| Large Numbers |
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Rules for one system extending up to 103000 are given in The Book of Numbers by Conway and Guy. This system was developed by John Conway and Allan Wechsler after significant research into Latin5 but Olivier Miakinen4 has refined it, as described below.
The name is built out of pieces representing powers of 103, 1030 and 10300 as shown by this table:
| 1's | 10's | 100's | |
| 0 | - | - | - |
| 1 | un | (n) deci | (nx) centi |
| 2 | duo | (ms) viginti | (n) ducenti |
| 3 | tre (*) | (ns) triginta | (ns) trecenti |
| 4 | quattuor | (ns) quadraginta | (ns) quadringenti |
| 5 | quin | (ns) quinquaginta | (ns) quingenti |
| 6 | se (sx) | (n) sexaginta | (n) sescenti |
| 7 | septe (mn) | (n) septuaginta | (n) septingenti |
| 8 | octo | (mx) octoginta | (mx) octingenti |
| 9 | nove (mn) | nonaginta | nongenti |
The rules are:
- Take the power of 10 you're naming and subtract 3.
- Divide by 3. If the remainder is 0, 1 or 2, put one, ten or one hundred at the beginning of your name (respectively).
- Break the quotient up into 1's, 10's and 100's. Find the appropriate name segments for each piece in the table. (NOTE: The original Conway-Wechsler system specifies quinqua for 5, not quin.)
- String the segments together, inserting an extra letter if the letter shown in parentheses at the end of one segment match a letter in parentheses at the beginning of the next. For example: septe(mn) + (ms)viginti = septemviginti; se(sx) + (mx)octoginta = sexoctoginta. For the special case of tre, the letter s should be inserted if the following part is marked with either an s or an x.
- If the result ends in a, change the a to i.
- Add llion at the end. You're done.
Many of the resulting names are only slightly different. For example 10261 is sexoctongintillion and 102421 is sexoctingentillion. Then there's 10309 = duocentillion and 10603 = ducentillion.
There are a few minor problems4,5 with the names quinquadecillion, sedecillion, and novendecillion (which are better known as quindecillion, sexdecillion and novemdecillion respectively). Miakinen explains that sedecillion and novendecillion are more true to the "rules of assimilation" in Latin, and thus the Conway-Wechsler version is better. But he also explains that quinquadecillion should be quindecillion because the Latin for 15 is "quindecim", not "quinquadecim", and proposes a similar change to all the Conway-Wechsler names involving the quin prefix; I have adopted his suggestion here.
Also of note is Wechsler's comment about an important omission in Conway's book:
The presentation in The Book of Numbers was designed to leave the impression that the system up to 999 was pre-existing, although in fact we invented a lot of it; you will note that Conway carefully never says the system is ancient. -- Allan Wechsler5
Undoubtedly The Book of Numbers will trip up one or two future etymology researchers, but the inconsistency with quindecillion, sexdecillion and novemdecillion should make the truth obvious.
Other Systems
A set of number names between vigintillion and centillion, differing somewhat from the above, was created by John Knoderer, and are listed on this page; a similar list is found here.
An alternative based on Greek prefixes has been proposed by Russ Rowlett, who suggests using the names tetrillion=1012, pentillion=1015, hexillion=1018, heptillion=1021, oktillion=1024, ennillion=1027, dekillion=1030, and so on.
These systems have plenty of differences, and some ambiguities. Going beyond 103000 produces even more confusion, mainly from the use of a large number of prefixes strung together and having to remember what order they go in. For example, in the system deveoped by Landon Curt Noll (see this page) you can make names for any power of 10 you wish. To give two examples, 1019683 is "one sexmilliaquingentsexagintillion" and 101010 = 1010000000000 is "ten tremilliamilliamilliatrecenttretriginmilliamilliatrecenttretriginmilliatrecentdotrigintillion".
