numbers

Large numbers are numbers that are large compared with the numbers used in everyday life. Very large numbers often occur in fields such as mathematics, cosmology and cryptography. Sometimes people refer to numbers as being "astronomically large". However, mathematically it is easy to define numbers that are much larger than occur even in astronomy.

This article is about large numbers themselves. See names of large numbers for information on how large numbers are named in English.

Large numbers in the everyday world

Large numbers are often found in science, and scientific notation was created to handle both these large numbers and also very small numbers. Some large numbers apply to things in the everyday world.

Examples of large numbers describing everyday real-world objects are:

• cigarettes smoked in the United States in one year, on the order of 10¹²
• bits on a computer hard disk (typically 10¹² to 10¹³)
• number of cells in the human body > 10¹⁴
• number of neuron connections in the human brain, maybe 10¹⁴
• Avogadro's number, approximately 6.022 × 10²³

Other examples are given in Orders of magnitude (numbers).

"Astronomically large" numbers

Other large numbers are found in astronomy:

• number of atoms in the visible universe, perhaps 10⁷⁹ to 10⁸¹, see Mass, Size, and Density of the Universe
• for astronomical numbers related to distance and time, see:
  • orders of magnitude (length)
  • orders of magnitude (time)

Large numbers are found in fields such as mathematics and cryptography.

The MD5 hash function generates 128-bit results. There are thus 2¹²⁸ (approximately 3.402×10³⁸) possible MD5 hash values. If the MD5 function is a good hash function, the chance of a document having a particular hash value is 2^-128, a value that can be regarded as equivalent to zero for most practical purposes. (But see birthday paradox.)

However, this is still a small number compared with the estimated number of atoms in the Earth, still less compared with the estimated number of atoms in the observable universe.

Even larger numbers

Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations of a set of unique objects, grows very rapidly with the number of objects.

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.

Examples:

• googol = $10^{100}$
• googolplex = $10^{\mbox{googol}}=10^{\,\!10^{100}}={\mbox{googol}}^{10^{98}}$ It is the number of states a system can be in that consists of $10^{98}$ particles which can each be in googol states.

• centillion = $10^{303}$ or $10^{600}$, depending on number naming system
• Skewes' numbers: the first is ca. $10^{\,\!10^{10^{34}}}$, the second $10^{\,\!10^{10^{1000}}}$

The total amount of printed material in the world is 1.6 × 10¹⁸ bits, therefore the contents can be represented by a number which is ca. $2^{1.6\times 10^{18}}\approx 10^{4.8\times 10^{17}}$

For a "power tower", the most relevant for the value are the height and the last few values. Compare with googolplex:

• $10^{\,\!100^{10}}=10^{10^{20}}$
• $100^{\,\!10^{10}}=10^{10^{10.3}}$

Also compare:

• $1.1^{\,\!1.1^{1.1^{1000}}}=10^{10^{1.02*10^{40}}}$
• $1000^{\,\!1000^{1000}}=10^{10^{3000.48}}$

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 (however, if these numbers are made 1 or less, that greatly changes the result). Comparing the last number with $10^{\,\!10^{10}}$, in the number 3000.48, the 1000 originates from the third number 1000 in the original power tower, a factor 3 comes from the second number 1000, and the minor term 0.48 comes from the first number 1000.

A very large number written with just three digits and ordinary exponentiation is $9^{\,\!9^{9}}\approx 10^{369,693,100}$.

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.

Tetration with base 10 can be used for very round numbers, each representing an order of magnitude in a generalized sense.

Numbers in between can be expressed with a power tower of numbers 10, with at the top a regular integer, possibly in scientific notation, e.g. $10^{\,\!10^{10^{10^{10^{4.829*10^{183230}}}}}}$, a number between $10\uparrow \uparrow 7$ and $10\uparrow \uparrow 8$ (if the exponent quite at the top is between 10 and $10^{10}$, like here, the number like the 7 here is the height).

If the height is too large to write out the whole power tower, a notation like $(10\uparrow )^{183}(3.12*10^{6})$ can be used, where $(10\uparrow )^{183}$ 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).

Various names are used for this representation:

• base-10 exponentiated tower form
• tetrated-scientific notation
• incomplete (power) tower

The notation $(10\uparrow )^{183}(3.12*10^{6})$ is in ASCII ((10^)^183)3.12e6; a proposed simplification is 10^^183@3.12e6

Thus googolplex = 10^^2@100 = 10^^3@2 = 10^^4@0.301; which notation is chosen may be considered on a number-by-number basis, or uniformly. In the latter case comparing numbers is easier. To standardize the range of the upper value (after the @), one can choose one of the ranges 0-1, 1-10, or 10-1e10:

• In the case of the range 0-1, an even shorter notation is like 10^^3.301; it provides at the same time a generalisation of 10^^x to real x>-2 (for this purpose the integer before the point is one less than in the previous notation). This may or may not be suitable depending on required smoothness and other properties; it is monotonically increasing and satisfies 10^^(x+1) = 10^(10^^x), but it is only piecewise differentiable. See also Extension of tetration to real numbers.
• The range 10-1e10 brings the notation closer to ordinary scientific notation, and the notation reduces to it if the number is itself in that range (the part "10^^0@" can be dispensed with).

