Large numbers are numbers that are significantly larger than those ordinarily used in everyday life, for instance in simple counting or in monetary transactions. The term typically refers to large positive integers, or more generally, large positive real numbers, but it may also be used in other contexts. For information on how large numbers are named in English, see names of large numbers. ... Big Nambas may refer to: Big Nambas people Big Nambas language Category: ... // Throughout this article, exponential or scientific notation is used. ... For other uses, see Number (disambiguation). ... The integers are commonly denoted by the above symbol. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Very large numbers often occur in fields such as mathematics, cosmology and cryptography. Sometimes people refer to numbers as being "astronomically large". However, it is easy to mathematically define numbers that are much larger even than those used in astronomy. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article is about the physics subject. ... The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ...

[edit] Using scientific notation to handle large and small numbers

See also: scientific notation, logarithmic scale, and orders of magnitude

Scientific notation was created to handle the wide range of values which occur in scientific study. 1.0 × 10⁹, for example, means one billion, a 1 followed by nine zeros: 1 000 000 000, and 1.0 × 10⁻⁹ means one billionth, or 0.000 000 001. Writing 10⁹ instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is. Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... A logarithmic scale is a scale of measurement that uses the logarithm of a physical quantity instead of the quantity itself. ... An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ... Scientific notation, also known as standard form, is a notation for writing numbers that is often used by scientists and mathematicians to make it easier to write large and small numbers. ... One thousand million (1,000,000,000) is the natural number following 999,999,999 and preceding 1,000,000,001. ...

Adding a 0 at the end of a number multiplies it by 10; 100 is 10 times 10. In scientific notation, however, the exponent only increases by one, from 10¹ to 10².

[edit] Large numbers in the everyday world

Examples of large numbers describing everyday real-world objects are:

• the number of bits on a computer hard disk (as of 2006, typically about 10¹², 125 GB)
• the number of cells in the human body (more than 10¹⁴)
• the number of neuronal connections in the human brain (estimated at 10¹⁴)
• Avogadro's number (i.e. the number of atoms in 12 grams of carbon-12, approximately 6.022 × 10²³)

2006 is a common year starting on Sunday of the Gregorian calendar. ... This article is about the unit of measurement. ... Avogadros number, also called Avogadros constant (NA), named after Amedeo Avogadro, is formally defined to be the number of carbon-12 atoms in 12 grams (0. ... Carbon 12 is a stable isotope of the element carbon. ...

[edit] Astronomically large numbers

Other large numbers, as regards length and time, are found in astronomy and cosmology. For example, the current Big Bang model of the Universe suggests that it is 13.7 billion years (4.3 × 10¹⁷ seconds) old, and that the observable universe is 46.5 billion light years across (4.65 × 10²⁶ metres), and contains about 5 × 10²² stars, organized into around 125 billion (1.25 × 10¹¹) galaxies, according to Hubble Space Telescope observations. There are about 10⁸⁰ atoms in the observable universe. Categories: | ... The pages linked in the right-hand column contain lists of times that are of the same order of magnitude (power of ten). ... For other uses, see Astronomy (disambiguation). ... Cosmology, from the Greek: κοσμολογία (cosmologia, κόσμος (cosmos) order + λογια (logia) discourse) is the study of the Universe in its totality, and by extension, humanitys place in it. ... For other uses, see Big Bang (disambiguation). ... See universe for a general discussion of the universe. ... A light year, abbreviated ly, is the distance light travels in one year: roughly 9. ... See universe for a general discussion of the universe. ...

Combinatorial 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 gives a precise asymptotic expression for this rate of growth. Combinatorics is a branch of mathematics that studies collections (usually finite) of objects that satisfy specified criteria. ... For factorial rings in mathematics, see unique factorisation domain. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ... In mathematics, Stirlings approximation (or Stirlings formula) is an approximation for large factorials. ...

Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms. Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ...