(Sources:
http://home.comcast.net/~igpl/Powers1000.html (complete table up to 10303)
http://home.comcast.net/~igpl/NWL.html (Different version of the same)
http://www.wikipedia.org/wiki/List_of_numbers (Another version that goes up to 10180)
http://www30.brinkster.com/manfear/office/large.html (Also gives names in German and proposes names based on Greek)
http://g42.org/MiscInfo/numbers.html (Much of the same info as on the previous page)
http://www.unc.edu/%7Erowlett/units/large.html (Much of the same info as on the previous page)
http://mathforum.org/library/drmath/view/59155.html (An introduction for kids)
http://www.grammarstation.com/KnowYourMath/numbers_symbols.html (An introduction for kids)
http://www.ex.ac.uk/cimt/dictunit/notesp.htm (Described the SI prefixes for powers of 1024, but now offline)
http://mathworld.wolfram.com/LargeNumber.html
http://home.att.net/~numericana/answer/units.htm (Lots of details about SI prefixes)
http://journals.iranscience.net:800/www.newscientist.com/www.newscientist.com/lastword/article.jsp@id=lw77 (comments on uses of zetta- and yotta-)
http://www.lewrockwell.com/orig/kinsella6.html (introduction to the SI units)
http://www.plexos.com/256_bit_CPUs_should_be_enough.htm (xona)
http://jimvb.home.mindspring.com/unitsystem.htm (xona)
http://www.omniglot.com/writing/latin.htm (Latin alphabet)
)
SI Prefixes: The original SI (Metric system) in 1793 had only the prefixes kilo, hecto, deca, deci, centi, and milli. In 1795 myria was added but later became deprecated (no longer officially allowed). mega dates from the late 1800's and was officially adopted in France in 1919. During the 1900's kilomega and megamega were used but it was eventually decided these needed their own prefixes; giga, nano and tera, pico were adopted in 1960, femto and atto in 1964, peta and exa in 1975, and zetta, zepto and yotta, yocto in 1991. These extensions have been added mainly for the convenience of scientists. For example, in 1993 some researchers had to refer to units of 10-21 volts but they didn't yet know about the prefix zepto so they called it "milliattovolt". There have always been fields where very small or very large values are expressed simply as big exponents of 10, or where a field-specific and somewhat arbitrary unit such as the parsec or electron-volt is used. The standard unit of atomic mass (1/12 the mass of a Carbon atom, or roughly the mass of a proton or a neutron) is 1.66×10-27 kg or 1.66 yg (1.66 yoctograms). Quasars have been discovered that are about 125 yottameters away. When volumes or weights are involved the units are even more often found to be insufficient — the Earth weighs about 6000 Yg (6000 yottagrams), and the mass of an electron is about 0.000 91 yg (yoctograms). By comparison, the diameter of the Earth and of an atom (both one-dimensional measurements) are both easily handled with the older kilo- and pico- prefixes.
All of the "SI prefixes" in italics are unofficial. The following prefixes can be dismissed immediately: hepa, ento, otta, fito, nea, syto, dea, tredo, una and revo; they were all apparently from a popular Internet rumor or hoax.
I have found no clear evidence from the CGPM (General Conference on Weights and Measures) as to the actual etymology of the prefixes Zetta- and Yotta-. However there have been several serious attempts to guess at their etymology and extend the pattern.
Oxford professor Jeff K. Aronson has suggested extending beyond zetta, zepto and yotta, yocto with xenta, xenno, wekta, weko, vendeka, vendeko, udeka, udeko based on the idea that the 'Z' and 'Y' prefixes would continue backwards through the English or modern Latin alphabet. Using a similar idea, Jim Blowers says:
The pattern here is that we go backwards from the beginning of the alphabet [shouldn't this be "end of the alphabet"? -ed], starting with z and y, and we follow it up with an alteration of the Greek or Latin for the next number. According to this pattern, the next ending should be xona-, since x comes before y in the alphabet, and 9 is noni- in Latin. Similarly, 1030 should be weka-, since w precedes x and 10 is deka in Greek.
He goes on to list a large number of prefixes starting with Xona-, Weka-, Vunda-, Uda-, Treda-, Sorta-, ... This is all based on the 26-letter "Latin" alphabet used in several European languages including English, although the original classical Latin language had no J, U or W.
I have also heard speculation that "peta", "exa", "zetta", "yotta" are derived from "pente", "hexa", "sette", "otto" (five and six in Greek, seven and eight in Italian) or similar number names. If that were true, any future SI extension will probably be more like Novetta, novemo, Decetta and decemo.
