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 See also
3 Even larger numbers
4 Uncomputably large numbers
5 To infinity, and beyond
6 Notations
7 See also
8 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:

cigarettes smoked in the United States in one year, on the order of 10¹²
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

Other examples are given in Orders_of_magnitude_(numbers).

Please add other examples to the appropriate Orders of magnitude article.

Other large numbers are found in astronomy:

number of atoms in the visible universe, perhaps 10⁷⁹ to 10⁸¹
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.

G�del 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 G�del numbers of typical mathematical propositions.

The Googol (10¹⁰⁰) was defined in order to create the concept of a Googolplex (10^googol).
Skewes' numbers
Graham's number
Moser's number
Steinhaus' Mega and Megiston

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 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.

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".

Notations

Many mathematicians have devised special notations to denote exactly how big a "large number" really is.

The Knuth arrow is a very simple notation that describes power towers. For larger power towers, it is preferrable to use hyper notation.

Steinhaus polygon notation and Moser polygon notation extend the idea further and use polygons to show this idea.