Large numbers are
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as
mathematics,
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
,
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
, and
statistical mechanics. They are typically large positive
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s, or more generally, large positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, but may also be other numbers in other contexts.
Googology is the study of nomenclature and properties of large numbers.
In the everyday world
Scientific notation
Scientific notation is a way of expressing numbers that are too large or too small (usually would result in a long string of digits) to be conveniently written in decimal form. It may be referred to as scientific form or standard index form, o ...
was created to handle the wide range of values that occur in scientific study. 1.0 × 10
9, for example, means one
billion
Billion is a word for a large number, and it has two distinct definitions:
*1,000,000,000, i.e. one thousand million, or (ten to the ninth power), as defined on the short scale. This is its only current meaning in English.
* 1,000,000,000,000, i. ...
, or a 1 followed by nine zeros: 1 000 000 000. The
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
, 1.0 × 10
−9, means one billionth, or 0.000 000 001. Writing 10
9 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is.
Examples of large numbers describing everyday real-world objects include:
* The number of
cells in the human body (estimated at 3.72 × 10
13)
* The number of
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s on a computer
hard disk (, typically about 10
13, 1–2
TB)
* The number of
neuronal connections in the human brain (estimated at 10
14)
* The
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
is the number of “elementary entities” (usually atoms or molecules) in one
mole
Mole (or Molé) may refer to:
Animals
* Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America
* Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
; the number of atoms in 12 grams of
carbon-12 approximately .
* The total number of
DNA base pairs within the entire
biomass on Earth, as a possible approximation of global
biodiversity
Biodiversity or biological diversity is the variety and variability of life on Earth. Biodiversity is a measure of variation at the genetic (''genetic variability''), species (''species diversity''), and ecosystem (''ecosystem diversity'') l ...
, is estimated at (5.3 ± 3.6) × 10
37
* The mass of Earth consists of about 4 × 10
51 nucleon
In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number).
Until the 1960s, nucleons were ...
s
* The estimated number of
atom
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, ...
s in the
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
(10
80)
* The lower bound on the game-tree complexity of chess, also known as the “
Shannon number
The Shannon number, named after the American mathematician Claude Shannon, is a conservative lower bound of the game-tree complexity of chess of 10120, based on an average of about 103 possibilities for a pair of moves consisting of a move for Wh ...
” (estimated at around 10
120)
Astronomical
Other large numbers, as regards length and time, are found in
astronomy
Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
and
cosmology
Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
. For example, the current
Big Bang model
The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
suggests that the universe is 13.8 billion years (4.355 × 10
17 seconds) old, and that the
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
is 93 billion
light years
A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 1012 ...
across (8.8 × 10
26 metres), and contains about 5 × 10
22 stars, organized into around 125 billion (1.25 × 10
11) galaxies, according to Hubble Space Telescope observations. There are about 10
80 atoms in the
observable universe
The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these ob ...
, by rough estimation.
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 Inflationism is a heterodox economic, fiscal, or monetary policy, that predicts that a substantial level of inflation is harmless, desirable or even advantageous. Similarly, inflationist economists advocate for an inflationist policy.
Mainstream ec ...
model with an
inflaton
The inflaton field is a hypothetical scalar field which is conjectured to have driven cosmic inflation in the very early universe.
The field, originally postulated by Alan Guth, provides a mechanism by which a period of rapid expansion from 10&m ...
whose mass is 10
−6 Planck masses.
[Information Loss in Black Holes and/or Conscious Beings?, Don N. Page, ''Heat Kernel Techniques and Quantum Gravity'' (1995), S. A. Fulling (ed), p. 461. Discourses in Mathematics and its Applications, No. 4, Texas A&M University Department of Mathematics. . .] This time assumes a statistical model subject to Poincaré recurrence. A much simplified way of thinking about this time is in a model where the universe's history
repeats itself arbitrarily many times due to
properties of statistical mechanics; this is the time scale when it will first be somewhat similar (for a reasonable choice of "similar") to its current state again.
Combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
processes rapidly generate even larger numbers. The
factorial function, which defines the number of
permutations on a set of fixed objects, grows very rapidly with the number of objects.
Stirling's formula
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less ...
gives a precise asymptotic expression for this rate of growth.
Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their
logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 of ...
s.
Gödel numbers, and similar numbers used to represent bit-strings in
algorithmic information theory
Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as str ...
