Notable Properties of Specific Numbers
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108.0723047260281×10153 = 4444
The highest value achievable in the "four 4's" puzzle if exponents are allowed. (This is the puzzle that asks you what numbers you can make using four 4's and the "common" operations on a calculator, for example, 1=44/44, 2=4/4+4/4, 3=4+4+4/4, 4=4√4/4)×4, etc.) See also 101.0979×1019.
Since it is 4↑↑4, or 4④4 using the hyper4 operator, it is 4⑤2 using the hyper5 operator.
A somewhat lower estimate of the number of possible universe histories given by Dave L. Renfro and calculated by a different method (he estimates the Planck-unit volume of the universe at 10123, the number of particles at 1080 and the universe age at 1041 times an "interval" of 10-24 seconds).39 See also 101.877×1054, 103.79×10281 and 105.7×10410.
(my "simple alternate universe count")
A highly-simplified formula to compute the number of possible universes. N = ev n where N is the number of possible universes, n is the number of fundamental particles in the universe and v is the number of particles that could be fit in the universe if it were packed full of particles. See also 101.877×1054.
This is ee661, an improved (but erroneous) upper-bound for the π(x) vs. li(x) problem (the higher "Skewes' number"), by Alan Turing in an unpublished manuscript. It was corrected to 102.6654...×10536 by Cohen and Mayhew in 1965.
logarithmica_numerus_lite.js is a JavaScript library meeting the needs of incremental games, developed by Aarex Tiaokhiao since 2017 December [236]. It represents numbers by their base-10 logarithm using IEEE double precision, and thus can handle values up to about 101.8×10308. However, Tiaokhiao's magna_numerus.js goes higher; and see 3pt1.0126×101656520 for more.
An estimate of the number of distinct universes in the "string theory landscape", given by [209] in section 5, and assuming that the "maximal number of observable e-folds" is about 290. 121
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.79×10281, 105.7×10410, 101010000000, 101010122, and 10101.51×103883775501690.
(my "alternate universe count")
The factorial of the single-perturbation count, a highly theoretical estimate of the number of different ways all the particles in the known universe could be randomly shuffled at each moment in time since the universe's creation. In quantum mechanics, it is comparable to the number of universe timeline wave-functions that exist simultaneously from the viewpoint of an observer outside our universe. Another way of saying the same thing is that, if the universe is being created over and over again, it would take (on average) this many repetitions before one would expect to get an exact recurrence of "our" universe.
This estimate is entirely arbitrary and omits many details (relativistic curvature, dark matter and dark energy, Pauli exclusion, etc.) because there is probably an even greater discrepancy between the "known" and "actual" size of the universe.
See also 101.877×1054, 1010166, 103.79×10281 and 10101.51×103883775501690.
This is ee1236, an improvement on the higher "Skewes' number" (and a correction of Alan Turing's 105.0867...×10286) published by Cohen and Mayhew in 1965.
The number of Adam Clarkson's lynz as of the 10th of August, 2003 (and due the following day) as referenced on a classmate's blog17.
This is the factorial of 10666, and is called the leviathan number by Clifford Pickover [193]. The word leviathan refers to a whale or sea-monster. Biblical references to "leviathan" are all in the Old Testament, although 666 is more commonly associated with the last book of the New Testament. See also 1000000000000066600000000000001.
105.78134...×102971 = 229866×203×1029867
(the ever-increasing number of Adam Clarkson's "Lynz")
The number of lines of text one Adam Clarkson will owe his (former) high school chemistry teacher on 22nd September 2025. The story runs as follows16:
In February (on the 26th, to be precise, since it's a good day to celebrate, num' sein?) of 1998, a chemistry teacher gave a set of lines to one of his students, Adam Clarkson. The lines read, "I must always tuck my shirt in whilst participating in a Chemistry Lesson".115 He had to do a hundred of them. However, the clever part was that if he didn't do them by the next day, they would double, and if they weren't done by the day after that, they would double again. We pointed out that the lines would become too great to do pretty soon, but this didn't stop the teacher giving them. . . .
Daily doubling is a very effective way to outpace anyone's ability at nearly any task; see the chess legend for another example. After one week the assignment had grown to 27×100 = 12,800 lines, which doesn't seem too bad, relatively speaking, but another week makes it 214×100 = 1,638,400. By the 26th March (one month after the initial assignment) it would have been 26,843,545,600 (more lines than in all the books of the school's library), and by 26th April Adam would have had to enlist the assistance of the entire world's population, writing over 8 billion lines apiece to complete the total 259×100 ≈ 5.7646×1019 lynz. This is slightly more than the number of combinations of a Rubik's cube, coincidentally the subject of another classic case of human innumeracy.
This alone would be a quintessential legend of the human struggle to understand large numbers, but the story doesn't end there16:
. . . [On the 17th] September 1998, 7 months late, due to a confused and old man apparently mishearing us (we tried to inform him of the vastness of the Lynz, but he didn't want to hear, and so he said "If they're not on my desk by tomorrow, they'll square!"), the lines were squared every day they weren't done. To this day, they still haven't been done.
Since the 17th September was 203 days after the 26th of February, the lines had been doubled 203 times — on that day the assignment was 2203×100 ≈ 1.286×1063 lines, a little over one vigintillion, and due the next day. This was more or less equal to the capacity of the known universe to produce handwritten lines, but only if all the galaxies were converted into paper and ink. There was no chance of doing these Lynz by the next day, and so the squaring commenced.
On the 18th and each day thereafter, the number was squared, so the assignment on the 18th September was (2203×100)2 = 22×203×104, about 1.65×10126: in one day they went from a vigintillion to being much larger than a googol. The day after that it was 24×203×108, or about 2.73×10252, and so forth. In general, the number is 22N×203×102(N+1) where N is the number of days since 17th September 1998.
Within days the Lynz blew past the short- and long-scale centillion and millillion; within a week or two more they had grown beyond the odds against Ford and Arthur's rescue, the Hamlet monkey number, a milli-millillion, and the various record-size primes. These numbers are many millions of digits long, meaning that it would have taken Adam millions of sheets of paper just to write down how many lines were due.
Because they are squared daily, the number of digits of the number of lynz doubles each day. By their first anniversary the Lynz were about 103.689×1050, just over G. H. Hardy's estimate of the number of possible chess games; by 8th August 1999 they surpassed a googolplex. By their second anniversary (26th February 2000), they had become so large that if you tried to write not the lines themselves, but merely the number of lines as a normal decimal number (that is, without using scientific notation) you'd be writing a number over 10160 digits long, a feat which could not be accomplished even if you could fit a vigintillion digits on each particle in the observable universe. By 2003 the length of the assignment had grown17 to the point where it was just "somewhat larger" than my rough estimate of the number of ways the universe's history could be shuffled. By their tenth anniversary116 the Lynz were in excess of 10101000, and the "Clarkkkkson" function based on them117 was generally regarded118 to be faster-growing than most anything out there. As the author points out,
[...] with the ticking time-bomb that is [The Lynz] squaring every day, it's got a Gmail-style claim to infinity.
To make clear the calculation of specific values of N in "22N×203×102(N+1)" above, this table shows the Lynz day-by-day, then by months, then by years, then 5 years at a time:
|
Most of the numbers were figured out by hand in a spreadsheet, then double-checked with a program which was then used to carry forward beyond 2018 (it knows to add a day every 4 years).
This real-life story invokes a similar respect for the innumeracy of common people to that described in the ancient chess legend; see also 2.315×1016, 43252003274489856000 and 10137.
10(2.62086...×106989) = .3-(.2-(.1-4)) = (10/3)5104
If decimal points are allowed in the digits 1 2 3 4 problem, this is the result. The subexpression .1-4 is equivalent to 104 = 10000; the subexpression .2-(.1-4) is equivalent to .2-10000 = 510000 = 5.01237274958×106989; similarly .3-x is equivalent to 3.3333...x. The idea for this was sent to me by Jim Denton (although his suggestion, 3.2-(.1-4), was slightly smaller).
magna_numerus.js is a JavaScript library meeting the needs of incremental games, developed by Aarex Tiaokhiao since 2017 December [236]. It represents numbers in mantissa-exponent format, with an integer exponent as large as 1065535, so it can handle values up to 101065535. However, Tiaokhiao's confractus_numerus.js goes higher; and see 3pt1.0126×101656520 for more.
In section 3 of their paper "How many universes are in the multiverse?" [209] Linde and Vanchurin estimate the number of "different types of universes" arising from a certain "eternal cosmic inflation" scenario, and limited by the fact that one of them is our universe, which is known to be spatially homogeneous and isotropic at large scales (a "Friedmann universe").