For completeness I'll also mention that in 1993, as a joke that was reportedly well received on USENET, Morgan Burke proposed harpo for 10-27 and groucho for 10-30 (and therefore harpi for 1027 and grouchi for 1030).
Donald Knuth invented a system that extends much further than the standard Latin-based system.
The word "thousand" is not used, and instead everything up to 9999 is named using the traditional names for numbers up to 99 plus "hundred", and no comma is used. For example:
127 = One hundred twenty-seven
1000 = Ten hundred
1356 = Thirteen hundred fifty-six
3000 = Thirty hundred
4192 = Forty-one hundred ninety-two
104 is called "myriad", a name that originally comes from ancient Egypt. It is written 1,0000 — note that the comma is added to separate the lowest four digits, not three. Numbers up to 9999,9999 are named like so:
1,2345 = One myriad twenty-three hundred forty-five
10,0000 = Ten myriad
26,0044 = Twenty-six myriad forty-four
100,0000 = One hundred myriad
1000,0000 = Ten hundred myriad
1400,2054 = Fourteen hundred myriad twenty hundred fifty-four
4309,8127 = Forty-three hundred nine myriad eighty-one hundred twenty-seven
108 is called "myllion" and is written 1;0000,0000. Notice a new punctuation mark is used to represent "myllion". Numbers up to 9999,9999;9999,9999 are named as in these examples:
9;0000,0000 = Nine myllion
100;0001,0000 = One hundred myllion one myriad
2000;0000,1234 = Twenty hundred myllion twelve hundred thirty four
4,0006;5020,0100 = Four myriad six myllion fifty hundred twenty myriad one hundred
Then 1016 is called "byllion", and a new punctuation mark is used. Notice each punctuation mark can be read exactly when it appears so it's easy to read off these numbers in words:
1844:6744,0737;0955,1616 = Eighteen hundred forty-four byllion sixty-seven hundred forty-four myriad seven hundred thirty-seven myllion nine hundred fifty-five myriad sixteen hundred sixteen
Each new number name is the square of the previous one — therefore, each new name allows us to name numbers with twice as many digits. This gives us a lot more mileage out of each name. Knuth continues borrowing the traditional names changing "illion" to "yllion" on each one. "vigintyllion" ends up being 104194304, a bit beyond the upper limit of class-2 numbers.
Class-3 numbers are those that can be represented inexactly using scientific notation, to within a given percentage of error. Numbers about the size of Googolplex are class-3 numbers, although Googolplex itself can be represented exactly. Class-3 numbers include (almost) all combinatoric enumerations of physical systems (i.e. the number of possible states of a system containing N particles, where N is a class-2 number, see googolplex notes). The limit of class-3 numbers depends on the limit of class-2 numbers and the base, for this discussion let's say that class-3 goes from 10106 to 10101000000.
Class-3 numbers are the largest which can effectively be compared to see if they are of comparable magnitude. For example, the following two numbers are class-3 (and are at the low end, as class-3 numbers go) :
A = 279641170620168673833
B = 350247984153525417450
Which is larger?
We cannot compute the exact values of these two numbers and compare directly — they have way too many digits to store the values on a computer. That is the nature of class-3 numbers. However, we can represent both in scientific notation with 10 digits of accuracy. This is accomplished in much the same way that your computer or a scientific calculator would do it. Starting with the logarithm of 2 (or 3), multiply by the exponent, then divide by the logarithm of 10, separating the integer from the fractional part, and use the fractional part to determine the first few digits of the answer. In this case we get:
A = 5.0760252191 × 1023974381246463762439
B = 5.0760252191 × 1023974381246463762439
Now you begin to see the problem. Using 10 decimal places, both values seem to be the same. (We know they are not, because one is a power of 2 and must be even, and the other, being a power of 3 is odd). As it turns out, you need at least 20 decimal places to see that B is slightly larger.
Now we move on to Class-4 numbers and higher classes. You have probably seen a pattern here, and we'll just continue the pattern:
class-4 numbers are those numbers that are larger than class-3, and whose logarithm can be represented as a class-3 number.