Gödel numbers, and similar numbers used to represent bit-strings in algorithmic information theory, are very large, even for mathematical statements of reasonable length. However, some pathological numbers are even larger than the Gödel numbers of typical mathematical propositions. This article or section may be confusing or unclear for some readers, and should be edited to rectify this. ... This article or section is in need of attention from an expert on the subject. ... In mathematics, a pathological example is one whose properties are (or should be considered) untypically bad. ...

[edit] Computers and computational complexity

Moore's Law, generally speaking, estimates that computers double in speed about every 18 months. This sometimes leads people to believe that eventually, computers will be able to solve any mathematical problem, no matter how complicated. This is not the case; computers are fundamentally limited by the constraints of physics, and certain upper bounds on what to expect can reasonably be formulated. Also, there are certain theoretical results which show that some problems are inherently beyond the reach of complete computational solution, no matter how powerful or fast the computation; see n-body problem. Gordon Moores original graph from 1965 Growth of transistor counts for Intel processors (dots) and Moores Law (upper line=18 months; lower line=24 months) For the observation regarding information retrieval, see Mooers Law. ... This article is about the n-body problem in classical mechanics. ...

Between 1980 and 2000, hard disk sizes increased from about 10 megabytes (1 × 10⁷) to over 100 gigabytes (10¹¹ bytes). A 100 gigabyte disk could store the given names of all of Earth's six billion inhabitants without using data compression. But what about a dictionary-on-disk storing all possible passwords containing up to 40 characters? Assuming each character equals one byte, there are about 2³²⁰ such passwords, which is about 2 × 10⁹⁶. In his paper Computational capacity of the universe,^[1] Seth Lloyd points out that if every particle in the universe could be used as part of a huge computer, it could store only about 10⁹⁰ bits, less than one millionth of the size such a dictionary would require. Seth Lloyd is a Professor of Mechanical Engineering at MIT. His research area is the interplay of information with complex systems, especially quantum systems. ...

Still, computers can easily be programmed to start creating and displaying all possible 40-character passwords one at a time. Such a program could be left to run indefinitely. Assuming a modern PC could output 1 billion strings per second, it would take one billionth of 2 × 10⁹⁶ seconds, or 2 × 10⁸⁷ seconds to complete its task, which is about 6 × 10⁷⁹ years. By contrast, the universe is estimated to be 13.7 billion (1.37 × 10¹⁰) years old. Computers will presumably continue to get faster, but the same paper mentioned before estimates that the entire universe functioning as a giant computer could have performed no more than 10¹²⁰ operations since the Big Bang. This is trillions of times more computation than is required for displaying all 40-character passwords, but computing all 50 character passwords would outstrip the estimated computational potential of the entire universe. For other uses, see Big Bang (disambiguation). ...

Problems like this grow exponentially in the number of computations they require, and are one reason why exponentially difficult problems are called "intractable" in computer science: for even small numbers like the 40 or 50 characters described earlier, the number of computations required exceeds even theoretical limits on mankind's computing power. The traditional division between "easy" and "hard" problems is thus drawn between programs that do and do not require exponentially increasing resources to execute. In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ...

Such limits are an advantage in cryptography, since any cipher-breaking technique which requires more than, say, the 10¹²⁰ operations mentioned before will never be feasible. Such ciphers must be broken by finding efficient techniques unknown to the cipher's designer. Likewise, much of the research throughout all branches of computer science focuses on finding efficient solutions to problems that work with far fewer resources than are required by a naïve solution. For example, one way of finding the greatest common divisor between two 1000 digit numbers is to compute all their factors by trial division. This will take up to 2 × 10⁵⁰⁰ division operations, far too large to contemplate. But the Euclidean algorithm, using a much more efficient technique, takes only a fraction of a second to compute the GCD for even huge numbers such as these. The German Lorenz cipher machine, used in World War II for encryption of very high-level general staff messages Cryptography (or cryptology; derived from Greek κρυπτός kryptós hidden, and the verb γράφω gráfo write or λεγειν legein to speak) is the study of message secrecy. ... This article is about algorithms for encryption and decryption. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In number theory, the Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (GCD) of two elements of any Euclidean domain (for example, the integers). ...