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.79×10281, 1010375, 105.7×10410, 101010122, and 10101.51×103883775501690.
The JavaScript library break_break_infinity.js was created by Patashu, (the author of break_infinity.js) in 2017 November, but was abandoned in 2019 March [237]. Meant to meet the needs of incremental games, It used the BigInteger arbitrary-precision integer library by Yaffle, and a mantissa-exponent format with the (integer) exponent stored exactly in available memory. This gave a practical limit of about 100,000,000 digits for the exponent, meaning that values up to about 1010108 could be handled. (Despite this, the magna_numerus.js author Aarex Tiaokhiao described it as being able to handle values up to "e(1.8e308)", which is the far smaller 101.8×10308.) See 3pt1.0126×101656520 for more on this topic.
102.0756×10121210694 = 3×23×2402653211+402653211-1 = 22223×3×23×3×223×3×23×3×2223×3×23×3×223×3×23×3-1
This is the highest base achieved in the "lower Goodstein sequence" iteration starting with 9=23+1. See 6.895×10121210694 for more.
This is the approximate value of 9999. Using Mathematica version 9, Robert G. Wilson was able to compute the first 100 digits and the last 100 digits of this huge number; see OEIS sequence A243913. See also 4.28...×10369693099.
109.35...×101414973347 = 1026109
In an answer on Philosophy stackexchange [225], I describe a hypothetical situation in which a person (who cannot age or want from any hardships) is watching a display that shows a number of the form 10N in normal notation (that is, a 1 followed by N zeros). As time goes on, N gets steadily larger (the number of zeros keeps increasing). The question is, "how big could the number get and the watcher still be able to directly perceive each displayed power of 10 in a distinct way?". I believe there is a limit to the mental capacity of this hypothetical observer. If that mental capacity is equivalent to a billion letters of writing using a 26-letter alphabet, then the person's mental capacity would be 261000000000. This person could "count the zeros" and fit the answer in their head until the count exceeds 261000000000 zeros. At this point the number being displayed would be 10261000000000. Rewritten in the form of 1010x, this comes out to about 109.35...×101414973347.
10101010 = 101010000000000 = 10④4
A power-tower of four 10's, written "10④4" using my hyper4 notation or "10↑↑4" using Knuth's up-arrow notation.
101.55×104342944819032 = ee1013
This is the value of the "inflation factor" in a model of the inflationary universe developed by Dr. Thanu Padmanabhan [151], resulting from the assumption that the cosmological constant lambda equals approximately 10-8, a value arising from grand unification theories.
confractus_numerus.js is a JavaScript library meeting the needs of incremental games, developed by Aarex Tiaokhiao since 2017 December [236]. It represents numbers by their base-10 logarithm using two IEEE double precision numbers, one for a mantissa and the other for an integer exponent. This means it can go up as high as 10 to the power of anything handled by Decimal.js and the like. However, Patashu's break_eternity.js goes higher; and see 3pt1.0126×101656520 for more.
103.5536897484442191...×108852142197543270606106100452735038 ≈ eee79 ≈ 10101034
(Skewes' number)
The Skewes' number, in the general sense refers to the lowest value x for which the prime-counting function π(x) is larger than the logarithmic integral function li(x).
eee79 is the original (higher) value of the first (Riemann hypothesis true) Skewes' number, published in a 1933 paper. It is normally written as "10101034". It was later reduced to ee27/4, which is "merely" 8.1847946207224960623437×10370. In 2005 numerical techniques were used to determine the actual value of the crossover, 1.397162914×10316. See also 1.53×101165, 10103.3×10963 and 4pt6.8880×1014.
10102.1485709110445×1038 = 22172912
In 2019 Harvey David and Joris van der Hoeven showed an algorithm for arbitrary-precision (bignum) multiplication that achieves the conjectured limit of efficiency, O(n log n) where n is the number of digits in the numbers being multiplied. It uses a lot of complicated techniques, including a 1729-dimensional Fourier transform. However, it is not better in practice than the Schönhage-Strassen algorithm ("FFT multiplication"), unless the numbers being multiplied are at least this large (i.e. have at least 2172912 digits when expressed in base 2).
This number was sent to me by a reader saying they found it on Wikipedia. The first part is easily identifiable as having come from a paper by Carroll and Chen [187]. It is the amount of time one would need to wait for "the spontaneous onset of eternal inflation" in a universe that has the familiar values for the Planck mass Mpl, de Sitter spacetime temperature T ≈ 10-43 GeV, and something they call the "mass of the scalar" m ≈ 1013 GeV.
The second and much smaller part of the expression is 1010115. This is probably meant to be the Tegmark Hubble volume repetition length, which can be interpreted as the distance one would need to travel (in any direction, and in any units e.g. metres) before encountering a universe indistinguishable from our own. As explained in that entry, such "travel" would violate relativistic causality.
In this case 1010115 is not a travel distance but a multiplier of the waiting time. The product of 10101056 and 1010115 would express the longer waiting time needed for the spontaneous onset of inflation to recreate a replica of our universe (in addition to a new "big bang", which merely guarantees recreating some universe). The product of the two numbers is effectively the same as the larger of the two because of the power tower paradox.
101010100-1 = 999999999999...(a total of googolplex 9's)...9999
This huge number, if written out, would have a googolplex 9's in a row. Many factors are known, based on simple facts of modulo arithmetic. Since it is all 9's, it is divisible by 3. Since the number of 9's is a multiple of 2, the whole number must be a multiple of 99, for the same reason that 1010-1 = 9999999999 = 99×101010101; and because it is divisible by 99, it is divisible by 11. The same principle lets us add factors of 9999 = 104-1, 99999 = 105-1, 99999999=108-1, and any number of the form 102a5b-1 where a and b can each be as high as a googol (such numbers are found in OEIS sequence A3592). Additional factors of 101010100-1 can be found via Fermat's Little Theorem (see 999999). As a result there are a huge number of known factors of 101010100-1, beginning with: 3, 11, 17, 41, 73, 101, 137, 251, 257, 271, 353, 401, 449, 641, 751, ... [186].
("googolplexplex")
10 to the power of googolplex.
Just as with -illion, there are many number names formed by folk etymological extension of the -plex suffix, such as millionplex for 101000000.
A common name for 10googolplex is googolplexian (seen in Internet searches) but I suspect googolplexplex is more commonly used by folks who try to come up with their own name. I have also seen googolduplex proposed by creators of (ad hoc googolisms), beginning a series that continues with googoltriplex=10googolduplex, googolquadriplex, googolquinplex and so on.
In [158] (page 8), Don Page estimates the Poincare recurrence time of a black hole of a mass equivalent to the visible universe to be "101010102.08 Planck times, millennia, or whatever." It can be understood as a period of time equal to the number of possible "macrostates" of the system inside the black hole, which is eN where N is the number of "microstates", and this in turn is equal to ees where s is the maximum possible entropy of the system. s is approximately (r/lp)2 where r is the radius of the universe and lp is the Planck length. Using a current (2014) value for the size of the visible universe we would get (4.45×1026/(1.616×10-35))2 or about 7.5×10122 in the exponent. This number is also described in Numberphile's video The LONGEST Time. See also 10101010101.1.
In [197] (page 12), speculating on the implications of one possible resolution of a paradox brought about by the Hartle-Hawking "no-boundary" model of the universe pointed out by Susskind, the size of the universe at the end of the inflationary period would be about 101010122. The author uses the units "Mpc" (megaparsecs) although the number is so large that any length unit (such as Planck lengths, inches or furlongs) or even a volume unit (cubic parsecs, drams or bushels) could be used and the number would still be just as accurate.
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.79×10281, 1010375, 105.7×10410, 101010000000, and 10101.51×103883775501690.
10103.29994322...×10963 = eeee7.705
(the Higher "Skewes' number")
In 1955, Skewes gave this as an upper bound of the first π(x) - li(x) crossing if the Riemann Hypothesis is false (see the first Skewes number entry for a fuller description). Sometimes the conservative overestimate 1010101000 is given. It was improved to 102.6654...×10536 by Cohen and Mayhew in 1965, before actual computational results (starting with Lehman's 1.53×101165) took over.
See also 4pt6.8880×1014.
10101.335740483×102184 = 55555
This number is the subject of one of the problems in "The Wohascum County Problem Book", published by the MAA in 1996. The book is a collection of problems for high school students. This particular problem reads: "What is the fifth digit from the end (the ten thousands digit) of the number 5^5^5^5^5?" The answer (0) is found by the method I describe in this article concerning modulo power tower computation.