Now we have another problem as before. Which of the following class-4 numbers is larger?
C = 22283
D = 33352
as before we take the logarithm of both but this time we must do it twice, and we find
ln(ln(C)) = ln(ln(2)) + [ln(2) * 9671406556917033397649408]
= 6703708186976009930559261.24579...
ln(ln(D)) = ln(ln(3)) + [ln(3) * 6461081889226673298932241]
= 7098223961595389530659098.10481...
so D is larger.
These numbers occur in the study of prime numbers, and particularly the frequency of occurrence of prime numbers. Gauss' well-known estimate of the number of prime numbers less than N is
For all values of n up to 1022 (which is as far as we've been able to compute so far) Li(n) is an overestimate. Littlewood showed that above some value of n it becomes an underestimate, then at an even higher value of n it becomes an overestimate again and so on. In 1933 Skewes showed that (if the Riemann Hypothesis is true) the first crossing cannot be greater than eee79. This is the first or "Riemann true" Skewes' Number; it is class-4. Converted to base 10, the value is normally approximated as 101010^34; a more accurate approximation is 10108.852142×1033 or 10108852142197543270606106100452735038.55.
Since then, others have improved the estimate dramatically. Conway and Guy (The Book of Numbers, page 61) cite the result of Lehman, who in 1966 gave an upper bound of about 101167. According to Eric W. Weisstein and Wikipedia, in 1987 H. J. J. te Riele reduced the upper bound of the first crossing to ee(27/4), a class 2 number approximately equal to 8.185×10370. In 2000 Bays and Hudson found an actual crossover point using numerical techniques — around 1.39822×10316. Most recently, in 2005 Patrick Demichel found a smaller crossover point near 1.397162914×10316. In any case, the original Skewes' Number is now just an interesting part of history.
In 1955 Skewes also defined an upper bound if the Riemann Hypothesis is false: 1010101000. This is the "second Skewes' Number"; it is much larger, but still class-4.
In a similar way, we can define higher classes:
class-5 numbers are those numbers that are larger than class-4, and whose logarithm can be represented as a class-4 number.
class-N numbers are those numbers that are larger than class N-1, and whose logarithm can be represented as a class N-1 number.
but as it turns out, these higher classes aren't too useful for representing the large numbers of abstract mathematics. Once we get into the really big numbers like the ones discussed below, exponents aren't being used anymore and we've moved on to operations and functions that grow much faster.
Here is a summary of what has been covered so far:
| class | from | to | distinguishing characteristics |
| 0 | 1 | 6 | unconscious awareness; animal brain |
| 1 | 6 | 1,000,000 = 106 | visual acuity; direct familiarity |
| 2 | 106 | 101,000,000 = 10106 | exact representation of integers |
| 3 | 10106 | 10101,000,000 = 1010106 | X indistinguishable from X+1 |
| 4 | 1010106 | 1010101,000,000 = 101010106 | X indistinguishable from X*2 |
| 5 | 101010106 | 101010101,000,000 = 10101010106 | X indistinguishable from X2 |
Uncomputably Larger and Uncomparable
At this point it is useful to define the concept of Uncomputably Larger.
Uncomputably Larger: A is Uncomputably Larger than B if A is larger than B, but it the difference does not show up when the numbers are expressed in the same system of representation, or if it is not possible to figure out the magnitude of the difference using ordinary numerical techniques. A system of representation is any system of digits and/or symbols used to express numbers in a standard form that lends itself well to seeing which is bigger. Standard scientific notation is an example; mathematical formulas in general are not.
If A is a class 3 number and K is class 2 or smaller, it is easy to distinguish A × K from A, but hard to distinguish A + K from A. A + K is uncomputably larger than A.
For an example of this, imagine A has trillions of digits. If you add some small number to it, only the last few digits will change — and all of the digits would have to be stored and examined to tell the difference. On the other hand, multiplying A by a small number N will change all the digits, and you can distinguish the difference by comparing the logarithm of A to that of A×N
If A is a class 4 number and K is class 3 or smaller, it is easy to distinguish AK from A, but hard to distinguish A × K from A. A × K is uncomputably larger than A.