As a general rule, then, PCs in 2005 can perform 2⁴⁰ calculations in a few minutes. A few thousand PCs working for a few years could solve a problem requiring 2⁶⁴ calculations, but no amount of traditional computing power will solve a problem requiring 2¹²⁸ operations (which is about what would be required to break the 128-bit SSL commonly used in web browsers, assuming the underlying ciphers remain secure). Limits on computer storage are comparable. Quantum computers may allow certain problems to become feasible, but have practical and theoretical challenges which may never be overcome. Secure Sockets Layer (SSL) and Transport Layer Security (TLS), its successor, are cryptographic protocols which provide secure communications on the Internet. ... The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. ...

See also: computation, computational complexity theory, algorithmic information theory, computability theory, and big O notation

Look up computation in Wiktionary, the free dictionary. ... As a branch of the theory of computation in computer science, computational complexity theory describes the scalability of algorithms, and the inherent difficulty in providing scalable algorithms for specific computational problems. ... This article or section is in need of attention from an expert on the subject. ... Computability theory is the branch of theoretical computer science that studies which problems are computationally solvable using different models of computation. ... For other uses, see Big O. In computational complexity theory, big O notation is often used to describe how the size of the input data affects an algorithms usage of computational resources (usually running time or memory). ...

[edit] Examples

• 10¹⁰ (10,000,000,000), called "10 billion" in the U.S., or (traditionally, but now rarely - see Long and short scales) 10,000 million in the U.K.
• googol = 10¹⁰⁰ (10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000)
• googolplex = It is the number of states a system can be in that consists of 10⁹⁸ particles, each of which can be in googol states. Alternatively, it is the number of states a system can be in that consists of a googol particles, each of which can be in 10 states, or a system of 3.32 googol particles, each of which has 2 possible states.
• centillion = 10³⁰³ or 10⁶⁰⁰, depending on number naming system
• Skewes' numbers: the first is ca. , the second

The total amount of printed material in the world is roughly 1.6 × 10¹⁸ bits; therefore the contents can be represented by a number which is roughly The long and short scales are two different numerical systems used throughout the world: Short scale is the English translation of the French term échelle courte. ... Not to be confused with Google, the Internet company, and Nikolai Gogol, the author. ... This article is about a number. ... The number centillion refers to different quantities based on locality of usage. ... In number theory, Skewes number can refer to several extremely large numbers used by the South African mathematician Stanley Skewes. ...

Compare:

The first number is much larger than the second, due to the larger height of the power tower, and in spite of the small numbers 1.1 (however, if these numbers are made 1 or less, that greatly changes the result). In comparing the magnitude of each successive exponent in the last number with , we find a difference in the magnitude of effect on the final exponent. In the number 3000.48, the final exponent, The overall magnitude of the final exponent is donated by the 2^nd exponent (1000). The 1^st exponent only adds a factor of 3 to the mix (). The base number only supports a factor of 1.00016 in the final exponent (). This clearly shows the importance of the highest exponent in the stack.

[edit] Systematically creating ever faster increasing sequences

Let a strictly increasing integer sequence f₀(n) (n≥1) be given (written as a function for conveniently writing the functional powers), with f₀(1) > 1. Then a sequence of sequences can be found by , from which we can select the "cross-sequence" f₀(10),f₁(10),.. ^[2] In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

Together this is a process of creating a new sequence from a given one. This can be repeated (i.e., we can apply recursionhttp://en.wikipedia.org/wiki/Recursion#Functional_recursion), and again we can select from the matrix of numbers a single sequence, by taking the 10th element of each. We can apply the same whole process again and again. This article is about the concept of recursion. ...

Starting from this process corresponds to adding an element 10 in the Conway chain before the variable n, which is at the end of the chain: we get , and the new sequence selected from the matrix is that of which the kth element is . (See also Knuth's up-arrow notation and Conway chained arrow notation). ^[3] In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ...