See also 10101.335740483×102184.
This is (((7!)!)!)!, given by Tibor Rado in the original 1962 paper[139] defining the "busy beaver" function. See also 107 and 1.29149×10865.
10[10(1.51×103883775501690)] = 10101010101.1
(Don Page's alternate universe count)
This is a quantity of time, estimated by Don Page in [158] (page 8) and 27, which he describes as the "quantum Poincare recurrence time for the quantum state of an extremely hypothetical rigid nonpermeable box containing a black hole with the mass of what may be the entire universe in one of Andrei Linde's stachastic inflationary models". The units could be Planck units, nanoseconds, fortnights, or centuries — it doesn't matter because the number is so large. Numberphile did a video on it: The LONGEST Time. See also 101.877×1054, 105.7×10410, and 10101.7×10120.
New "pt" Notation
Note the uncommon representation "3pt..." in the entries to follow ("3pt6.8880×1014", "3pt2.069...×1036305", and so on). The pt stands for "Powers of Ten", and signifies the fact that a number, expressed as a power tower, can be described as "3 powers of ten with (whatever) at the top".
Alternatively, "3pt6.8880×1014" could be described as 4pt14.838, or "4 powers of ten with 14.838 at the top": 14.838 is approximately the logarithm to base 10 of 6.8880×1014.
1010106.8880×1014 = 3pt6.8880×1014 = eeee35
Wolf's incorrect value [220] of the Knapowski number.
This is the number of 1's produced by a particular 6-state, 2-symbol Turing machine described by Shawn Ligocki on the 27th May 2022. The machine implements a Collatz-like iteration, like an earlier one found by Pavel Kropitz in June 2010 (see 7.412...×1036534) but exponentiates three more times before halting. For a few days it held the record for the 6-state 2-symbol busy beaver function, but was soon surpassed by Kropitz with machine "t15".
1010102.069197...×1036305 = 3pt2.069...×1036305 = 666666
(Pickover's superfactorial of 3)
The higher-valued "superfactorial" function, defined by Pickover in 1995, is:
n$ = n!④n! = n!↑↑n! = n! ^ n! ^ n! ^ ... ^ n!
where "$" is "superfactorial", ④ or ↑↑ represents the higher hyper4 operator also called tetration, and there are n! copies of "n!" on the right hand side. According to this definition, 3 superfactorial is:
3$ = 3!3!3!3!3!3! = 666666
See also 288 and 22pt1.84×1033.
1010101.0126×101656520 = 3pt1.0126×101656520 = eeeeeee
(range of level-index representaton)
D. W. Lozier and P. R. Turner have published papers describing a number format called level-index in which numbers are stored in the form +e+ee...X, where X is a fraction from 0.000 to 0.999... and there are as many e's as necessary (up to a proven maximum of six). For example, 10 would be ee0.834032..., 143 would be eee0.471239... and so on. This system has as its main advantage the property that there is no overflow or underflow if you perform a finite number of the operations + - × and /. The reason it never overflows is that for sufficiently high X, roundoff causes the operation X2=X×X to give X as an answer (see my uncomparably larger discussion and note the paragraph on "If A is a class 5 number". The power tower paradox discussion is also relevant.).
In their article "Error-Bounding in Level-Index Computer Arithmetic" they propose a format that uses a 3-bit level field with 2 "sign" bits and the remaining bits (59, if it's a 64-bit word) for the fraction. This allows representing numbers as high as the number shown above, a "power-tower" of seven e's. For more about the symmetric level-index system, go here. There are lots of nice properties, such as greater precision for the most commonly used ranges of values, progressive gradual degradation of mantissa precision as the exponent grows (a major advantage over normal floating-point formats), reduction of all common operations to a single invertible monotonic function y=ln(x+1) which has an efficient hardware implementation, and more.
Incremental Games
For many years the Lozier/Turner level-index system had the highest limit for any published or remotely serious computer number-representation system with a full software library (apart from my own Hypercalc which I don't count). However, that has changed with the advent of incremental games, particularly since 2014 (the year of the well-known Cookie Clicker). Such games motivate the player as long as they can achieve ever higher values of some quantity. Typically an exponential rate of progress will keep people playing until the quantities involved becomes too high for their abilities of mental abstraction, and this happens somewhere in the class 2 range. Some games overcome this by replacing the main goal with another (rather than words, you now accumulate books, and later it will be libraries), but many games just go to scientific notation with ever-larger exponents. Faster rates of growth, ("tetrational" or beyond) are often needed to keep things interesting.
