Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These 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).[citation needed] Googology studies the naming conventions and properties of these immense numbers.[1][2]
Since the customary decimal format of large numbers can be lengthy, other systems he been devised that allows for shorter representation. For example, a billion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 (1012) in exponential. Values that vary dramatically can be represented and compared graphically via logarithmic scale.
Natural language numbering[edit] Main article: Names of large numbersA natural language numbering system represents large numbers using names rather than a series of digits. For example "billion" may be easier to comprehend for some readers than "1,000,000,000". Sometimes it is shortened by using a suffix, for example 2,340,000,000 = 2.34 B (B = billion). A numeric value can be lengthy when expressed in words, for example, "2,345,789" is "two million, three hundred forty five thousand, seven hundred and eighty nine".
Scientific notation[edit]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. A value is represented as a decimal fraction times a multiple power of 10. The factor is intended to make reading comprehension easier than a lengthy series of zeros. For example, 1.0×109 expresses one billion – 1 followed by nine zeros. The reciprocal, one billionth, is 1.0×10−9. Sometimes the letter e replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0×109.
Examples[edit] googol = 10 100 {\displaystyle 10^{100}} centillion = 10 303 {\displaystyle 10^{303}} or 10 600 {\displaystyle 10^{600}} , depending on number naming system millinillion = 10 3003 {\displaystyle 10^{3003}} or 10 6000 {\displaystyle 10^{6000}} , depending on number naming system The largest known Smith number = (101031−1) × (104594 + 3×102297 + 1)1476 ×103913210 The largest known Mersenne prime = 2 136 , 279 , 841 − 1 {\displaystyle 2^{136,279,841}-1} [3] googolplex = 10 googol = 10 10 100 {\displaystyle 10^{\text{googol}}=10^{10^{100}}} Skewes's numbers: the first is approximately 10 10 10 34 {\displaystyle 10^{10^{10^{34}}}} , the second 10 10 10 964 {\displaystyle 10^{10^{10^{964}}}} Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation. Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number. Rayo's number 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 on 26 January 2007.
centillion =
10
303
{\displaystyle 10^{303}}
or
10
600
{\displaystyle 10^{600}}
, depending on number naming system
millinillion =
10
3003
{\displaystyle 10^{3003}}
or
10
6000
{\displaystyle 10^{6000}}
, depending on number naming system
The largest known Smith number = (101031−1) × (104594 + 3×102297 + 1)1476 ×103913210
The largest known Mersenne prime =
2
136
,
279
,
841
−
1
{\displaystyle 2^{136,279,841}-1}
[3]
googolplex =
10
googol
=
10
10
100
{\displaystyle 10^{\text{googol}}=10^{10^{100}}}
Skewes's numbers: the first is approximately
10
10
10
34
{\displaystyle 10^{10^{10^{34}}}}
, the second
10
10
10
964
{\displaystyle 10^{10^{10^{964}}}}
Graham's number, larger than what can be represented even using power towers (tetration). However, it can be represented using layers of Knuth's up-arrow notation.
Kruskal's tree theorem is a sequence relating to graphs. TREE(3) is larger than Graham's number.
Rayo's number 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 on 26 January 2007.