Repeating this process we get for successive values of n, and selecting k=10 we get a single sequence .

Repeating this whole process we get ever longer chains. Selecting n=10 we get the sequence of (10→10), (10→10→10), (10→10→10→10),... This can be used as starting sequence to apply the process again, etc. Even the value f₁(2) for this sequence is already a Conway chain of length 10 billion plus one.

Each sequence in this whole process can be identified by its order type in the process: In mathematics, especially in set theory, ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, and so on) and to measure the length of the whole set by the least...

• (10→n→k) is the sequence with index n with order type k - 1
• (10→10→n→k) is the sequence with index n with order type ω + k - 1
• (10→10→10→n→k) is the sequence with index n with order type 2ω + k - 1
• (10→10), (10→10→10), (10→10→10→10),... is the sequence with order type ω²

In a similar way this can be continued, and we get a set of sequences, well-ordered by the procedure of construction. In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...

• zero case: f₀(n) = 10ⁿ
• successor ordinal:
• for limit ordinals: for a suitable sequence of ordinals a_{n − 1} tending to a.

Thus we have transfinite recursion as far as we define for each limit ordinal a suitable sequence of ordinals tending to it. When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. ... A limit ordinal is an ordinal number which is not a successor ordinal. ... Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. ...

Consider the Cantor normal form , where k is a natural number, are positive integers, and are ordinal numbers. The limit ordinals cover the case β_k > 0. For order types less than ε₀ (epsilon nought) β_k is less than the ordinal itself. The following rules provide for each limit ordinal a suitable sequence of ordinals: In mathematics, ε0 is the smallest transfinite ordinal number which cannot be reached from ω (the smallest transfinite ordinal) with a finite number of the ordinal operations of addition, multiplication and exponentiation. ...

If where p ≥ 0, u ≥ v, and v is a limit ordinal, we take for a suitable sequence of ordinals v_n tending to v.

For where p ≥ 0, and v is a limit ordinal, we take the sequence for a suitable sequence of ordinals v_n tending to v.

For where p ≥ 0 and a ≥ 0, we take the sequence .

For example, this procedure defines for the order type a particular, very rapidly increasing, sequence of integers, and to specify a particular large integer, we can refer e.g. to the 37th element in this sequence. The sequence is defined in 578 steps from the sequence with order type , which in turn is defined from the sequences with order types . The twelfth element in this sequence is defined from the sequences with order types , etc.

For ε₀ we can take the sequence , and for ε₀(k+1) the sequence ε₀k + , etc. We cannot reach omega-onehttp://en.wikipedia.org/wiki/Ordinal_number#Initial_ordinal_of_a_cardinal (ω₁), the set of all countable ordinal numbers, and the smallest uncountable ordinal number: no sequence of ordinal numbers below ω₁ has that ordinal as limit. It is also clear from the fact that we define sequences of which each element is defined in a finite number of steps, so making use of only a finite number of auxiliary sequences. Therefore for any of our sequences the set of auxiliary sequences is countable. In set theory, ordinal, ordinal number, and transfinite ordinal number refer to a type of number introduced by Georg Cantor in 1897, to accommodate infinite sequences and to classify sets with certain kinds of order structures on them. ...

We have f_a(n) < f_{a + 1}(n), except that we have "=" for n = 1. Note however that for a < b we do not always have f_a(n) < f_b(n). For example:

• f₃(n) < f_ω(n) for n = 1, 3, 4, 5, 6, .., but we have "=" for n = 2.
• f₄(n) < f_ω(n) for n = 1, 4, 5, 6, .., but we have ">" for n = 2, 3.

[edit] Standardized system of writing very large numbers

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.

Scientific notation with the mantissa between 1 and 10 positions any number with respect to the numbers 10ⁿ, with the mantissa providing the detail where between two consecutive powers the number is positioned. Numeric order is lexicographical order with respect to the pair (n, mantissa). The mantissa is irrelevant if n is not given exactly. In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ...