To calculate scores and rewards, etc. in such games, a variety of number formats with suitable software libraries has been developed, mainly in JavaScript. Some of the more modest software libraries are described in the entries for 109×1015, 101.797...×10308, 101065535, and 1010100000000. Those limits are all far smaller than the Lozier/Turner level-index format being discussed here, but incremental games have a voracious appetite for more. Since early 2019 we have had the break_eternity.js library by Patashu, handling numbers as large as 10↑↑(1.8×10308) in Knuth's up-arrow notation. Not to be outdone, Naruyoko began work on OmegaNum.js which can go up as high as 10↑↑↑...↑(9×1015) with 1000 up-arrows. This is nowhere near as large as Graham's number, but fortunately, since early 2020 we have had Naruyoko's ExpantaNum.js for that.
1010101010000000 = 4pt10000000
Another example of a number bigger than Skewes' that has been published in a journal (in 1994). The generalised Poincaré conjecture of topology relates to the "smoothness" of multidimensional space. Curiously, 4-dimensional space has been shown to be unique among all dimensions in having an uncountably infinite number of topological structures that are equivalent to simple flat space in a very important way58,60. In work related to this47, Zarko Bizaca shows how to construct a "level 7 embedded Casson tower" and estimates the number of "kinks" or "kinky handles" in the "core" of said tower to be around 10101010107.
In another paper 59, Bizaca says the number of links on each "kinky handle" of a Casson tower's level 1, 2, 3, 4, ... is: 1, 2, 2, 2, 200, 2×10101010, 2×10↑↑12, 2×10↑↑20, ..., where ↑↑ is tetration, the higher hyper4 operator. For higher levels the number is 2×10↑↑(8n-44).
See also 101010106.8880×1014.
As of March 2014, this is the record for the 7-state busy beaver Turing machine. It is based closely on the 2010 6-state record and takes this many steps (squared) before halting with this many ones on the tape. The machine was developed by "Wythagoras" on Googology, based on the 6-state machine by Pavel Kropitz and its analysis by Pascal Michel. The number shown here was later double-checked and proven by "Cloudy176". See also 107, 4098, and 47176870.
101010106.8880×1014 = 4pt6.8880×1014 = eeeee35
The Knapowski number
This number appears in the 1962 paper "On sign-changes of the difference π(x) - li x", by S. Knapowski [138]. Knapowski proves that for all x larger than this value, the number of crossings of π(x) and li x (see Skewes') is greater than ln(ln(ln(ln(x))))/e35.
Thus it is bigger than both of Skewes' numbers, though the various versions of Graham's number are larger. As Skewes' numbers were published in 1933 and 1955 and the Graham-Rothschild_number was published in 1971, there was a period (1962-1971) during which Knapowski's number "held the record" for largest number explicitly mantioned in a published academic paper. (All have since been exceeded by TREE(3) and SCG(2); see also 1010101010000000, which is from 1994 and never held the record).
The Knapowski paper is mentioned briefly by Wolf [220] who erroneously says that it is eeee35, a power-tower with one fewer e.
10101010104.8293×10183230 = 5pt4.8293×10183230
This is 10^(9^(8^(7^(6^(5^(4^(3^(2^(1^0))))))))), where each ^ is the exponent operator, also referred to as the exponential factorial of 10. It is cited on Frank Pilhofer's Googolplex page as an example of something larger than Googolplexplex.
See also 6pt1.86×103148880079.
10101010103.58259...×103010299956639812
The value of "latinlatinlatinbyllionyllionyllionyllion" under Knuth's "latin{name-of-N-with-spaces-deleted}yllion" extension of his -yllion naming system, described on pages 310-312 of [146]. See also 104194304.
1010101010101.2826×1082 = 6pt1.86×103148880079
This 2^(3^(4^(5^(6^(7^(8^(9^10))))))), where each ^ is the exponent operator. It is the largest value you can get using one of each of the ten digits 0 through 9, without any symbols or punctuation: 2345678910. (An improvement on Epstein's suggestion84 by replacing 8910 with the larger 8910). See also 101.0979×1019 and 5pt4.8293×10183230.