, are very large, even for mathematical statements of reasonable length. However, some
pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
numbers are even larger than the Gödel numbers of typical mathematical propositions.
Logician
Harvey Friedman
__NOTOC__
Harvey Friedman (born 23 September 1948)Handbook of Philosophical Logic, , p. 38 is an American mathematical logician at Ohio State University in Columbus, Ohio. He has worked on reverse mathematics, a project intended to derive the axi ...
has done work related to very large numbers, such as with
Kruskal's tree theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
History
The theorem was conjectured by Andrew Vázsonyi and proved b ...
and the
Robertson–Seymour theorem
In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering. Equivalently, every family of graphs that is c ...
.
"Billions and billions"
To help viewers of ''
Cosmos
The cosmos (, ) is another name for the Universe. Using the word ''cosmos'' implies viewing the universe as a complex and orderly system or entity.
The cosmos, and understandings of the reasons for its existence and significance, are studied in ...
'' distinguish between "millions" and "billions", astronomer
Carl Sagan stressed the "b". Sagan never did, however, say "
billions and billions
''Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium'' is a 1997 book by the American astronomer and science popularizer Carl Sagan. The last book written by Sagan before his death in 1996, it was published by Rando ...
". The public's association of the phrase and Sagan came from a ''
Tonight Show
''The Tonight Show'' is an American late-night talk show that has aired on NBC since 1954. The show has been hosted by six comedians: Steve Allen (1954–1957), Jack Paar (1957–1962), Johnny Carson (1962–1992), Jay Leno (1992–2009 and 2010 ...
'' skit. Parodying Sagan's effect,
Johnny Carson quipped "billions and billions". The phrase has, however, now become a humorous fictitious number—the
Sagan. ''Cf.'',
Sagan Unit
Carl Edward Sagan (; ; November 9, 1934December 20, 1996) was an American astronomer, planetary scientist, cosmologist, astrophysicist, astrobiologist, author, and science communicator. His best known scientific contribution is research on extr ...
.
Examples
*
googol =
*
centillion
Two naming scales for large numbers have been used in English and other European languages since the early modern era: the long and short scales. Most English variants use the short scale today, but the long scale remains dominant in many non-Eng ...
=
or
, depending on number naming system
*
millinillion =
or
, depending on number naming system
*The largest known
Smith number
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the given number base. In the case of numbers that are not square-f ...
= (10
1031−1) × (10
4594 + 3 + 1)
1476
*The largest known
Mersenne prime
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th ...
=
(''as of December 21, 2018'')*
googolplex
A googolplex is the number 10, or equivalently, 10 or 1010,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 . Written out in ordinary decimal notation, it is 1 fol ...
=
*
Skewes's number
In number theory, Skewes's number is any of several large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number x for which
:\pi(x) > \operatorname(x),
where is the prime-counting function ...
s: the first is approximately
, the second
*
Tritri on the lower end of BEAF (Bowers Exploding Array Function). It can be written as 33, 3^^^3 or 3^^(3^^3), the latter 2 showing how
Knuth's up-arrow notation
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
begins to build grahams number.
*
Graham's number, larger than what can be represented even using power towers (
tetration
In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common.
Under the definition as rep ...
). However, it can be represented using layers of Knuth's up-arrow notation.
*
Supertet , example of the numbers that can be generated through BEAF (Bowers Exploding Array Function). It can be written as 44, a more clear representation of the denotetration used to generate the number.
*
Kruskal's tree theorem
In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under homeomorphic embedding.
History
The theorem was conjectured by Andrew Vázsonyi and proved b ...
is a sequence relating to graphs. TREE(3) is larger than
Graham's number.
*
Rayo's number
Rayo's number is a large number named after Mexican philosophy professor Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at Massachusetts Institute of Technology, MIT on 26 Janua ...
is a large number named after Agustín Rayo which has been claimed to be the largest named number. It was originally defined in a "big number duel" at
MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
on 26 January 2007.
Standardized system of writing
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.
To compare numbers in scientific notation, say 5×10
4 and 2×10
5, compare the exponents first, in this case 5 > 4, so 2×10
5 > 5×10
4. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×10
4 > 2×10
4 because 5 > 2.
Tetration with base 10 gives the sequence
, the power towers of numbers 10, where
denotes a
functional power
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
of the function
(the function also expressed by the suffix "-plex" as in googolplex, see
the googol family).
