Large number

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".

Table of contents 1 Large numbers in the everyday world 2 Even larger numbers 3 Uncomputably large numbers 4 To infinity, and beyond 5 External links

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:

• bits on a computer hard disk (typically 10¹¹ to 10¹²)
• 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
• please add more to this list

Other large numbers are found in astronomy:

• distance to the nearest star
• distance to the nearest galaxy
• diameter of the observable universe
• number of atoms in the visible universe, perhaps 10⁷⁹ to 10⁸¹

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 function is 2^-128, a value that can be regarded as equivalent to zero for most practical purposes.

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.

• The Googol (10¹⁰⁰) was defined in order to create the concept of a Googolplex (10^googol).
• Skewes' number
• Graham's number
• Moser's number[?]

Uncomputably large numbers

The busy beaver function Σ is an example of a function which grows faster than any computable function. Consequently, its value for even relatively small input is huge. The first few values of Σ(n) for n = 1, 2, ... are 1, 4, 6 and 13. Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 1.29×10⁸⁶⁵.

To infinity, and beyond

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
• 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".

...to be written... this is a stub article

See also:

• Orders of magnitude
• history of large numbers
• law of large numbers
• Ackermann's function[?]
• bignums
• small numbers
• continuum hypothesis
• human scale

External links

• http://home.earthlink.net/~mrob/pub/math/largenum
• http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm