Large numbers


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 13 characters in decimal format, but is only 3 characters when expressed in exponential format. A trillion is 17 characters in decimal, but only 4 in exponential. Values that vary dramatically can be represented and compared graphically via 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. 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

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 number naming system
  • millinillion = or, depending on number naming system
  • The largest known Smith number = × 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. However, it can be represented using layers of Knuth's up-arrow notation.
  • Kruskal's tree theorem is a sequence relating to graphs. TREE 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:
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. The observable universe spans 93 billion light years and hosts around stars, organized into roughly 125 billion galaxies. As a rough estimate, there are about atoms within the observable universe.
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
, roughly 10^10^1.288*10^3.884 T 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 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". 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 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 .
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 .
Thus googolplex is.
Another example:
Thus the "order of magnitude" of a number, can be characterized by the number of times 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 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.
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 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:
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. . 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:
An advantage of the first is that when considered as function of b, there is a natural notation for powers of this function :. 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 with decreasing values of n 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".
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.:
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. Introducing a function =, 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. If n is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the form fm =. 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.
If m in 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. Using the functional power notation of f this gives multiple levels of f. Introducing a function these levels become functional powers of g, 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 = gm. If n is large any of the above can be used for expressing it. Similarly a function h, etc. can be introduced. If many such functions are required, they can be numbered instead of using a new letter every time, e.g. as a subscript, such that there are numbers of the form where k and m are given exactly and n is an integer which may or may not be given exactly. Using k=1 for the f above, k=2 for g, etc., obtains =. If n is large any of the above can be used to express it. Thus is obtained a nesting of forms where going inward the k decreases, and with as inner argument a sequence of powers with decreasing values of n with at the end a number in ordinary scientific notation.
When k is too large to be given exactly, the number concerned can be expressed as = with an approximate n. Note that the process of going from the sequence = to the sequence = is very similar to going from the latter to the sequence =: it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again. Numbering the subsequent versions of this function a number can be described using functions, nested in lexicographical order with q the most significant number, but with decreasing order for q and for k; as inner argument yields a sequence of powers with decreasing values of n with at the end a number in ordinary scientific notation.
For a number too large to write down in the Conway chained arrow notation it size can be described by the length of that chain, for example only using elements 10 in the chain; in other words, one could specify its position in the sequence 10, 10→10, 10→10→10,.. If even the position in the sequence is a large number same techniques can be applied again.