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Text of the page (random words):
refers to large positive integers or more generally large positive real numbers but it may also be used in other contexts the study of nomenclature and properties of large numbers is sometimes called googology 1 2 contents 1 in the everyday world 2 astronomical 2 1 billions and billions 3 examples 4 standardized system of writing 4 1 examples 4 2 other notations 5 comparison of base values 6 accuracy 6 1 for very large numbers 6 2 classes 6 3 approximate arithmetic 7 systematically creating ever faster increasing sequences 8 in some noncomputable sequences 9 infinite numbers 10 see also 11 references in the everyday world edit see also scientific notation logarithmic scale and orders of magnitude scientific notation was created to handle the wide range of values that occur in scientific study 1 0 10 9 for example means one billion or a 1 followed by nine zeros 1 000 000 000 the reciprocal 1 0 10 9 means one billionth or 0 000 000 001 writing 10 9 instead of nine zeros saves readers the effort and hazard of counting a long series of zeros to see how large the number is examples of large numbers describing everyday real world objects include the number of cells in the human body estimated at 3 72 10 13 3 the number of bits on a computer hard disk as of 2022 update typically about 10 13 1 2 tb the number of neuronal connections in the human brain estimated at 10 14 the avogadro constant is the number of elementary entities usually atoms or molecules in one mole the number of atoms in 12 grams of carbon 12 approximately 6 022 10 23 the total number of dna base pairs within the entire biomass on earth as a possible approximation of global biodiversity is estimated at 5 3 3 6 10 37 4 5 the mass of earth consists of about 4 10 51 nucleons the estimated number of atoms in the observable universe 10 80 the lower bound on the game tree complexity of chess also known as the shannon number estimated at around 10 120 6 astronomical edit other large numbers as regards length and time are found in astronomy and cosmology for example the current big bang model suggests that the universe is 13 8 billion years 4 355 10 17 seconds old and that the observable universe is 93 billion light years across 8 8 10 26 metres and contains about 5 10 22 stars organized into around 125 billion 1 25 10 11 galaxies according to hubble space telescope observations there are about 10 80 atoms in the observable universe by rough estimation 7 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 10 10 10 10 10 1 1 years displaystyle 10 10 10 10 10 1 1 mbox years which corresponds to the scale of an estimated poincaré recurrence time for the quantum state of a hypothetical box containing a black hole with the estimated mass of the entire universe observable or not assuming a certain inflationary model with an inflaton whose mass is 10 6 planck masses 8 9 this time assumes a statistical model subject to poincaré recurrence a much simplified way of thinking about this time is in a model where the universe s history repeats itself arbitrarily many times due to properties of statistical mechanics this is the time scale when it will first be somewhat similar for a reasonable choice of similar to its current state again combinatorial processes 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 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 logician harvey friedman has done work related to very large numbers such as with kruskal s tree theorem and the robertson seymour theorem billions and billions edit 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 affect johnny carson quipped billions and billions 10 the phrase has however now become a humorous fictitious number the sagan cf sagan unit examples edit googol 10 100 displaystyle 10 100 centillion 10 303 displaystyle 10 303 or 10 600 displaystyle 10 600 depending on number naming system millinillion 10 3003 displaystyle 10 3003 or 10 6000 displaystyle 10 6000 depending on number naming system the largest known smith number 10 1031 1 10 4594 3 10 2297 1 1476 10 3 913 210 the largest known mersenne prime 2 82 589 933 1 displaystyle 2 82 589 933 1 as of december 21 2018 googolplex 10 googol 10 10 100 displaystyle 10 text googol 10 10 100 skewes s numbers the first is approximately 10 10 10 34 displaystyle 10 10 10 34 the second 10 10 10 964 displaystyle 10 10 10 964 graham s number larger than what can be represented even using power towers tetration however it can be represented using knuth s up arrow notation kruskal s tree theorem is a sequence relating to graphs tree 3 is larger than graham s number rayo s number is a large number named after agustín rayo which has been claimed to be the largest named number it was originally defined in a big number duel at mit on 26 january 2007 standardized system of writing edit 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 10 4 and 2 10 5 compare the exponents first in this case 5 4 so 2 10 5 5 10 4 if the exponents are equal the mantissa or coefficient should be compared thus 5 10 4 2 10 4 because 5 2 tetration with base 10 gives the sequence 10 n 10 n 2 10 n 1 displaystyle 10 uparrow uparrow n 10 to n to 2 10 uparrow n 1 the power towers of numbers 10 where 10 n displaystyle 10 uparrow n denotes a functional power of the function f n 10 n displaystyle f n 10 n the function also expressed by the suffix plex as in googolplex see the googol family these are very round numbers each representing an order of magnitude in a generalized sense a crude way of specifying how large a number is is specifying between which two numbers in this sequence