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:
Other large numbers are found in astronomy:
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.
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.
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 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.
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