Tetration with base 10 gives the sequence , the power towers of numbers 10, where denotes a functional power of the function f(n) = 10ⁿ (the function also expressed by the suffix "-plex" as in googolplex, see the Googol family). Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... This article is about a number. ... // Throughout this article, exponential or scientific notation is used. ...

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. An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. ...

More accurately, numbers in between can be expressed in the form , i.e., with a power tower of numbers 10, with a regular number at the top, possibly in scientific notation, e.g. , a number between and (if the exponent quite at the top is between 10 and 10¹⁰, like here, the number like the 7 here is the height).

If the height is too large to write out the whole power tower, a notation like can be used. Various names are used for this representation:

• base-10 exponentiated tower form
• tetrated-scientific notation
• incomplete (power) tower

The notation is in ASCII ((10^)^183)3.12e6; a proposed simplification is 10^^183@3.12e6; the notations 10^^1@3.12e6 and 10^^0@3.12e6 are not needed, one can just write 10^3.12e6 and 3.12e6.

Thus googolplex = 10^^2@100 = 10^^3@2 = 10^^4@0.301; which notation is chosen may be considered on a number-by-number basis, or uniformly. In the latter case comparing numbers is sometimes a little easier. For example, comparing 10^^2@23.8 with 10^6e23 requires the small computation 10^.8=6.3 to see that the first number is larger.

To standardize the range of the upper value (after the @), one can choose one of the ranges 0–1, 1–10, or 10–10¹⁰:

• In the case of the range 0–1, an even shorter notation is (here for googolplex) like 10^^3.301 (proposed by William Elliot). This is not only a notation, it provides at the same time a generalisation of 10^^x to real x>-2 (10^^4@0=10^^3, hence the integer before the point is one less than in the previous notation). This function may or may not be suitable depending on required smoothness and other properties; it is monotonically increasing and continuous, and satisfies 10^^(x+1) = 10^(10^^x), but it is only piecewise differentiable. The inverse function is a super-logarithm or hyper-logarithm, defined for all real numbers, also negative numbers. See also Extension of tetration to real numbers.
• The range 10–1e10 brings the notation closer to ordinary scientific notation, and the notation reduces to it if the number is itself in that range (the part "10^^0@" can be dispensed with).

Another example: In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...

(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 log₁₀ to get a number between 1 and 10. Thusly, the number is between and . As explained, a more accurate 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 we can make the power tower 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 we can use the double-arrow notation, e.g. . 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 we can, instead of using the power notation of , add 1 to the exponent of , so we get 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 we can, instead of using the power notation of , use the triple arrow operator, e.g. .

If the right-hand argument of the triple arrow operator is large the above applies to it, so we have e.g. (between and ). This can be done recursively, so we can have a power of the triple arrow operator.

We can proceed with operators with higher numbers of arrows, written .

Compare this notation with the hyper operator and the Conway chained arrow notation: The hyper operators forming the hypern family are as follows: hypern (a, b) = (See Knuths up-arrow notation and Conway chained arrow notation. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ...

= ( 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 b = 1 we get:

= ( 10 → 3 → 3 )

Since the b can also be very large, in general we write a number with a sequence of powers with decreasing values of n (with exactly given integer exponents k_n) with at the end a number in ordinary scientific notation. Whenever a k_n is too large to be given exactly, the value of k_{n + 1} 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 we have 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 acts. We can 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, and we can also use the chain notation.

The above can be applied recursively for this n, so we get the notation in the superscript of the first arrow, etc., or we have 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 f^m(n) where m is given exactly and n is an integer which may or may not be given exactly (for the example: . If n is large we can use any of the above 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 G < f⁶⁴(4) < f⁶⁵(1). Grahams number, named after Ronald Graham, is often described as the largest number that has ever been seriously used in a mathematical proof. ...