10101010101010101010 = 9pt10 = 10pt1
A power-tower of ten 10's. This is 10④10 using the higher hyper4 operator, or 10↑↑10 using Knuth's up-arrow notation.
This is 4$ = 4!④4! = 24↑↑24, or "4 superfactorial". It is a power tower of 24's of height 24. See 1010102.069×1036305.
Using the 195 digits in Numberphile's can of Numberetti, one can go far higher than 10195 by building a power-tower of exponents.
It makes sense to use the 2s, 3s, etc. through 9s as single digits, with the lower digits at the bottom of the power tower (for the same reason that 234 is greater than 432) but it's not so clear how to best use the 0s and 1s. One viewer suggested to pair the 1s to make 11s, then use the last 1 with all the 0s to make a big 100000... at the top.
But viewer inperangua then suggested pairing 1s with 0s to make 10s and then "end with leftover 0s or 1s". This is in fact better, because (for example) given the digits 11100000 you get higher with 10101000 than with 11100000; or given 11111000 you're better with 10101011 than with 11111000.
There are 147 digits 2 through 9, and sixteen 1's and thirty-two 0s; so the highest result comes from having 147 single digits, fifteen 10s and 1017 at the top: a power tower 163 numbers high, starting with 222... at the bottom and ...1010100000000000000000 at the top. This is not quite as high as the power-tower of 256's used to make Steinhaus's "Mega".
Steinhaus's "Mega"
Using the original Steinhaus notation, the number represented by "2 inside a square" is "triangle(triangle(2))", where "triangle(x)" is xx. Thus, 2 = triangle(22) = triangle(4) = 44 = 256.
Steinhaus defined the "Mega" as "2 inside a circle". In general, "n inside a circle" is n inside n concentric squares, so "2 inside a circle" is "square(square(2))". We already know that square(2) is 256, so "2 inside a circle" is square(256) or "256". This is 256 inside 256 concentric triangles.
To compute this, we begin with x0=256. Let x1 = x0 to the power of itself. Then let x2 = x1 to the power of itself. Continue until reaching the value x256. Using a Hypercalc BASIC program shown here, it's easy to find that the answer, converted to base 10, is a power-tower with 255 repetitions of the number 10, and 1.9923739...×10619 (which is approximately the logarithm to base 10 of 256256257) at the top.
Later, Moser became involved and generalised the notation to include pentagons, hexagons, etc. The pentagon replaced the old circle, and circles no longer had any meaning. Steinhaus's "Mega" would be represented as "2 inside a pentagon" using the Steinhaus-Moser notation.
995pt6.031226×1019727 ≈ 2↑↑1000
This is 2↑↑1000. There is a "meta-proof" by Harvey Friedman that any proof of the finiteness of TREE[3] using "finite arithmetic" would take at least this many symbols. It is discussed in a video by Numberphile (which is a supplement to this video about TREE[3]).
See also 103.5775080127201×1028.
break_eternity.js is a JavaScript library meeting the needs of incremental games, developed by Patashu since 2019 March [238]. It represents numbers in a level-index representation, using an IEEE double precision number for the "height" of the power tower. This means it can handle a power tower of height 1.797×10308, represented here as "10↑↑(1.797...×10308) using the Knuth up-arrow notation. This is far higher then earlier incremental-oriented libraries like break_break_infinity.js and confractus_numerus.js. However, others go far higher; see 3pt1.0126×101656520 for a discussion.
Beyond
For a discussion of more and larger numbers, particularly those that are so large that their values are difficult to express in any form, proceed to the hyper5 discussion on my large numbers page.
Note. I try to explain things at least a little bit, and to give suitable references. I definitely do not follow my own First Law of Mathematics. If you suggest an improvement for these pages, I'll probably be able to do something to make it better — just let me know (contact links at the bottom of the page).
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Quick index: if you're looking for a specific number, start with whichever of these is closest: 0.065988... 1 1.618033... 3.141592... 4 12 16 21 24 29 39 46 52 64 68 89 107 137.03599... 158 231 256 365 616 714 1024 1729 4181 10080 45360 262144 1969920 73939133 4294967297 5×1011 1018 5.4×1027 1040 5.21...×1078 1.29...×10865 1040000 109152051 101036 101010100 — — footnotes Also, check out my large numbers and integer sequences pages.
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