This page covers all the huge numbers I have seen discussed in books
and web pages, and it actually does so in numerical order
(as near as I can tell!).
One important thing to notice is that all discussions like this
ultimately lead to issues from the
theory of algorithms and computation. Appropriately
enough, this very page ends with Turing machines just before
crossing over to the transfinite numbers. If you want to
learn something about the theory of algorithms and computation, get
two or more fairly knowledgeable people to compete at describing the
highest finite number they can describe, and then stand back!
Classes
First of all, I'm going to define what I call "classes" of numbers.
This is a somewhat refined and more precise version of the "levels of
perceptual realities" presented by Douglas Hofstadter in his 1982
article "On Number Numbness" (Scientific American, May 1982,
reprinted in Hofstadter's 1985 book Metamagical Themas). It is a
powerful and basic concept but usually goes unsaid. I think you'll
agree that the classes make sense and are a useful way to distinguish
numbers. Almost all numbers that are easy to make simple statements
about (such as which of two numbers is larger) can be
put into the class system.
All numbers that anyone ever has to deal with in any practical
application (unless you count abstract mathematics and nerdy
one-upmanship contests as practical :-) are members of one of the
first four classes. Googol and
googolplex are examples from class 2 and class
3, respectively.
Class-0 Numbers
Class-0 numbers are those that are small enough to have an immediate
intuitive or perceptual impact. Perceiving such a number is called
subitizing. (After Kaufman, et. al., 1949: The Discrimination of
visual number; a work that is widely quoted, notably by George
Miller in his 1956 paper The Magical Number Seven Plus or Minus
Two: Some Limits on Our Capacity for Processing Information). I'll
be a bit conservative here and place the limit at six. So, the numbers
0 through 6 are class 0.
Experiments with animals, when sufficiently well set up and conducted,
show that animals are able to identify numbers of objects and exhibit
different behavior based on whether the number of objects is equal to
some specific value — for example, pressing a lever only when five
objects are present. Such experiments also show that the animal's
ability to perform the feat falls off sharply between 4 and 8: the
task can almost always be performed reliably when the number is 4, and
can seldom be performed reliably when the number is 8 (with
intermediate results in-between).
It is a widespread belief (perhaps myth) that there are/were some
primitive tribes which distinguished the concept of number but
couldn't count any higher than three or some other small number. (A
tribe called the Hottentots was said to have only four words for
numbers, "one" "two" "three" and "many" — but not having a word for
it is different from not being able to distinguish it). Whether or not
this is true, it reflects the basic truth of the fact that the human
mind requires some additional abstraction or understanding to go
beyond the first few or several small numbers.
One way to see this phenomenon for yourself is to use flash cards (or a
computer program set up to simulate flash cards) that present pictures
of objects that can be counted and placed in random arrangements —
but look at the picture only long enough to see it, and not long
enough to start counting. Then, after the picture is hidden, ask how
many objects there were. You then try to count the number of objects
in your mental image of the picture you've just seen. If the number of
objects is a class 0 number, you'll usually be able to give the right
answer. As you increase the numbers of objects, your counts will be
less and less likely to be correct. Obviously, this gives a rather
fuzzy definition of "class 0", but the value you get will almost
always be "around" 6.
Class-1 Numbers
Class-1 numbers are those that are small enough to be perceived as a
bunch of objects seen directly by the human eye. What I mean by "seen
directly" is that it is possible to see the number as a set of
separate, distinct objects in a single scene (no time limit, but the
observer and the objects cannot move). 100 is a
class-1 number because it is possible to see 100 objects (goats for
example) in a single scene. The limit for class-1 numbers is around a
million, 1,000,000 or 106. You can just barely
put 1,000,000 dots on a large piece of paper and stand at a distance
such that you can perceive each individual dot as a distinct dot, and
at the same time be within viewing distance of the other 999,999 dots.
(I have actually done this, just for fun!) As with Class-0 the
definition is fuzzy, some people have better vision and could manage
10,000,000 dots or even more.
The earliest conscious communication of numbers between humans was
probably limited to class-0 and very low class-1 numbers, because of
simple physical methods of counting (like fingers and toes). The first
written number systems consisted of tally marks and extended into the
class-1 range.
Class-1 numbers include all quantities that people can comfortably
handle or perceive. For values in class 1, it is easy to distinguish
the magnitude of the value just by looking at it. Most people have
realized that, if they walk into a room with 85 people, although they
can't tell it's exactly 85, they know right away it's somewhere around
75 to 100. No thought or calculation is necessary. This is an
immediate perception of magnitude, and the ability extends to numbers
up into the thousands and tens of thousands, but drops off after that.