If m in f^m(n) is too large to give exactly we can 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 g(n) = fⁿ(1) these levels become functional powers of g, allowing us to write a number in the form g^m(n) where m is given exactly and n is an integer which may or may not be given exactly. We have (10→10→m→3) = g^m(1). If n is large we can use any of the above for expressing it. Similarly we can introduce a function h, etc. If we need many such functions we can better number them instead of using a new letter every time, e.g. as a subscript, so we get 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., we have (10→10→n→k) = . If n is large we can use any of the above for expressing it. Thus we get a nesting of forms where going inward the k decreases, and with as inner argument a sequence of powers with decreasing values of n (where all these numbers are exactly given integers) 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 f_n(10)=(10→10→10→n) with an approximate n. Note that the process of going from the sequence 10ⁿ=(10→n) to the sequence =(10→10→n) is very similar to going from the latter to the sequence f_n(10)=(10→10→10→n): it is the general process of adding an element 10 to the chain in the chain notation; this process can be repeated again (see also the previous section). 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 we have a sequence of powers with decreasing values of n (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation. In mathematics, the lexicographic or lexicographical order, (also known as dictionary order, alphabetic order or lexicographic(al) product), is a natural order structure of the Cartesian product of two ordered sets. ...

For a number too large to write down in the Conway chained arrow notation we can describe how large it is by the length of that chain, for example only using elements 10 in the chain; in other words, we specify its position in the sequence 10, 10→10, 10→10→10, .. If even the position in the sequence is a large number we can apply the same techniques again for that.

[edit] Examples of numbers, in numerical order

• ( 1 → Y ) = 1 for any subchain Y
• ( 2 → 2 → Y ) = 4 for any subchain Y
• = ( 2 → 3 → 2 ) = 16
• 3³ = ( 3 → 2 → 2 ) = 27
• 4⁴ = ( 4 → 2 → 2 ) = 256
• 5⁵ = ( 5 → 2 → 2 ) = 3125
• 6⁶ = ( 6 → 2 → 2 ) = 46,656
• = ( 2 → 3 → 3 ) = ( 2 → 4 → 2 ) = 65,536
• 7⁷ = ( 7 → 2 → 2 ) = 823,543
• 8⁸ = ( 8 → 2 → 2 ) = 16,777,216
• 9⁹ = ( 9 → 2 → 2 ) = 387,420,489
• 10¹⁰ = ( 10 → 2 → 2 ) = 10,000,000,000
• = ( 3 → 3 → 2 ) = ( 3 → 2 → 3 ) = 7,625,597,484,987
• googol = 10¹⁰⁰
• = ( 4 → 3 → 2 ) =
• ( 2 → 5 → 2 ) =
• = (10 → 3 → 2)
• = ( 3 → 4 → 2 ) =
• googolplex =
• ( 2 → 6 → 2 ) =
• = ( 10 → 4 → 2 )
• ( 2 → 7 → 2 ) =
• = ( 10 → 5 → 2 )
• = ( 10 → 6 → 2 )
• = ( 10 → 7 → 2 )
• = ( 10 → 8 → 2 )
• = ( 10 → 9 → 2 )
• 
• = ( 10 → 2 → 3 ) = ( 10 → 10 → 2 )
• ( 2 → 3 → 4 ) = ( 2 → 4 → 3 ) = ( 2 → 65,536 → 2 )
• 
• 
• = ( 10 → 3 → 3 )
• 
• 
• = ( 10 → 4 → 3 )
• 
• = ( 10 → 5 → 3 )
• = ( 10 → 6 → 3 )
• = ( 10 → 7 → 3 )
• = ( 10 → 8 → 3 )
• = ( 10 → 9 → 3 )
• = ( 10 → 2 → 4 ) = ( 10 → 10 → 3 )
• = (10 → 3 → 4)
• = ( 4 → 4 → 4 )
• = ( 10 → 4 → 4 )
• = ( 10 → 5 → 4 )
• = ( 10 → 6 → 4 )
• = ( 10 → 7 → 4 )
• = ( 10 → 8 → 4 )
• = ( 10 → 9 → 4 )
• = ( 10 → 2 → 5 ) = ( 10 → 10 → 4 )
• ( 2 → 3 → 2 → 2 ) = ( 2 → 3 → 8 )
• ( 3 → 2 → 2 → 2 ) = ( 3 → 2 → 9 ) = ( 3 → 3 → 8 )
• ( 10 → 10 → 10 ) = ( 10 → 2 → 11 )
• ( 10 → 2 → 2 → 2 ) = ( 10 → 2 → 100 )
• ( 10 → 10 → 2 → 2 ) = ( 10 → 2 → 10¹⁰ ) =
• ( 10 → 10 → )
• ( 10 → 10 → 3 → 2 ) = (10 → 10 → (10 → 10 → 10¹⁰) ) =
• ( 10 → 10 → 10 → 2 )
• ( 10 → 10 → 64 → 2 )
• Graham's number^[4]
• ( 10 → 10 → 65 → 2 )
• ( 10 → 10 → 10 → 3 )
  