Examples of large numbers describing real-world things:
The number of cells in the human body (estimated at 3.72×1013), or 37.2 trillion[4] The number of bits on a computer hard disk (as of 2024[update], typically about 1013, 1–2 TB), or 10 trillion The number of neuronal connections in the human brain (estimated at 1014), or 100 trillion The Avogadro constant is the number of "elementary entities" (usually atoms or molecules) in one mole; the number of atoms in 12 grams of carbon-12 – approximately 6.022×1023, or 602.2 sextillion. The total number of DNA base pairs within the entire biomass on Earth, as a possible approximation of global biodiversity, is estimated at (5.3±3.6)×1037, or 53±36 undecillion[5][6] The mass of Earth consists of about 4 × 1051, or 4 sexdecillion, nucleons The estimated number of atoms in the observable universe (1080), or 100 quinvigintillion The lower bound on the game-tree complexity of chess, also known as the "Shannon number" (estimated at around 10120), or 1 novemtrigintillion.[7] Note that this value of the Shannon number is for Standard Chess. It has even larger values for larger-board chess variants such as Grant Acedrex, Tai Shogi, and Taikyoku Shogi. Astronomical[edit]In astronomy and cosmology large numbers for measures of length and time are encountered. For instance, according to the prevailing Big Bang model, the universe is approximately 13.8 billion years old (equivalent to 4.355×1017 seconds). The observable universe spans 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). As a rough estimate, there are about 1080 atoms within the observable universe.[8]
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
10 10 10 10 10 1.1 years {\displaystyle 10^{10^{10^{10^{10^{1.1}}}}}{\mbox{ years}}}
(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), roughly 10^10^1.288*10^3.884 T [9][10] 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 processes give rise to astonishingly large numbers. The factorial function, which quantifies permutations of a fixed set of objects, grows superexponentially as the number of objects increases. Stirling's formula provides a precise asymptotic expression for this rapid growth.
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. Remarkably, certain pathological numbers surpass even the Gödel numbers associated with typical mathematical propositions.
Logician Harvey Friedman has made significant contributions to the study of very large numbers, including work related to Kruskal's tree theorem and the Robertson–Seymour theorem.
"Billions and billions"[edit]To help viewers of Cosmos distinguish between "millions" and "billions", astronomer Carl Sagan stressed the "b". Sagan never did, however, say "billions and billions". The public's association of the phrase and Sagan came from a Tonight Show skit. Parodying Sagan's effect, Johnny Carson quipped "billions and billions".[11] The phrase has, however, now become a humorous fictitious number—the Sagan. Cf., Sagan Unit.
Standardized system of writing[edit]
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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 5×104 and 2×105, compare the exponents first, in this case 5 > 4, so 2×105 > 5×104. If the exponents are equal, the mantissa (or coefficient) should be compared, thus 5×104 > 2×104 because 5 > 2.
Tetration with base 10 gives the sequence 10 ↑↑ n = 10 → n → 2 = ( 10 ↑ ) n 1 {\displaystyle 10\uparrow \uparrow n=10\to n\to 2=(10\uparrow )^{n}1} , the power towers of numbers 10, where ( 10 ↑ ) n {\displaystyle (10\uparrow )^{n}} denotes a functional power of the function f ( n ) = 10 n {\displaystyle f(n)=10^{n}} (the function also expressed by the suffix "-plex" as in googolplex, see the googol family).
, the power towers of numbers 10, where
(
10
↑
)
n
{\displaystyle (10\uparrow )^{n}}
denotes a functional power of the function
f
(
n
)
=
10
n
{\displaystyle f(n)=10^{n}}
(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 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 ( 10 ↑ ) n a {\displaystyle (10\uparrow )^{n}a} , i.e., with a power tower of 10s, and a number at the top, possibly in scientific notation, e.g. 10 10 10 10 10 4.829 = ( 10 ↑ ) 5 4.829 {\displaystyle 10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829} , a number between 10 ↑↑ 5 {\displaystyle 10\uparrow \uparrow 5} and 10 ↑↑ 6 {\displaystyle 10\uparrow \uparrow 6} (note that 10 ↑↑ n ↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑
, i.e., with a power tower of 10s, and a number at the top, possibly in scientific notation, e.g.
10
10
10
10
10
4.829
=
(
10
↑
)
5
4.829
{\displaystyle 10^{10^{10^{10^{10^{4.829}}}}}=(10\uparrow )^{5}4.829}
, a number between
10
↑↑
5
{\displaystyle 10\uparrow \uparrow 5}
and
10
↑↑
6
{\displaystyle 10\uparrow \uparrow 6}
(note that
10
↑↑
n
↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