These are very round numbers, each representing an
order of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually 10, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic di ...
in a generalized sense. A crude way of specifying how large a number is, is specifying between which two numbers in this sequence it is.
More precisely, numbers in between can be expressed in the form
, i.e., with a power tower of 10s and a number at the top, possibly in scientific notation, e.g.
, a number between
and
(note that
if
). (See also
extension of tetration to real heights.)
Thus googolplex is
Another example:
:
(between
and
)
Thus the "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (''n'') one has to take the
to get a number between 1 and 10. Thus, the number is between
and
. As explained, a more precise description of a number also specifies the value of this number between 1 and 10, or the previous number (taking the logarithm one time less) between 10 and 10
10, or the next, between 0 and 1.
Note that
:
I.e., if a number ''x'' is too large for a representation
the power tower can be made 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 large, the various representations for large numbers can be applied to the height itself. If the height is given only approximately, giving a value at the top does not make sense, so the double-arrow notation (e.g.
) can be used. If the value after the double arrow is a very large number itself, the above can recursively be applied to that value.
Examples:
:
(between
and
)
:
(between
and
)
Similarly to the above, if the exponent of
is not exactly given then giving a value at the right does not make sense, and instead of using the power notation of
, it is possible to add
to the exponent of
, to obtain e.g.
.
If the exponent of
is large, the various representations for large numbers can be applied to this exponent itself. If this exponent is not exactly given then, again, giving a value at the right does not make sense, and instead of using the power notation of
it is possible use the triple arrow operator, e.g.
.
If the right-hand argument of the triple arrow operator is large the above applies to it, obtaining e.g.
(between
and
). This can be done recursively, so it is possible to have a power of the triple arrow operator.
Then it is possible to proceed with operators with higher numbers of arrows, written
.
Compare this notation with the
hyper operator
In mathematics, the hyperoperation sequence is an infinite sequence of arithmetic operations (called ''hyperoperations'' in this context) that starts with a unary operation (the successor function with ''n'' = 0). The sequence continues with the ...
and the
Conway chained arrow notation
Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g. 2\to3\to4\to5\to6.
As wit ...
:
:
= ( ''a'' → ''b'' → ''n'' ) = hyper(''a'', ''n'' + 2, ''b'')
An advantage of the first is that when considered as function of ''b'', there is a natural notation for powers of this function (just like when writing out the ''n'' arrows):
. For example:
:
= ( 10 → ( 10 → ( 10 → ''b'' → 2 ) → 2 ) → 2 )
and only in special cases the long nested chain notation is reduced; for
obtains:
:
= ( 10 → 3 → 3 )
Since the ''b'' can also be very large, in general it can be written instead a number with a sequence of powers
with decreasing values of ''n'' (with exactly given integer exponents
) with at the end a number in ordinary scientific notation. Whenever a
is too large to be given exactly, the value of
is increased by 1 and everything to the right of
is rewritten.
For describing numbers approximately, deviations from the decreasing order of values of ''n'' are not needed. For example,
, and
. Thus is obtained the somewhat counterintuitive result that a number ''x'' can be so large that, in a way, ''x'' and 10
x are "almost equal" (for arithmetic of large numbers see also below).
If the superscript of the upward arrow is large, the various representations for large numbers can be applied to this superscript itself. If this superscript is not exactly given then there is no point in raising the operator to a particular power or to adjust the value on which it act, instead it is possible to simply use a standard value at the right, say 10, and the expression reduces to
with an approximate ''n''. For such numbers the advantage of using the upward arrow notation no longer applies, so the chain notation can be used instead.
The above can be applied recursively for this ''n'', so the notation
is obtained in the superscript of the first arrow, etc., or a nested chain notation, e.g.:
:(10 → 10 → (10 → 10 →
) ) =
If the number of levels gets too large to be convenient, a notation is used where this number of levels is written down as a number (like using the superscript of the arrow instead of writing many arrows). Introducing a function
= (10 → 10 → ''n''), these levels become functional powers of ''f'', allowing us to write a number in the form
where ''m'' is given exactly and n is an integer which may or may not be given exactly (for example:
). If ''n'' is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form ''f''
''m''(1) = (10→10→''m''→2). For example,
Compare the definition of Graham's number: it uses numbers 3 instead of 10 and has 64 arrow levels and the number 4 at the top; thus
, but also