Large numbers are numbers far larger than those encountered in everyday life, such as simple counting or financial transactions. These quantities appear prominently inmathematics,cosmology,cryptography, andstatistical mechanics.Googology studies the naming conventions and properties of these immense numbers.[1][2]
Since the customarydecimal format of large numbers can be lengthy, other systems have been devised that allows for shorter representation. For example, abillion is represented as 13 characters (1,000,000,000) in decimal format, but is only 3 characters (109) when expressed inexponential format. Atrillion is 17 characters in decimal, but only 4 (1012) in exponential. Values that vary dramatically can be represented and comparedgraphically vialogarithmic scale.
Anatural 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 was devised to represent the vast range of values encountered inscientific research in a format that is more compact than traditional formats yet allows for high precision when called for. A value is represented as adecimal fractiontimes a multiplepower 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. Thereciprocal, one billionth, is 1.0×10−9. Sometimes the lettere replaces the exponent, for example 1 billion may be expressed as 1e9 instead of 1.0×109.
Examples of large numbers describing real-world things:
Inastronomy andcosmology large numbers for measures of length and time are encountered. For instance, according to the prevailingBig Bang model, the universe is approximately 13.8 billion years old (equivalent to4.355×1017 seconds). Theobservable universe spans 93 billionlight years (approximately8.8×1026 meters) and hosts around5×1022 stars, organized into roughly 125 billion galaxies (as observed by the Hubble Space Telescope). As a rough estimate, there are about1080 atoms within the observable universe.[8]
According toDon 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 estimatedPoincaré 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 certaininflationary model with aninflaton whose mass is 10−6Planck 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 historyrepeats itself arbitrarily many times due toproperties 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. Thefactorial function, which quantifiespermutations 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 usinglogarithms.
Gödel numbers, along with similar representations of bit-strings inalgorithmic information theory, are vast—even for mathematical statements of moderate length. Remarkably, certainpathological numbers surpass even the Gödel numbers associated with typical mathematical propositions.
LogicianHarvey Friedman has made significant contributions to the study of very large numbers, including work related toKruskal's tree theorem and theRobertson–Seymour theorem.
To help viewers ofCosmos distinguish between "millions" and "billions", astronomerCarl Sagan stressed the "b". Sagan never did, however, say "billions and billions". The public's association of the phrase and Sagan came from aTonight Show skit. Parodying Sagan's effect,Johnny Carson quipped "billions and billions".[11] The phrase has, however, now become a humorous fictitious number—theSagan.Cf.,Sagan Unit.
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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, the power towers of numbers 10, where denotes afunctional power of the function (the function also expressed by the suffix "-plex" as in googolplex, seethe googol family).
These are very round numbers, each representing anorder 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 alsoextension of tetration to real heights.)
Thus googolplex is.
Another example:
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 numberx is too large for a representation the power tower can be made one higher, replacingx by log10x, or findx 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:
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 thehyper operator and theConway chained arrow notation:
An advantage of the first is that when considered as function ofb, there is a natural notation for powers of this function (just like when writing out then arrows):. For example:
and only in special cases the long nested chain notation is reduced; for obtains:
Since theb can also be very large, in general it can be written instead a number with a sequence of powers with decreasing values ofn (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 ofn are not needed. For example,, and. Thus is obtained the somewhat counterintuitive result that a numberx can be so large that, in a way,x and 10x 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 approximaten. 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 thisn, 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 (like using the superscript of the arrow instead of writing many arrows). Introducing a function = (10 → 10 →n), these levels become functional powers off, allowing us to write a number in the form wherem is given exactly and n is an integer which may or may not be given exactly (for example:). Ifn is large, any of the above can be used for expressing it. The "roundest" of these numbers are those of the formfm(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.
Ifm in is too large to give exactly, it is possible to use a fixedn, e.g.n = 1, and apply the above recursively tom, i.e., the number of levels of upward arrows is itself represented in the superscripted upward-arrow notation, etc. Using the functional power notation off this gives multiple levels off. Introducing a function these levels become functional powers ofg, allowing us to write a number in the form wherem is given exactly and n is an integer which may or may not be given exactly. For example, if (10→10→m→3) =gm(1). Ifn is large any of the above can be used for expressing it. Similarly a functionh, 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 wherek andm are given exactly and n is an integer which may or may not be given exactly. Usingk=1 for thef above,k=2 forg, etc., obtains (10→10→n→k) =. Ifn is large any of the above can be used to express it. Thus is obtained a nesting of forms where going inward thek decreases, and with as inner argument a sequence of powers with decreasing values ofn (where all these numbers are exactly given integers) with at the end a number in ordinary scientific notation.
