2.1.9
Number Ranges
' We're talking numbers so huge that "giant" , "super" , and "catastrophic" do them no justice, we would need new words like "gongulacious" or "kungulacious" ' -- Jonathan Bowers
Go Back Home
Return to Section 2 - 1
PREV>> 2.1.8 - Larger Numbers in probability, Statistics, and Combinatorics
CLASSIFICATION
One problem with the study of large numbers is that it is difficult to convey to the ordinary person just how large these numbers are. These numbers are described in many ways as "large numbers" , "big numbers", "very large" , "very big" , "gigantic" , "super gigantic", to name just a few. This can be confusing, especially when one is using such words as search terms. Sometimes I here other authors using "large numbers", and I discover that they are discussing a much smaller set than I'm expecting. The problem is that in large numbers there is no such thing as too large. So if we have "large numbers", there must also be "very large numbers", but beyond this must lie " very very large numbers". Finally, no amount of very's properly conveys their size. This shows just how limited our language really is for describing the size of these numbers. In fact Jonathan Bowers makes the point that the these numbers are so big that we need to invent new terminology, and invents "gongulacious" and "kungulacious" on the spot in quasi-seriousness.
Obviously if people want to be clear on what "large numbers" they are talking about ( can it be expressed as a simple exponent, does it require repeated exponents, etc.) then we need some kind of classification system. Unfortunately there is no widely accepted or well-known classification. However I have my own provisional classification system...
The Standard Hierarchy
In section I , I occasionally mention the "counting range" but I don't really explain what this is related to. It is part of a classification system that breaks up the counting numbers (ie. 1,2,3, ...) into "ranges" or "classes". I use the terms interchangably here.
The counting range or counting class, is simply the first class of numbers I deal with. Specifically they are the smallest of the counting numbers, so they include all the initial ones like 1,2,3,4,5, etc. But to make this range useful it is extended to the limits of counting. As we learned, human beings are probably limited to counting no higher than about a few hundred million in a life time, and there seems to be a fairly improbable limit of about a billion. Beyond this , humans have no chance of reaching such numbers by mere counting. For this reason the counting range ends somewhere around here. However my ranges are not so percise, and there is some degree of flexability in their definitions. The limit of the counting range varies from as little as a million (10^6) to usually no higher than ten billion (10^10). The reason is that I want to give the range some space, without treading too far into the next class of numbers...
The next class of numbers we encountered in the previous articles in this chapter. I call these numbers the "exponential class numbers" which are part of the "exponential range". These numbers are very common in science. The exponential range isn't really related to science though, its related to the operation of exponentiation. Once we get past the first few orders of magnitude, we start dealing with numbers that only result when studying "exponential growth", and "large scale to small scale comparisons". These are the numbers that most people traditionally have and still do think of as "large numbers". Alot of the time when people , ordinary or otherwise, speak of large numbers , they are actually talking about numbers in this range. For example, a google search for "large numbers" will bring up alot of things like "large number of reports of such and such " which have nothing to do with the abstractions of large numbers, but are rather talking about actual statistical data. Practical applications outway the purely abstract ones.
The Exponential range, as I usually define it, starts somewhere around 10^10, and goes all the way up to about 10^1,000,000. That may seem very large for an exponential number, but in light of the next range it's actually quite tame. Most of the exponential class numbers that arrise however never get much past 10^303 though, as we saw in the earlier articles. But again, these ranges are created from a mathematical point not a purely scientific one.
These numbers seem huge, until one confronts the next class of numbers ...
The third class is called the "hyper exponential class" which is part of the "hyper exponential range".
I usually define hyper exponential numbers in this way ...
" A hyper exponential number is a number that when expressed as a power of 10 has an exponent which itself is a exponential class number "
So by this reasoning a number like 10^1,000,000,000,000 would definitely be hyper-exponential because it's exponent is a trillion which is a low level exponential class number. In actuality I start the hyper-exponentials after 10^(10^6) even though technically values between 10^6 and 10^10 are ambiguous.
Hyper-exponentials occur alot when considering all the possible configurations of large complex systems. These numbers are therefore not quite as remote as most people think.
The hyper-exponentials usually extend all the way until the exponent itself is a hyper-exponential number. Thus the range ends at about 10^(10^1,000,000).
At this point the pattern should be apparent. We can continue to create new ranges by considering numbers whose exponents are the previous class.
