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The largest finite number indirectly referred to in any published music (as far as I know). My Hero, Zero, the Schoolhouse Rock song about how the digit '0' is used to multiply any number by powers of 10, includes the lines:
Place a zero after one,
and you've got yourself a ten --
see how important that is!
When you run out of digits
you can start all over again --
see how convenient that is!
That's why with only ten digits, including zero,
you can count as high you could ever go --
forever, towards infinity.
No-one ever gets there, but you could try ...
with ten billion zeros.
It doesn't exactly say what is being done with those "ten billion zeros" (1010), but the picture on-screen during the lines "forever, towards infinity / no-one ever gets there, but you could try" shows a pyramid made up of the numbers 9, 80, 700, 6000, 50000, and so on — the screen ends up filled with small zeros — so I imagine they were implying the idea of writing some (nonzero) digit(s) followed by 10,000,000,000 zeros in a row — and then you'd get at least 101010.
See also 525600, 8675309, 1010, 1011 and 1010100.
5.1843×1022652507173 = 275250000000
An upper limit of the volume of the universe, if one makes the following assumptions: 1) Not all of the universe can be observed directly (because of hyperinflation), 2) Life originated purely by the chance meeting of particles to form a single original cell, and 3) That event has happened only once, and all extant life is the result of it. A size of about 1022000000000 is the size necessary to guarantee that each possible chance meeting of 75250000000 particles has occurred somewhere at least once; an additional factor of about 1050 has to be added to ensure this happens in a hospitable environment (a habitable planet).
This number is based on the possibility of a living cell forming through a thermodynamic coincidence. Complex structures can also appear spontaneously through a quantum-mechanical event called a de Sitter fluctuation, and the possibility of such events is important in arguments such as [100] and [106] that attempt to narrow down the possibilities for how the universe might have begun. de Sitter fluctuations can happen either in "normal" universes like our own, or in vast "empty" universes that are predicted by various string-theory models of the beginning of the universe. The important difference is that in an "empty" universe, any spontaneously-appearing life has no chance of continuing to survive, whereas in a normal universe such life can survive if it happens to occur on a habitable planet. This makes it possible to prove that the theories that predict huge amounts of empty space are unlikely to reflect how our own universe originated. See 101010122.
This G43, the first element in the Göbel sequence Gn that is not an integer, where Gn is given by:
G0 = G1 = 1
Gn+1 = 1/n × (SUMi=0..nGi2) (for n>1)
the sequence starts: 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, 267593772160, 7160642690122633501504, ... (Sloane's A003504). The "1/n" in the formula makes it look like there should be fractional terms, but the sequence doesn't actually have any fractional terms until the 43rd term.
This number is given by [106] (in section 7) as an estimate of the number of possible information "configurations" of a human brain (which is not quite the same as this). This presents a limit on the number of possible universes that can be perceived by human observers contained within them, which can present a limit to certain anthropic explanations of the origin of the universe and to the interpretation of "multiple-universe" implications of cosmic inflation models.90
See also 1040, 10500, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.
1080000000000000000 = 108×1016
The highest value defined within the counting system set out by Archimedes in "The Sand Reckoner". See here and here. See also 1063.
101000000000000000000 = 101018
A very rough estimate of the number of possible life-experiences a person can have (which is not quite the same as this). This is based on a sensory bandwidth of 1010 bits per second.
106000000000000000000 = 106×1018
In 2003 Y. Cheng showed that there is a prime between every pair of consecutive cubes N3 and (N+1)3 for all values of N less than 102000000000000000000 (or N3 less than 106000000000000000000). Proving this for all integers seems like it ought to be easy, but it isn't. See also 1.3063778838.
The largest number that can be formed from the digits 1, 2, 3 and 4 using the ordinary functions addition, multiplication and/or exponents. See also 163, 10460353203, 4.281248×10369693099, 108.0723047260281×10153, 10(2.62086×106989) and 6 PT 1.2826×1082.
103.0056206947796095239×1029 = (27!)!
Very large factorials like this one can be computed with Stirling's series, a more accurate form of the better-known "Stirling's formula". The series gives a value for the logarithm of the Gamma function.
The Gamma function
The Gamma function comes up in lots of different places in mathematics, and is defined in terms of an integral10. For positive integers, the value of the Gamma function is equal to the factorial of the integer plus 1.
The Gamma function can be computed by the following series (which gives its logarithm)
ln gamma(z) = 1/2 ln(2 π) + (z + 1/2) ln z - z
+ SUMn=1...inf [ B2n / (2 n (2n-1) z^(2n-1) ) ]
= 1/2 ln(2 π)
+ (z + 1/2) ln z
- z
+ 1/(12 z)
- 1/(360 z^3)
+ 1/(1260 z^5)
- ...
where B2n is a Bernoulli number.
The Barnes' G-function has a similar relationship to the superfactorials, as does the K-function to the hyperfactorials.
The approximate odds against a person living at least 1000 years, as given by life insurance tables quoted by William Feller, in "Probability Theory and its Applications". (The tables don't actually go up that far; they simply give an extrapolation formula for ages above a certain point.)
1037218383881977644441306597687849648128 = 107×2122 ≅ 103.7218×1037
This number is described in the Mahayana Buddhist scripture Buddha-avatamsaka-nama-vaipulya-sutra (Flower Garland Sutra of Great Universal Buddha, in book 30, "the Incalculable"), which dates from about 420 CE. In Japanese its name is pronounced hukasetsuhukasetsuten (ふかせつふかせつてん). 55,56,57.
See also 10421.
5.45431...×1051217599719369681875006054625051616349 ≅ 10(5.1217599719369×1037) = 2170141183460469231731687303715884105727-1 = 2(2127-1)-1
This is C5 in Catalan's sequence and conjectured to be prime. It's a little too big to test.
2.604233075698...×10634704607339355474571695927232512278791 ≅ 10(6.3470460733936×1038) = 272727
This is 272727 calculated to 50 significant digits with Hypercalc. It has over 1038 digits, which is enough to pretty much guarantee that we will never find out, for example, whether its digits include a run of 40 consecutive 0's. Nevertheless, it is quite easy to figure out its first and last digits. The initial digits are found using logarithms: The logarithm to base 10 of 272727 is log1027×2727, quite easily calculated to 50 decimal places as 634704607339355474571695927232512278791.41567985046... The integer part (to the left of the decimal point) tells us what power of 10 it has, and the fractional part (.41567985046...) tells us that the first few digits are 26042...
Perhaps more surprising, the last digits can be calculated by "modulo arithmetic". Modulo arithmetic exploits repeating patterns such as the alternating 125/625 in successive powers of 5. Modulo arithmetic shows that the last five digits of 272727 are 03683: 272727 mod 100000 = 27(2727 mod 5000) mod 100000 = 272803 mod 100000 = 03683.
By extending this method recursively (by the method described here) it can be shown that 2727 ends in 9892803, 272727 ends in 0403683, 27272727 ends in 7450083, 2727272727 ends in 1242083, 272727272727 ends in 7002083, 27272727272727 ends in 9802083, and all higher power towers of 27's end in 3802083. Each time you add another 27 to the power tower, another final digit becomes constant.
Also, because 27 is a factor of 999 we know that if we add the digits of 272727 in groups of 3 the result will also be a multiple of 27.
Did I mention that I like the number 27?
An estimate of the number of possible chess games, given by G. H. Hardy ("Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work", 1999). See also 1.15x1042 and 10120.
101.877...×1054 = (e1037)(1.37×1010)
In [101] (page 12), cosmologist Alan Guth suggests that "each second the number of universes that exist is multiplied" by approximately e1037. If this has been happening for the entire 13.7-billion-year history of our universe, then there are over 101054 other universes out there (or more, if the process began before our universe was created). Such a calculation does not actually have much meaning, because of general relativity, the passage of time in one universe is not comparable to the passage of time in the false vacuum that generates all these hypothesized universes. Nevertheless, it shows that current inflationary cosmology provides for a possibility similar to the "alternate universe count" I describe here. See also 101016, 101077, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.
In section 2 of [106] the authors imagine that our universe came about after 60 "e-folds of the slow-roll inflation", and give this as a rough estimate of the number of "universes with different geometrical properties" which will have been created in such a process. The general form of this number is ec e3N, where c is a constant substantially smaller than eN (not the speed of light and N is the number of e-folds. In this case we have ec e3×60 ≅ 101077. Because e3N is so large c can be ignored, and because N is at best a rough guess, it doesn't even matter that ee180 is actually closer to 106.4683×1077. 90
See also 1040, 10500, 101016, 101.877×1054, 101082, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.
Another estimate of the total number of different universes in the multiverse, given in section 4 of [106]. 90
See also 1040, 10500, 101016, 101.877×1054, 101077, 1010166, 103.67×10281, 1010375, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.
Main article: Googol and Googolplex
Googolplex, for many people is the largest number with a name. Credit for the invention of the -plex suffix is indeterminate. See also 101010100.
googolplex plus one. This number is known to not be prime, with the smallest known factor being 316912650057057350374175801344000001 = 210456+1. Several other larger prime factors are known. Factors of many numbers of the form googolplex+n for small n are listed here.
Results like these are found using modulo arithmetic, similar to my description of how to find the last few digits of 272727. See [Crandall 1997]. See also 4.57936×10917
Factorial of a googol. Notice how this appears to be only "a little larger" than googolplex.
10(3.4677786443...×10130) = 2786!
This is an example of a calculation that can be performed easily either with Maxima5 or Hypercalc. Both require a special command or syntax to get a full precision exponent. In Maxima:
In Hypercalc:
Maxima, which is based on the original MIT MACSYMA, performs many of the same functions as the commercial programs Maple, Mathematica and MATLAB, but is free and open-source. It can do exact integer calculations up to about 101000000 and floating-point up to about 10101000000.
108.0723047260281×10153 = 4444
The highest value achievable in the "four 4's" puzzle. (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.
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" if 10-24 seconds).39 See also 101.877×1054, 103.67×10281 and 105.5×10405.
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.
An estimate of the number of distinct universes in the "string theory landscape", given by [106] in section 5, and assuming that the "maximal number of observable e-folds" is about 290. 90
See also 1040, 10500, 101016, 101.877×1054, 101077, 101082, 1010166, 103.67×10281, 105.5×10405, 101010000000, 101010122, and 10101.51×103883775501690.
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 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 recurrance of "our" universe. See also 101.877×1054, 1010166, 103.67×10281 and 10101.51×103883775501690.
This is (10666)!, called the leviathan number. The word leviathan refers to a whale or sea-monster. Its Biblical references are all in the Old Testament, although 666 is more commonly associated with the last book of the New Testament.
102.62230310839×101322 = 224387×203×1024388
The number of lines of text one Adam Clarkson will owe his high school chemistry teacher on September the 22nd 2010. 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".
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.
[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 of September was 203 days after the 26th of February, the lines had been doubled 203 times — on that day the assignment was 100×2203 ≅ 1.2855504354...×1063 lines, a little over one vigintillion, and due the next day. (This quantity is somewhat larger than the capacity of the known universe, even if all the galaxies were converted into paper and ink.) Each day after that the number is squared, so the assignment on the 18th of September was (100×2203)2 = 1002×22×203, about 1.65×10126. The day after that it was 1004×24×203, or about 2.73×10252, and so forth:
| on day: | the assignment was: | due on: | on day: | the assignment was/will be: | due on: | ||
| 19980226 | 100 | 19980227 | 19980921 | 224×203×1025 | 19980922 | ||
| 19980227 | 200 | 19980228 | 19990921 | 224+365×203×1025+365 | 19990922 | ||
| 19980228 | 4×100 | 19980301 | 20000921 | 22369+366×203×102370+366 | 20000922 | ||
| 19980301 | 23×100 | 19980302 | 20010921 | 22735+365×203×102736+365 | 20010922 | ||
| 19980302 | 24×100 | 19980303 | 20020921 | 221100+365×203×1021101+365 | 20020922 | ||
| 19980401 | 23+31×100 | 19980402 | 20030921 | 221465+365×203×1021466+365 | 20030922 | ||
| 19980501 | 234+30×100 | 19980502 | 20040921 | 221830+366×203×1021831+366 | 20040922 | ||
| 19980601 | 264+31×100 | 19980602 | 20050921 | 222196+365×203×1022197+365 | 20050922 | ||
| 19980701 | 295+30×100 | 19980702 | 20060921 | 222561+365×203×1022562+365 | 20060922 | ||
| 19980801 | 2125+31×100 | 19980802 | 20070921 | 222926+365×203×1022927+365 | 20070922 | ||
| 19980901 | 2156+31×100 | 19980902 | 20080921 | 223291+366×203×1023292+366 | 20080922 | ||
| 19980917 | 2187+16×100 = 2203×100 | 19980918 | 20090921 | 223657+365×203×1023658+365 | 20090922 | ||
| 19980918 | 2406×104 | 19980919 | 20100921 | 224022+365×203×1024023+365 | 20100922 | ||
| 19980919 | 24×203×1023 | 19980920 | 20110921 | 224387+365×203×1024388+365 | 20110922 | ||
| 19980920 | 223×203×1024 | 19980921 | 20120921 | 224752+366×203×1024753+366 | 20120922
|
So, on the 22nd September 2010 the number of lines due will be 224387×203×1024388, which is about 10(3.48926916371×101212).
The length of the assignment, as the author notes17, is "somewhat larger" than the just-mentioned count of the number of ways the universe's history could be shuffled. It is so large that if you tried to write the number of lines as a normal decimal number (that is, without using scientific notation) you'd be writing a number over 101322 digits long, a feat which could not be accomplished even if you could fit a googol digits on each particle in the observable universe.
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.
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