Whenk is too large to be given exactly, the number concerned can be expressed as=(10→10→10→n) with an approximaten. Note that the process of going from the sequence=(10→n) to the sequence=(10→10→n) is very similar to going from the latter to the sequence=(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 inlexicographical order withq the most significant number, but with decreasing order forq and fork; as inner argument yields a sequence of powers with decreasing values ofn (where all these numbers are exactly given integers) 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.
Numbers expressible in decimal notation:
Numbers expressible in scientific notation:
Numbers expressible in (10 ↑)nk notation:
Bigger numbers:
Some notations for extremely large numbers:
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.
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.
The following illustrates the effect of a base different from 10, base 100. It also illustrates representations of numbers and the arithmetic.
, with base 10 the exponent is doubled.
, ditto.
, the highest exponent is very little more than doubled (increased by log102).
For a number, one unit change inn changes the result by a factor 10. In a number like, with the 6.2 the result of proper rounding using significant figures, the true value of the exponent may be 50 less or 50 more. Hence the result may be a factor 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).
In the case of an approximation of an extremely large number, therelative error may be large, yet there may still be a sense in which one wants to consider the numbers as "close in magnitude". For example, consider
The relative error is
a large relative error. However, one 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%.
The point is thatexponential functions magnify relative errors greatly – ifa andb have a small relative error,
the relative error is larger, and
will have an even larger relative error. The question then becomes: on which level of iterated logarithms to compare two numbers? There is a sense in which one may want to consider
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:
Such comparisons of iterated logarithms are common, e.g., inanalytic number theory.
One solution to the problem of comparing large numbers is to define classes of numbers, such as the system devised by Robert Munafo,[14] which is based on different "levels" of perception of an average person. Class 0 – numbers between zero and six – is defined to contain numbers that are easilysubitized, that is, numbers that show up very frequently in daily life and are almost instantly comparable. Class 1 – numbers between six and 1,000,000=106 – is defined to contain numbers whose decimal expressions are easily subitized, that is, numbers who are easily comparable not bycardinality, but "at a glance" given the decimal expansion.
Each class after these are defined in terms of iterating this base-10 exponentiation, to simulate the effect of another "iteration" of human indistinguishability. For example, class 5 is defined to include numbers between 101010106 and 10101010106, which are numbers whereX becomes humanly indistinguishable fromX2[15] (taking iterated logarithms of suchX yields indistinguishibility firstly between log(X) and 2log(X), secondly between log(log(X)) and 1+log(log(X)), and finally an extremely long decimal expansion whose length can't be subitized).
There are some general rules relating to the usual arithmetic operations performed on very large numbers:
Hence:
Given a strictly increasing integer sequence/function (n≥1), it is possible to produce a faster-growing sequence (where the superscriptn denotes thenthfunctional power). This can be repeated any number of times by letting, each sequence growing much faster than the one before it. Thus it is possible to define, which grows much faster than any for finitek (here ω is the first infiniteordinal number, representing the limit of all finite numbers k). This is the basis for the fast-growing hierarchy of functions, in which the indexing subscript is extended to ever-larger ordinals.
For example, starting withf0(n) =n + 1:
Thebusy beaver function Σ is an example of a function which grows faster than anycomputable function. Its value for even relatively small input is huge. The values of Σ(n) forn = 1, 2, 3, 4, 5 are 1, 4, 6, 13, 4098[16] (sequenceA028444 in theOEIS). Σ(6) is not known but is at least 10↑↑15.
Although all the numbers discussed above are very large, they are all stillfinite. Certain fields of mathematics defineinfinite andtransfinite numbers. For example,aleph-null is thecardinality of theinfinite set ofnatural numbers, andaleph-one is the next greatest cardinal number. is thecardinality of the reals. The proposition that is known as thecontinuum hypothesis.