The next class we can call "hyper hyper exponentials" (for lack of a better name) and these numbers can be defined as having hyper exponential exponents. Thus their range would be from 10^(10^1,000,000) to 10^(10^(10^1,000,000))
Then would be hyper hyper hyper exponentials, hyper hyper hyper hyper exponentials , etc.
I have also considered shortening them by using the replication words. For example "hyper hyper exponential" can simply be "double-hyper exponential", "hyper hyper hyper exponential" can be "triple-hyper exponential", etc.
Most of these terms I never use. As the numbers get larger and larger these classes become less and less distinguishable. In any case, this conventional system of ranges we can call the "standard hierarchy".
Actually I'm not the first person to implement such a system. Robert Munafo uses a very similiar system for classifying numbers . He Talks about it in the introduction of his large number discussion [1] . He says that his system 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). "
Munafo uses various "classes" to seperate the numbers. First are the Class-0 numbers. As he describes it Class-0 numbers are those "that are small enough to have an immediate intuitive or perceptual impact" . This is very closely related to the concept of "number sense" that I began the first article with. Munafo says that class-0 numbers go from 0 through 6 ( although I myself would hesitate to add zero in my classification system because 0 is not, strictly speaking, a counting number).
Next comes class-1 numbers. He says that these are numbers which are small enough to be percieved directly by the human eye. He puts an upper limit of about 1,000,000 for class-1 numbers. In otherwords the range is from 6 to 10^6
Class-2 numbers are those that can be represented exactly in decimal notation. Again he puts the upper limit at about 10^1,000,000.
After this he generalizes the concept and basically defines the limits of class-N as 10 to the power of the largest class-(N-1) number. And these match up perfectly with my own ranges.
The Counting range cooresponds to Class-0 and Class-1 numbers. The exponential range cooresponds to Class-2 numbers. Hyper exponential numbers are class-3, and so on.
At first this classification system seems quite handy, but eventually it becomes useless when the numbers get very large. In fact we can prove this.
If we can imagine a sequence of classes, class-1 , class-2 , class-3, then we can extrapolate this and jump to class-100 , class-1000, or any large counting number N. Now define a large number in class-N and call it M, we can now speak of class-M numbers. Then define a large number in class-M and repeat the process. Eventually it will not be possible to distinguish between the number and the class that it falls under, and thus the classification will become effectively useless.
Actually this proof doesn't just show that my classification is flawed. It shows that any prefabricated hierarchy of classes will eventually become effectively useless. In otherwords, if we create a completed class system, where there are potentially an indefinite number of classes which are well defined, the system will be inadequate for expressing certain very large numbers.
Any system which does not fall under this must therefore always be provisional. In actuality it is not possible to have an all purpose classification system. Each system is useful only for a certain range of values. The best stratedgy is to construct your system as you go along, and avoid trying to anticipate the next class, because this would be tantamount to attempting to add sequentiality to your system.
There is also a broader classification that I use. As mentioned in the previous article I also speak about the corporeal numbers. The corporeal numbers are those loosely defined to be real by virtue of the fact that they have some tangible relation to known reality.
counting class, exponential, and hyper exponential numbers are all within the corporeal range. Usually I consider the corporeal range as ending at Promaxima ( 10^(10^343) ). But perhaps one can argue that if Poincare recurrence times represent something in reality, that numbers even larger than this are corporeal. Even so, even the largest numbers in science do not yet venture far beyond triple-hyper exponential numbers ( Class-5 numbers in Munafo's system). Beyond this then , numbers don't have much relation to reality. They shed their physical shell and become what they were all along, pure abstractions. These "pure abstractions" I call "ethereal numbers". "Ethereal" here is a blanket term and doesn't refer to a specific range. There is no "largest" ethereal number. Instead it merely covers all existing numbers beyond the corporeal range. Ethereal numbers will eventually become our main topic of interest in articles to come. However, before we do that, we still have alot to say about the corporeals.
Now that we have familiarized ourselves with the kinds of numbers that DO arrise in our known world, we will begin to discuss how mathematicians, scientists, and most importantly , ordinary people can and do deal with these kinds of numbers. This discussion will begin in the next chapter entitled "Chapter II - The tools of science". Return to the Chapter Homepage, and then you can use the link on the bottom to jump to the next chapter.
Go Back Home
Return to Section 2 - 1
Source material :
[1] http://www.mrob.com/pub/math/largenum.html : This is the first page of Robert Munafo's large number discussion. He begins by introducing his concept of number classes.