Large numbers, far beyond those encountered in everyday life—such as simple counting or financial transactions—play a crucial role in various domains.
These expansive quantities appear prominently in mathematics, cosmology, cryptography, and statistical mechanics.
While they often manifest as large positive integers, they can also take other forms in different contexts (such as P-adic number).
Googology delves into the naming conventions and properties of these immense numerical entities.
[1][2] Since the customary, traditional (non-technical) decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation.
Values that vary dramatically can be represented and compared graphically via logarithmic scale.
Standard notation is a variation of English's natural language numbering, where it is shortened into a suffix.
Scientific notation was devised to represent the vast range of values encountered in scientific research in a format that is more compact than traditional formats yet allows for high precision when called for.
The factor is intended to make reading comprehension easier than a lengthy series of zeros.
For instance, according to the prevailing Big Bang model, our universe is approximately 13.8 billion years old (equivalent to 4.355×1017 seconds).
The observable universe spans an incredible 93 billion light years (approximately 8.8×1026 meters) and hosts around 5×1022 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope).
[7] 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 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.
[8][9] This time assumes a statistical model subject to Poincaré recurrence.
Combinatorial processes give rise to astonishingly large numbers.
In statistical mechanics, combinatorial numbers reach such immense magnitudes that they are often expressed using logarithms.
Gödel numbers, along with similar representations of bit-strings in algorithmic information theory, are vast—even for mathematical statements of moderate length.
To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b".
the power tower can be made one higher, replacing x by log10x, or find x from the lower-tower representation of the log10 of the whole number.
If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g.
If the right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g.
Then it is possible to proceed with operators with higher numbers of arrows, written
For example: and only in special cases the long nested chain notation is reduced; for
obtains: Since the b can also be very large, in general it can be written instead a number with a sequence of powers
For describing numbers approximately, deviations from the decreasing order of values of n are not needed.
= (10 → 10 → n), these levels become functional powers of f, allowing us to write a number in the form
is too large to give exactly, it is possible to use a fixed n, e.g. n = 1, and apply the above recursively to m, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc.
these levels become functional powers of g, allowing us to write a number in the form
, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more.
There are some general rules relating to the usual arithmetic operations performed on very large numbers: Hence: Given a strictly increasing integer sequence/function
This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.