Online Encyclopedia
Large numbers
- For information on how large numbers are named in English, see names of large numbers.
Large numbers are numbers that are large compared with the numbers used in everyday life. Very large numbers often occur in fields such as mathematics, cosmology and cryptography. Sometimes people refer to numbers as being "astronomically large". However, mathematically it is easy to define numbers that are much larger than occur even in astronomy.
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Writing and thinking about large numbers
Large numbers are often found in science, and scientific notation was created to handle both these large numbers and also very small numbers. 1.0 × 10^{9}, for example, means one billion, a 1 followed by nine zeros: 1,000,000,000, and 1.0 × 10^{-9} means one billionth, or 0.0000000001. Writing 10^{9} instead of nine zeros saves the reader the effort and hazard of counting a long string of zeros to see how large the number is.
Adding a 0 to a large number multiplies it by ten: 100 is ten times 10. In scientific notation, however, the exponent only increases by one, from 10^{1} to 10^{2}. Remember then, when reading numbers in scientific notation, that small changes in the exponent equate to large changes in the number itself: 2.5 × 10^{5} dollars ($250,000) is a common price for new homes in the U.S., while 2.5 × 10^{10} dollars ($25 billion) would make you one of the world's richest people.
Large numbers in the everyday world
Some large numbers apply to things in the everyday world.
Examples of large numbers describing everyday real-world objects are:
- cigarettes smoked in the United States in one year, on the order of 10^{12} (one trillion)
- bits on a computer hard disk (typically 10^{12} to 10^{13})
- number of cells in the human body > 10^{14}
- number of neuron connections in the human brain, 10^{14} (estimated)
- Avogadro's number, approximately 6.022 × 10^{23}
Other examples are given in Orders of magnitude (numbers).
Large numbers and computers
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 we can expect can be reasonably formulated.
First, a rule of thumb for converting between scientific notation and powers of two, since computer-related quantities are frequently stated in powers of two. Since the logarithm of 10 in base 2 is a little more than 3, multiplying a scientific notation exponent by 3 gives its approximate value as an exponent with a base of 2. For example, 10^{3} (1000) is somewhere in the neighborhood of 2^{9} (512). (But remember that when dealing with very large numbers, such "neighborhoods" will themselves be quite large).
Between 1980 and 2000, hard disk sizes increased from about 10 megabytes (1 × 10^{7}) to over 100 gigabytes (1 × 10^{11}). A 100 gigabyte disk could store the 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 2^{320} such passwords, which is about 2 × 10^{96}. This paper http://arxiv.org/abs/quant-ph/0110141 points out that if every particle in the universe could be used as part of a huge computer, it could store only about 10^{90} bits, less than one millionth of the size our dictionary would require.
Of course, even if computers can't store all possible 40 character strings, they can easily programmed to start creating and displaying them one at a time. As long as we don't try to store all the output, our program could run indefinitely. Assuming a modern PC could output 1 billion strings per second, it would take one billionth of 2 × 10^{96} seconds, or 2 × 10^{87} seconds to complete its task, which is about 6 × 10^{79} years. By contrast, the universe is estimated to be 13.7 billion (1.37 × 10^{10}) years old. Of course, 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 10^{120} operations since the big bang. This is trillions of times more computation than is required for our string-displaying problem, but simply by raising the stakes to printing all 50 character strings instead of all 40 character strings we can outstrip the estimated computational potential of even the universe itself.
Problems like our simple string-displaying example grow exponentially in the number of computations they require, and are one reason why exponentially difficult problems are called "intractible" in computer science: for even small numbers like the 40 or 50 characters we used in our example, 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.
Such limits work to our advantage in cryptography, since we can safely assume that any cipher-breaking technique which requires more than, say, the 10^{120} operations mentioned before will never be feasible. Of course, many ciphers have been broken by finding efficient techniques which require only modest amounts of computing power and exploit weaknesses unknown to the cipher's designer. Likewise, much of the research throughout all branches of computer science focuses on finding new, efficient solutions to problems that work with far fewer resources than are required by a naďve 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 × 10^{500} 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.
As a general rule, then, PCs in 2004 can perform 2^{40} calculations in a few minutes. A few thousand PCs working for a few years could solve a problem requiring 2^{64} calculations, but no amount of traditional computing power will solve a problem requiring 2^{128} 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 as of 2004 it is far too soon to tell.
"Astronomically large" numbers
Other large numbers are found in astronomy:
- number of atoms in the visible universe, perhaps 10^{79} to 10^{81}, see Mass, Size, and Density of the Universe http://www.sunspot.noao.edu/sunspot/pr/answerbook/universe.html#q70
- for astronomical numbers related to distance and time, see:
Large numbers are found in fields such as mathematics and cryptography.
The MD5 hash function generates 128-bit results. There are thus 2^{128} (approximately 3.402×10^{38}) possible MD5 hash values. If the MD5 function is a good hash function, the chance of a document having a particular hash value is 2^{-128}, a value that can be regarded as equivalent to zero for most practical purposes. (But see birthday paradox.)
However, this is still a small number compared with the estimated number of atoms in the Earth, still less compared with the estimated number of atoms in the observable universe.
Even larger numbers
Combinatorial processes rapidly generate even larger numbers. The factorial function, which defines the number of permutations of a set of unique objects, grows very rapidly with the number of objects.
Combinatorial processes generate very large numbers in statistical mechanics. These numbers are so large that they are typically only referred to using their logarithms.
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.
Examples
- googol = 10^{100}
- googolplex = It is the number of states a system can be in that consists of 10^{98} particles which can each be in googol states.