Not to be confused with Google, the Internet company, and Nikolai Gogol, the author. ... This article is about a number. ... Grahams number, named after Ronald Graham, is often described as the largest number that has ever been seriously used in a mathematical proof. ...

[edit] Comparison of base values

The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers, and the arithmetic.

100¹² = 10²⁴, with base 10 the exponent is doubled.

, ditto.

, the highest exponent is very little more than doubled.

• 
• 
• 
• (thus if n is large it seems fair to say that is "approximately equal to" )
• 
• 
• (compare ; thus if n is large it seems fair to say that is "approximately equal to" )
• (compare )
• (compare )
• (compare ; if n is large this is "approximately" equal)

[edit] Accuracy

Note that for a number 10ⁿ, one unit change in n changes the result by a factor 10. In a number like , with the 6.2 the result of proper rounding, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor 10⁵⁰ too large or too small. This seems like extremely poor accuracy, but for such a large number it may be considered fair (a large error in a large number may be "relatively small" and therefore acceptable).

[edit] Accuracy for very large numbers

With extremely large numbers, the relative error may be large, yet there may still be a sense in which we want to consider the numbers as "close in magnitude". For example, consider

10¹⁰ and 10⁹

The relative error is

a large relative error. However, we can also consider the relative error in the logarithms; in this case, the logarithms (to base 10) are 10 and 9, so the relative error in the logarithms is only 10%. In mathematics, if two variables of bn = x are known, the third can be found. ... In mathematics, if two variables of bn = x are known, the third can be found. ...

The point is that exponential functions magnify relative errors greatly – if a and b have small relative error, The exponential function is one of the most important functions in mathematics. ...

10^a and 10^b

may not have small relative error, and

and

will have even larger relative error. The question then becomes: on which level of iterated logarithms do we wish to compare two numbers? There is a sense in which we may want to consider

and

to be "close in magnitude". The relative error between these two numbers is large, and the relative error between their logarithms is still large; however, the relative error in their second-iterated logarithms is small:

and

Such comparisons of iterated logarithms are common, e.g., in analytic number theory. Analytic number theory is the branch of number theory that uses methods from mathematical analysis. ...

[edit] Approximate arithmetic for very large numbers

There are some general rules relating to the usual arithmetic operations performed on very large numbers:

• The sum and the product of two very large numbers are both "approximately" equal to the larger one.
• 

Hence:

• A very large number raised to a very large power is "approximately" equal to the larger of the following two values: the first value and 10 to the power the second. For example, for very large n we have (see e.g. the computation of mega) and also . Thus , see table.

In mathematics, Mosers polygon notation is a means of expressing certain extremely large numbers. ... In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ...