Another example:

$2\uparrow \uparrow \uparrow 4={\begin{matrix}\underbrace {2_{}^{2^{{}^{.\,^{.\,^{.\,^{2}}}}}}} \\\qquad \quad \ \ \ 65,536{\mbox{ copies of }}2\end{matrix}}\approx (10\uparrow )^{65,531}(6.0\times 10^{19,728})$ (between $10\uparrow \uparrow 65,533$ and $10\uparrow \uparrow 65,534$)

The "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (say n) one has to take the $log_{10}$ to get a number between 1 and 10. Then the number is between $10\uparrow \uparrow n$ and $10\uparrow \uparrow (n+1)$

An obvious property that is yet worth mentioning is:

$10^{(10\uparrow )^{n}x}=(10\uparrow )^{n}10^{x}$

I.e., if a number x is too large for a representation $(10\uparrow )^{n}x$ we can make the power tower one higher, replacing x by $log_{10}x$, or find x from the lower-tower representation of the $log_{10}$ 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 not exactly given then giving a value at the top does not make sense, and a notation like $10\uparrow \uparrow (7.21*10^{8})$ can be used.

If the value after the double arrow is a very large number itself, the above can recursively be applied on that value.

Examples:

$10\uparrow \uparrow 10^{\,\!10^{10^{3.81*10^{17}}}}$ (between $10\uparrow \uparrow \uparrow 2$ and $10\uparrow \uparrow \uparrow 3$)

$10\uparrow \uparrow 10\uparrow \uparrow (10\uparrow )^{497}(9.73*10^{32})$ (between $10\uparrow \uparrow \uparrow 4$ and $10\uparrow \uparrow \uparrow 5$)

Some large numbers which one may try to express in such standard forms include:

• Graham's number
• Steinhaus' Mega and Megiston, Moser's number

External link: Notable Properties of Specific Numbers (last page of a series which treats the numbers in ascending order, hence the largest numbers in the series)

Accuracy

Note that for a number $10^{p}$, one unit change in p changes the result by a factor 10. In a number like $10^{\,\!6.2\times 10^{3}}$, with the 6.2 the result of proper rounding, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor $10^{50}$ too large or too small. This is seemingly an extremely poor accuracy, but for such a large number it may be considered fair. The idea that it is the relative error that counts (a large error in a large number may be relatively small and therefore acceptable), is taken a step further here: the number is so large that even a large relative error may be acceptable. Perhaps what counts is the relative error in the exponent.

Approximate arithmetic for very large numbers

In this context approximately equal may for example mean that two numbers are both written $10^{\,\!10^{10^{10^{10^{4.829*10^{183230}}}}}}$, with the true values instead of 4.829 being e.g. 4.8293 and 4.8288.

• The sum and the product of two very large numbers are both approximately equal to the larger one.
• $(10^{a})^{\,\!10^{b}}=10^{a10^{b}}=10^{10^{b+\log _{10}a}}$

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 $n^{n}\approx 10^{n}$ (see e.g. the computation of mega) and also $2^{n}\approx 10^{n}$. Thus $2\uparrow \uparrow 65536>10\uparrow \uparrow 65533$, see table.

Uncomputably large numbers

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 1.29×10⁸⁶⁵.

Infinite numbers

See main article cardinal number

Although all these numbers above are very large, they are all still finite. Some fields of mathematics define infinite and transfinite numbers.

• aleph-null is the cardinality of the infinite set of the integers
• aleph-one is the next highest cardinal number.
• C is the cardinality of the reals. The proposition that C=Aleph-one is known as the continuum hypothesis.
• Mahlo cardinals
• Indescribable cardinals

Beyond all these, Georg Cantor's conception of the Absolute Infinite surely represents the absolute largest possible concept of "large number".

Notations

Some notations for extremely large numbers:

• Knuth's up-arrow notation / hyper operators / 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.

See also

• Orders of magnitude
• Orders of magnitude (numbers)
• History of large numbers
• Law of large numbers
• Names of large numbers
• Exponential growth
• Bignums
• Small numbers
• Continuum hypothesis
• Large cardinals
• Human scale
• Number names

External links

• "Large Numbers" and "Notable Properties of Specific Numbers" - Robert Munafo
• Big numbers - Susan Stepney
• Hamlet is a Big Number - Jim Blowers (1), (2)
• How to calculate with large numbers - Dave L. Renfro
• Graham's number and rapidly growing functions - Dave L. Renfro
• Discussion and modification of notation
• Ackermann’s function and new arithmetical operations - Rubtsov and Romerio (pdf)