- centillion = 10^{303} or 10^{600}, depending on number naming system
- Skewes' numbers: the first is ca. , the second
The total amount of printed material in the world is 1.6 × 10^{18} bits, therefore the contents can be represented by a number which is ca.
For a "power tower", the most relevant for the value are the height and the last few values. Compare with googolplex:
Also 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). Comparing the last number with , in the number 3000.48, the 1000 originates from the third number 1000 in the original power tower, a factor 3 comes from the second number 1000, and the minor term 0.48 comes from the first number 1000.
A very large number written with just three digits and ordinary exponentiation is .
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.
Tetration with base 10 can be used for very round numbers, each representing an order of magnitude in a generalized sense.
Numbers in between can be expressed with a power tower of numbers 10, with at the top a regular number, possibly in scientific notation, e.g. , a number between and (if the exponent quite at the top is between 10 and 10^{10}, 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, where denotes a functional power of the function f(n) = 10^{n} (the function also expressed by the suffix "-plex" as in googolplex, see the Googol family).
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^^[email protected]; the notations 10^^[email protected] and 10^^[email protected] are not needed, one can just write 10^3.12e6 and 3.12e6.
Thus googolplex = 10^^[email protected] = 10^^[email protected] = 10^^[email protected]; 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^^[email protected] 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 10-1e10:
- In the case of the range 0-1, an even shorter notation is (here for googolplex) like 10^^3.301 (proposed by William Elliot http://groups.google.com/groups?hl=en&lr=&newwindow=1&safe=off&selm=200204110321
13.I50801-100000%40agora.rdrop.com ). This is not only a notation, it provides at the same time a generalisation of 10^^x to real x>-2 (10^^[email protected]=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^^[email protected]" can be dispensed with).
Another example:
- (between and )
The "order of magnitude" of a number (on a larger scale than usually meant), can be characterized by the number of times (say n) one has to take the log_{10} to get a number between 1 and 10. Then the number is between and
An obvious property that is yet worth mentioning is:
I.e., if a number x is too large for a representation we can make the power tower one higher, replacing x by log_{10}x, or find x from the lower-tower representation of the log_{10} 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 not exactly given then giving a value at the top does not make sense, and a notation like can be used.
If the value after the double arrow is a very large number itself, the above can recursively be applied on that value.
Examples:
- (between and )
- (between and )
Some large numbers which one may try to express in such standard forms include:
External link: Notable Properties of Specific Numbers http://home.earthlink.net/~mrob/pub/math/numbers-14.html (last page of a series which treats the numbers in ascending order, hence the largest numbers in the series)
Accuracy
Note that for a number 10^{p}, one unit change in p 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 10^{50} too large or too small. This is seemingly an extremely poor accuracy, but for such a large number it may be considered fair. The idea that it is the relative error that counts (a large error in a large number may be relatively small and therefore acceptable), is taken a step further here: the number is so large that even a large relative error may be acceptable. Perhaps what counts is the relative error in the exponent.
Approximate arithmetic for very large numbers
In this context approximately equal may for example mean that two numbers are both written , with the true values instead of 4.829 being e.g. 4.8293 and 4.8288.
- 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.
Uncomputably large numbers
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. Σ(5) is not known but is definitely ≥ 4098. Σ(6) is at least 1.29×10^{865}.
Infinite numbers
See main article cardinal number
Although all these numbers above are very large, they are all still finite. Some fields of mathematics define infinite and transfinite numbers.
- aleph-null is the cardinality of the infinite set of the integers
- aleph-one is the next highest cardinal number.
- C is the cardinality of the reals. The proposition that C=Aleph-one is known as the continuum hypothesis.
- Mahlo cardinals
- Indescribable cardinal s
Beyond all these, Georg Cantor's conception of the Absolute Infinite surely represents the absolute largest possible concept of "large number".
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.
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.
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.
See also
- Orders of magnitude
- Orders of magnitude (numbers)
- History of large numbers
- Law of large numbers
- Names of large numbers
- Exponential growth
- Bignums
- Small numbers
- Continuum hypothesis
- Large cardinals
- Human scale
- Number names
External links
- "Large Numbers" and "Notable Properties of Specific Numbers" - Robert Munafo http://home.earthlink.net/~mrob/pub/math/largenum.html
- Big numbers - Susan Stepney http://www-users.cs.york.ac.uk/~susan/cyc/b/big.htm
- Hamlet is a Big Number - Jim Blowers (1) http://jimvb.home.mindspring.com/hamlet.htm , (2) http://jimvb.home.mindspring.com/ComNum.htm
- How to calculate with large numbers - Dave L. Renfro http://groups.google.com/groups?selm=28ae5e5e.0204081449.3b8479c4%40posting.goog
le.com&output=gplain - Graham's number and rapidly growing functions - Dave L. Renfro http://mathforum.org/discuss/sci.math/m/394644/394645
- Discussion and modification of notation http://groups.google.com/groups?hl=en&lr=&newwindow=1&safe=off&threadm=28ae5e5e.
0204081449.3b8479c4%40posting.google.com&rnum=1&prev=/groups%3Fnum%3D100%26hl%3D
en%26lr%3D%26newwindow%3D1%26safe%3Doff%26q%3D%2B%2522tetration%2522%2B%253A%2B%
2522scientific%2Bnotation%2522 - Ackermann’s function and new arithmetical operations - Rubtsov and Romerio http://www.rotarysaluzzo.it/filePDF/Iperoperazioni%20(1).pdf (pdf)
- wiki:ReallyBigNumbers
- http://michaelhalm.tripod.com/mathematics_beyond_the_googol.htm (various terminology for large numbers; contains errors)