[edit] Large numbers in some noncomputable sequences

The busy beaver function Σ is an example of a function which grows faster than any computable function. Its value for even relatively small input is huge. The values of Σ(n) for n = 1, 2, 3, 4 are 1, 4, 6, 13 (sequence A028444 in OEIS). Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 1.29×10⁸⁶⁵. In computability theory, a Busy Beaver (from the colloquial expression for industrious person) is a Turing machine that, when given an empty tape, does a lot of work, then halts. ... In computer science, computability theory is the branch of the theory of computation that studies which problems are computationally solvable using different models of computation. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is an extensive searchable database of integer sequences, freely available on the Web. ...

Some of the work by Harvey Friedman also involve sequences that grow faster than any computable function.^[5] Harvey Friedman is a mathematical logician at The Ohio State University. ...

[edit] Infinite numbers

Main article: cardinal number

See also: large cardinal, Mahlo cardinal, and totally indescribable cardinal

Although all these numbers above are very large, they are all still finite. Certain fields of mathematics define infinite and transfinite numbers. For example, aleph-null is the cardinality of the infinite set of natural numbers, and aleph-one is the next greatest cardinal number. is the cardinality of the reals. The proposition that is known as the continuum hypothesis. Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set, known as its cardinality. ... In mathematics, a cardinal is called a large cardinal if it belongs to a class of cardinals, the existence of which provably cannot be proved within the standard axiomatic set theory ZFC, if one assumes ZFC itself is consistent. ... In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. ... In mathematics, a totally indescribable cardinal is a certain kind of large cardinal number. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... Transfinite numbers, also known as infinite numbers, are numbers that are not finite. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In set theory, an infinite set is a set that is not a finite set. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality (or size) of infinite sets. ... In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ... In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...

[edit] Notations

Some notations for extremely large numbers:

• Knuth's up-arrow notation / hyper operators / Ackermann function, including tetration
• Conway chained arrow notation
• Steinhaus-Moser notation; apart from the method of construction of large numbers, this also involves a graphical notation with polygons; alternative notations, like a more conventional function notation, can also be used with the same functions.
• Jonathan Bowers' array notation allows considerably larger numbers to be represented than with Knuth's up-arrow notation.

These notations are essentially functions of integer variables, which increase very rapidly with those integers. Ever faster increasing functions can easily be constructed recursively by applying these functions with large integers as argument. In mathematics, Knuths up-arrow notation is a notation for very large integers introduced by Donald Knuth in 1976. ... The hyper operators forming the hypern family are as follows: hypern (a, b) = (See Knuths up-arrow notation and Conway chained arrow notation. ... In recursion theory, the Ackermann function or Ackermann-Péter function is a simple example of a computable function that is not primitive recursive. ... Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ... Conway chained arrow notation, created by mathematician John Conway, is a means of expressing certain extremely large numbers. ... In mathematics, Mosers polygon notation is a means of expressing certain extremely large numbers. ... A polygon (from the Greek poly, for many, and gonos, for angle) is a closed planar path composed of a finite number of sequential straight line segments. ...

Note that a function with a vertical asymptote is not helpful in defining a very large number, although the function increases very rapidly: one has to define an argument very close to the asymptote, i.e. use a very small number, and constructing that is equivalent to constructing a very large number, e.g. the reciprocal.

[edit] See also

// Throughout this article, exponential or scientific notation is used. ... Different cultures used different traditional numeral systems for naming large numbers. ... // The law of large numbers (LLN) is any of several theorems in probability. ... In mathematics, exponential growth (or geometric growth) occurs when the growth rate of a function is always proportional to the functions current size. ... Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ... A bignum package in a computer or program allows internal representation of very large integers, rational numbers, decimal numbers, or floating-point numbers (limitted only by available memory), and provides a set of arithmetic operations on such numbers. ... Within a set of positive numbers, a number is small if it is close to zero. ... Human scale means of a scale comparable to a human being. A number of characteristic physical quantities can be associated with the human body, the human mind, and the preservation of human life. ... Different cultures have different traditional numeral systems used for writing numbers and for naming large numbers. ... The term zillions can seriously mean all the possible -illions, as in million, billion, trillion, etc. ... It has been suggested that this article or section be merged with Indefinite and fictitious large numbers. ...

[edit] Notes and references