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. Googology studies the naming conventions and properties of these immense numbers.
Since the customary decimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, a billion is represented as 10 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed in exponential format. A trillion is 13 characters in decimal, but only four (1012) in scientific notation. Values that vary dramatically can be represented and compared graphically via a logarithmic scale.
Natural language numbering
A 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 = 2.34B (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 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 expresses one billion 1 followed by nine zeros. The reciprocal, one billionth, is 1.0. Sometimes the letter e replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0.
Examples googol = centillion = or , depending on the number naming system millinillion = or , depending on the number naming system The largest known Smith number = (101031−1) × (104594 + 3 + 1)1476 The largest known Mersenne prime = googolplex = Skewes's numbers: the first is approximately , the second 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, or 37.2 trillion The number of bits on a computer hard disk (typically 1–2 TB as of 2024), about 1013, 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 , 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 , or 53±36 undecillion The 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 10120), or 1 novemtrigintillion. 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 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 seconds). The observable universe spans 93 billion light years (approximately meters) and hosts around stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about atoms within the observable universe.Atoms in the Universe. Universe Today. 30-07-2009. Retrieved 02-03-13.
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), roughly 10^10^1.288*10^3.884 T 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. . .How to Get A Googolplex 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.
"Millions and billions" 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".Carl Sagan takes questions more from his 'Wonder and Skepticism' CSICOP 1994 keynote, Skeptical Inquirer The phrase has, however, now become a humorous fictitious number—the Sagan. Cf., Sagan Unit.
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 5×104 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 , the power towers of numbers 10, where denotes a functional power 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 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 1010, 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 log10x, or find x from the lower-tower representation of the log10 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 and the Conway chained arrow notation: = ( 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 10x are "almost equal" (for arithmetic of large numbers see also below).