it is more precisely numbers in between can be expressed in the form 10 n a displaystyle 10 uparrow n a i e with a power tower of 10s and a number at the top possibly in scientific notation e g 10 10 10 10 10 4 829 10 5 4 829 displaystyle 10 10 10 10 10 4 829 10 uparrow 5 4 829 a number between 10 5 displaystyle 10 uparrow uparrow 5 and 10 6 displaystyle 10 uparrow uparrow 6 note that 10 n 10 n a 10 n 1 displaystyle 10 uparrow uparrow n 10 uparrow n a 10 uparrow uparrow n 1 if 1 a 10 displaystyle 1 a 10 see also extension of tetration to real heights thus googolplex is 10 10 100 10 2 100 10 3 2 displaystyle 10 10 100 10 uparrow 2 100 10 uparrow 3 2 another example 2 4 2 2 2 65 536 copies of 2 10 65 531 6 10 19 728 10 65 533 4 3 displaystyle 2 uparrow uparrow uparrow 4 begin matrix underbrace 2_ 2 2 qquad quad 65 536 mbox copies of 2 end matrix approx 10 uparrow 65 531 6 times 10 19 728 approx 10 uparrow 65 533 4 3 between 10 65 533 displaystyle 10 uparrow uparrow 65 533 and 10 65 534 displaystyle 10 uparrow uparrow 65 534 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 l o g 10 displaystyle log_ 10 to get a number between 1 and 10 thus the number is between 10 n displaystyle 10 uparrow uparrow n and 10 n 1 displaystyle 10 uparrow uparrow n 1 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 10 10 or the next between 0 and 1 note that 10 10 n x 10 n 10 x displaystyle 10 10 uparrow n x 10 uparrow n 10 x i e if a number x is too large for a representation 10 n x displaystyle 10 uparrow n x 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 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 10 7 21 10 8 displaystyle 10 uparrow uparrow 7 21 times 10 8 if the value after the double arrow is a very large number itself the above can recursively be applied to that value examples 10 10 10 10 3 81 10 17 displaystyle 10 uparrow uparrow 10 10 10 3 81 times 10 17 between 10 2 displaystyle 10 uparrow uparrow uparrow 2 and 10 3 displaystyle 10 uparrow uparrow uparrow 3 10 10 10 497 9 73 10 32 10 2 10 497 9 73 10 32 displaystyle 10 uparrow uparrow 10 uparrow uparrow 10 uparrow 497 9 73 times 10 32 10 uparrow uparrow 2 10 uparrow 497 9 73 times 10 32 between 10 4 displaystyle 10 uparrow uparrow uparrow 4 and 10 5 displaystyle 10 uparrow uparrow uparrow 5 similarly to the above if the exponent of 10 displaystyle 10 uparrow 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 10 displaystyle 10 uparrow add 1 to the exponent of 10 displaystyle 10 uparrow uparrow so we get e g 10 3 2 8 10 12 displaystyle 10 uparrow uparrow 3 2 8 times 10 12 if the exponent of 10 displaystyle 10 uparrow uparrow 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 10 displaystyle 10 uparrow uparrow use the triple arrow operator e g 10 7 3 10 6 displaystyle 10 uparrow uparrow uparrow 7 3 times 10 6 if the right hand argument of the triple arrow operator is large the above applies to it so we have e g 10 10 2 10 497 9 73 10 32 displaystyle 10 uparrow uparrow uparrow 10 uparrow uparrow 2 10 uparrow 497 9 73 times 10 32 between 10 10 4 displaystyle 10 uparrow uparrow uparrow 10 uparrow uparrow uparrow 4 and 10 10 5 displaystyle 10 uparrow uparrow uparrow 10 uparrow uparrow uparrow 5 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 n displaystyle uparrow n compare this notation with the hyper operator and the conway chained arrow notation a n b displaystyle a uparrow n b 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 a n k b displaystyle a uparrow n k b for example 10 2 3 b displaystyle 10 uparrow 2 3 b 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 10 2 3 1 displaystyle 10 uparrow 3 3 10 uparrow 2 3 1 10 3 3 since the b can also be very large in general we write a number with a sequence of powers 10 n k n displaystyle 10 uparrow n k_ n with decreasing values of n with exactly given integer exponents k n displaystyle k_ n with at the end a number in ordinary scientific notation whenever a k n displaystyle k_ n is too large to be given exactly the value of k n 1 displaystyle k_ n 1 is increased by 1 and everything to the right of n 1 k n 1 displaystyle n 1 k_ n 1 is rewritten for describing numbers approximately deviations from the decreasing order of values of n are not needed for example 10 10 5 a 10 6 a displaystyle 10 uparrow 10 uparrow uparrow 5 a 10 uparrow uparrow 6 a and 10 10 3 10 10 10 1 10 3 displaystyle 10 uparrow 10 uparrow uparrow uparrow 3 10 uparrow uparrow 10 uparrow uparrow 10 1 approx 10 uparrow uparrow uparrow 3 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 10 n 10 10 10 n displaystyle 10 uparrow n 10 10 to 10 to n 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 n displaystyle uparrow n in the superscript of the first arrow etc or we have a nested chain notation e g 10 10 10 10 3 10 5 displaystyle 3 times 10 5 10 10 3 10 5 10 10 displaystyle 10 uparrow 10 uparrow 3 times 10 5 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 f n 10 n 10 displaystyle f n 10 uparrow n 10 10 10 n these levels become functional powers of f allowing us to write a number in the form f m n displaystyle f m n where m is given exactly and n is an integer which may or may not be given exactly for example f 2 3 10 5 displaystyle f 2 3 times 10 5 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 10 10 3 2 10 10 10 10 10 10 displaystyle 10 to 10 to 3 to 2 10 uparrow 10 uparrow 10 10 10 10 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 g 3 3 65 2 10 10 65 2 f 65 1 displaystyle g 3 rightarrow 3 rightarrow 65 rightarrow 2 10 to 10 to 65 to 2 f 65 1 but also g f 64 4 f 65 1 displaystyle g f 64 4 f 65 1 if m in f m n displaystyle 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...