If A is a class 5 number and K is class 4 or smaller, it is easy to distinguish A④K from A, but hard to distinguish AK from A. (④ is the hyper4 operator). AK is uncomputably larger than A.
This pattern does not continue with higher operators, because the "class" system is based on exponents. For example, if A is a class 10 number and K is class 9 or smaller, it is still easy to distinguish A④K from A, and hard to distinguish AK from A.
Uncomparable: A and B are said to be Uncomparable if it is unknown which is larger.
As you go to higher and higher operators and functions it becomes quite difficult to determine which of the large numbers is larger than which others. It is easy to see that Skewes' Number is bigger than googolplex, but not nearly so easy to figure out which of Graham's Number and the Moser is bigger. Graham's Number and the Moser are defined with different systems of representation, and the two systems cannot be readily converted into each other. They are called uncomparable until the two systems are studied and a method is developed to show which number is larger. They have been presented in order here only after much careful study.
A tower of exponents like eee79 mentioned above in the discussion of Skewes' numbers is often called a power tower. Notice how in the examples of class-3 numbers there are three numbers in the power tower, and in the class-4 examples there are 4 numbers in the power tower. But this isn't necessarily always the case, for example 2222} is a power tower with 4 numbers but its value is 65536, a class-1 number.
Problem: Start with the 3-level power tower 2210. Consider two different ways to make it bigger: Increase the bottom-most number, making it H210 where H is something really huge like 1000000, or make the power tower higher by making it S2210, where S is something really small like 1.001. Determine which is biggest: the original power tower X = 2210, or the two altered versions, A = 1000000210 or B = 1.0012210 ?
First we show that A and B are both bigger than X. A>X is obvious. For B it's less obvious. We're comparing:
B = 1.0012210 ⋛ 2210 = X
Convert both to a power of 2, by taking the log base 2 of 1.001, and repeat. We get:
B = (20.001442)2210 ⋛ 2210 = X
B = 2(0.001442 × 2210) ⋛ 2210 = X
log2B = 0.001442 × 2210 ⋛ 210 = log2X
log2B = 2(-9.4377 + 210) ⋛ 210 = log2X
log2(log2B) = -9.4377 + 210 ⋛ 10 = log2(log2X)
log2(log2B) is about 1014.56, much bigger than log2(log2X) which is 10. So B>X.
Now we need to compare A to B. Let's rewrite A as a power of 1.001:
A = (1.001(log1.0011000000))210 = 1.001(log1.0011000000) × (210)
Then it is a question of which of these is larger: A' = log1.0011000000 × 210 or B' = 2210. Substituting 210 = 1024, we're comparing A' = (log1.0011000000)×1024 to B' = 2^1024, so A'/B' = (log1.0011000000)×1024 / 21024. Cancelling powers of 2, we remove 1024 from the numerator and reduce the denominator to 21014: A'/B' = log1.0011000000 / 2^1014. log1.0011000000 is pretty large (it's a little over 13822) but 2^1014 is much much larger! So B' is larger, and therefore B is the biggest of our three power towers.
We could have used any really big number H in place of 1000000 and any small number S in place of 1.001 and B would still be the biggest, as long as logSH is less than 21014. 21014 is about 1.7556×10305, a class 2 number. To show how extreme this is, let H be a googol and S be 1+1/googol. logSH would be ln(10)×100×googol = 2.3026×10102, still much less than 21014. So even with this really huge H and really small S, the power tower S2210 is still bigger than H210.
In general, if you have a tower of exponents and you want to make it larger, you'll make it much larger by adding another exponent to the bottom than by increasing the size of the bottom exponent, as long as the tower of exponents has something fairly big at the top and the numbers involved are all class 1 or on the lower end of class 2. This leads to the (somewhat nonintuitive) result that if you're comparing two towers of exponents, you can look at how many exponents are in the tower and know right away which is larger. For example,
1.11.11.11000
is much larger than
100010001000
and
1.11.11.11.11.11.11.11000
is much larger than
1000100010001000100010001000
If you're interested in trying some of these out, my Perl-based calculator program can handle everything discussed so far, up to power towers thousands of numbers high.
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