If you're in a stadium with 10,000 people, your magnitude perception
will be fuzzier (you'll probably guess from 3,000 to 30,000). By the
time you get to numbers like 108 (the number of blades of grass in
an acre) you'd probably be just as likely to say "10 million" (107)
as "a trillion" (1012) unless you take the time to do some
calculations.
Class-1 numbers also include most types of things that people
aggregate or count with the passage of time. If you have kept count of
how many times you have done something (e.g. jogging) or the number of
things in a collection (e.g. stamps) it probably numbers in the class
1 range. The actual act of counting usually wears out before exceeding
class 1, partly because of the difficulty of accurately remembering
the digits. (While counting the number of days you have jogged is
fairly easy, most people would not be able to persist in keeping count
of how many steps they had taken once that number gets into 6 or 7
digits!) When I was a child, I was sufficiently obsessed with numbers
and bored while waiting in the lunch line that I began counting while
waiting in line. My count continued from one day to the next, always
picking up where I had left off. I got up to around 35,000 after a few
months, at which point I had lost my place so many times I decided it
was not worth the effort.
Symbolic representations of numbers soon became common. The earlier
systems were just tally-marks with lots of different symbols, like one
symbol to represent 1's and another to represent 10's, etc. Roman
numerals are the last surviving example of this. Often, different
types of physical objects (like round and flat stones) were used for
counting. With symbolic systems it became easy for people to express,
write, and do arithmetic with numbers throughout the class-1 range.
Such representation systems usually reached their limit right around
1,000,000 for the same reasons that class-0 perceptive abilities are
limited to 6: it is difficult to keep track of lots of different types
of symbols/objects at once, and 5 or 6 types of symbols/objects is a
practical limit.
Class-2 Numbers
Class-2 numbers are those that can be represented in exact form using
decimal place-value notation (or another small integer base, e.g. base
2, 16 or 60).
Place-value notation was popularized in the Arabic culture (but came
from India, and perhaps from China before that). It opened up the
range of class-2 numbers to anyone who wanted to use them. It was no
longer necessary to come up with new symbols for each successive power
of 10. Generalizations in arithmetic rules were obvious: adding
2000+7000 was not only analogous to adding 2+7, it was essentially the
same thing. Handling huge numbers became easy.
Googol is a class-2 number, as are the various large
prime numbers used in cryptography, all of the known
Perfect numbers, the
Fermat numbers with known factorization, etc.
All of the large physical constants like
6.02×1023 (Avogadro's number) and 1080
(the number of protons in the universe) are class-2. So are most of
the numbers with names ending in -illion, like
vigintillion (1063),
centillion (10303), and on up to the
somewhat contrived
milli-millillion (103000003) (which,
I have arbitrarily decided, is actually a bit beyond the class-2
range).
As with class-0 and class-1, the limit for class-2 numbers depends on
where and how they are recorded, which in turn depends on what you
want to do with the number. If you need to perform a very highly
repeated iteration on a number, such as factoring an N-digit integer
or evaluating a definite integral to N-digit accuracy, the limit is
quite small — perhaps a few hundred digits. Such operations are
essentially time-limited. Other operations, like a single or a few
multiplications, are space-limited and can work with much bigger
numbers as long as it fits in the computer's main memory. This limits
the size to something like 10 million digits. (For examples of these
types of calculations,
look here). If
you just want to store the exact value of a number and not do anything
with it, you can keep it on a tape or disk, which has much more
capacity — perhaps as much as 1011 digits.
So, the limit for class 2 could be anywhere from 103 to 1011
digits, depending on the desired operation. We'll just continue the
pattern and say that class 2 ends at 1 million digits, i.e. numbers up
to 101000000 or 10106.
Almost any number that has a proven property (such as being prime) is
class 2, because it must be represented in exact form in order to test
for the property. For larger numbers such properties can only be
proven for special cases (for example, it's easy to determine if
1012345678012345678-1 is prime, but not
1012345678012345678+1).
Big Number Names
The word million comes from around 12702, and entered the
English language around 13707. The names
billion, trillion, ...
nonillion first appear in the late 15th century, in writing by
Nicolas Chuquet, a French mathematician living in Lyons from 1480
until his death in 1488. (There were also the longer forms bymillion
and trimillion used as early as 1475 by Jehan Adam, but these never
caught on).
The following quotation has been refuted by none other than Chuquet's
great-nephew Michael Chuquet. I mention it here just to make its
contested status clear: Au lieu de dire mille milliers, on dira
million, au lieu de dire mille millions, on dira byllion, etc..., et
tryllion, quadrilion ... octylion, nonyllion, et ainsi des autres si
plus oultre on voulait proceder. (Now believed to originate not
from Chuquet but from a later writer, this passage means, "Instead of
saying one thousand thousand, one may say million; instead of saying
one thousand million, one may say billion, — and trillion,
quadrillion, ... octillion, nonillion, and others as far beyond as you
wish to go.")
Regardless of its origin, this passage proposes the system now
familiar to Americans (which will hereafter be called the 109
system). This system was probably put forward by someone who saw
Chuquet's number-names but did not understand (or agree) that they
should express powers of 106.
Now let's see the real Chuquet quote, from his Triparty en la
Science des Nombres dated 1484, which presents the original "powers
of a million" system (hereafter called the 1012 system)
familiar to many in Europe:
original words of Chuquet apparently transcribed Source:
http://www-history.mcs.st-andrews.ac.uk/history/Bookpages/Chuquet4.gif [...]
pr[oc]eder. Item lon doit savoir que ung million vault mille
milliers de unitez, et ung byllion vault mille milliers de millions,
et [ung] tryllion vault mille milliers de byllions, et ung quadrillion
vault mille milliers de tryllions et ainsi des aultres. Et de ce en
est pose ung exemple : nombre divise et punctoye ainsi que devant est
dit, tout lequel nombre monte 745324 tryllions 804300 byllions 700023
millions 654321. Exemple : 745324,8043000,700023,654321.
French: [...]
to go. Item: one should know that a million
is worth a thousand thousand units, and a byllion is worth a thousand
thousand millions, and tryillion is worth a thousand thousand
byllions, and a quadrillion is worth a thousand thousand tryllions,
and so on for the others. And an example of this follows, a number
divided up and punctuated as previously described, the whole number
being 745324 trillion 804300 billion 700023 million 654321.
Example : 745324,8043000,700023,654321.
As you can see, Chuquet intended the names to represent powers of
1000000. It is also clear that he was using prefixes that came from
Latin, and to the more scientific mind it is more logical to have
billion mean 10000002, trillion be 10000003 and so on. In
his example number there is an extra 0 that does not belong, 8043000
should be 804300. I suspect that the text pictured is a copy and that
the error was introduced during transcription.
Triparty en la Science des Nombres was copied and published by
Estienne de La Roche in 1520, without attribution to Chuquet. 1520 is
the date given by French dictionaries for the origin of the word
billion9, but trillion is properly dated as coming from 1484.
In 1549 Jacques Peletier repeated the suggestion that billion should
be one million million = 1012, and trillion for 1018 and so
on. He also introduced1,2 the use of milliart, billiart
and so on to represent the skipped-over powers of 1000, like 109
and 1015.
However, in the 17th century, French mathematicians decided to
switch to the 109 system because they found it easier.2
The misquote above might be from around this time.
These number names were adopted throughout Europe during the next
century (with minor spelling changes for each language), and used for
both systems causing ambiguity when texts were translated. The 1012
system was adopted in England, Germany, Spain, Scandanavia, and
eastern Europe except Russia8. During the 19th century the
French usage of the 109 system gained influence in the United
States. As the world became larger and nations more interdependent,
the ambiguity became an ever-greater problem, particularly when large
amounts of money were being discussed.
Chuquet's manuscript was discovered by Aristide Marre in the late
1870s and published in 18802,6. In France, the 1948 General
Conference on Weights and Measures deprecated the 109-system senses
of the words billion through sextillion in favor of the 1012
system and this suggestion became official in 1961 throughout the
French-speaking world (septillion and the higher number names were
never part of the French language). France had finally2 switched
back to the original version of the number names she had created. But
the influence of the 109 system, primarily from the United States,
was so great that by 1974, the Prime Minister of the U.K. announced
that the 109 system was to be used for all official communications,
effectively completing a transition that had long been taking place
among the general speaking population.
Here are the big number names, using the 109 system. I have
included examples of longer and more complex ones taken from several
different sources; further discussion of these follows the table:
Chuquet left it to others to work out the details of extending the
names beyond nonillion. An obvious start is to continue using the
Latin number names as prefixes, which goes smoothly until you get to
vigintillion. There is no good Latin prefix for twenty-one. Just
as in English, starting with 21 the name for the number goes to two
words, with the smaller part as the second word. Those who have
entended Chuquet names beyond vigintillion usually go to something
like unvigintillion, breaking away from the Latin but keeping the
similarity to undecillion. As you might expect, it gets messier when
you reach the name for 10366, which requires adapting the Latin
name for 121, the first number in Latin with a three-word name.
