1.2.1 - Introduction
1.2.1
Introduction
Introduction
As we learned in the previous chapter, human beings can recognize and understand fundamental numerical concepts without the aid of any language or notation whatsoever, but man would not have progressed very far in his understanding in this manner. In order for us to progress further in our quest of large numbers therefore, we must, just as early man, develop systems of numeration and ways of handling numbers. In this chapter we will do just that, and by the end of it, you will have a good understanding of what kinds of numeration systems can be constructed and what their relative strengths and weaknesses are. As we study the art of developing numeration systems in the abstract we will also get a tour of the history of early mathematics.
The Main Sequence
Many accounts of the development of number can be found in popular literature on mathematics and on the internet. We will be making all the popular stops on our tour of early numeration, but there will be one key difference to my account that you aren't likely to read elsewhere, namely, that I'll be viewing that history through the lens of googology. Our primary interest here is to understand how different types of numeration systems offer different opportunities to express larger and larger numbers. Not all numeration systems are made equal. Some are simply more powerful than others, and better at expressing a larger range of values. The second main difference between my account and the standard account, is that I'm not interested in simply recounting the history. Here I will present a theory of how numeration systems typically evolve from simple tally systems to more elaborate place value systems. There appears to be a main sequence, that most numeration systems undergo. The rest of this chapter is structured around that theoretical development. This does, for the most part, end up reading much like the history of mathematics, however because numeration systems often developed in different places and times independently, we'll find that taken as a whole the history of numeration systems does not look very developmental. Only when we look at a single time and place do we see, again and again, similar lines of development. It is that development we are most interested in, because it parallels our own development of large numbers.
There are four major phases of the Main Sequence of numeration development. They are:
Tally Systems
Denominational Systems
Rank-and-File Systems
Place Value Systems
Each new system offers advantages and improvements over it's predecessors. There are basic distinguishing marks for each major phase, and each can be broken into more subcategories.
The first, Tally Systems, are distinguished by having each mark represent only a single object at a time. To represent any number of a particular thing, that number of the symbol representing it must be used. There are three sub-phases to tally systems. The first phase is qualitative tally systems. In these systems there are technical NO symbols for numbers whatsoever, only symbols for objects. The number of repetitions of the object symbol is the count of that type of object. This is the simplest and least abstract of all the notations, but also the weakest in terms of expressibility. The second phase is quantative tally systems. In such systems there is only a single symbol used to represent numbers, and each number is simply represented by that number of the symbol being written out. This is more abstract than qualitative tally systems, because the count can now represent anything, not just one particular type of object. The tally mark, divorced of any qualitative element, can now be streamlined, resulting in a more compact notation. However the advantage is quite miniscule and doesn't represent more than a few fold improvement. The third phase is a grouping tally system. In such a system, tally marks will form groups, which can then be counted as whole units themselves, aiding in the counting of larger numbers. This last kind of tally system leads naturally to the development of the next major phase: Denominational systems.
Denominational Systems are just a little more advanced than tally systems, but they are far more powerful. The basic idea is to use several different symbols which act as short hand for different size groups. To describe any number, simply take the sum of all of these group symbols. The result is often an exponential improvement in the ability to express numbers. There are two sub-phases. The first is simple Denominational Systems. In these, a finite number of symbols are used, each representing a different denomination. The inherent limitation of any such system is that there must be a largest grouping symbol which the scribe can only reproduce a finite number of times. Thus there is a pretty strict limit to how far such a system can extend. This issue is mitigated somewhat by the next sub-phase: Special Denominational Systems. In these systems, special rules are in place, in addition to the rule of summing the symbols, which can truncate expressions. Some way to keep adding new denominations may also be built into the system. This results in an even a larger range of exponential numbers that can easily be expressed. The denominational systems, while probably the most crucial step forward, have a few minor issues. Perhaps the most glaring is that expressions can be quite long and unweildly and aren't too easy to manipulate. These issues are addressed by the next major phase: Unti-Denomination systems.
Rank-and-File systems introduce the concept of unit symbols into the mix. Most Rank-and-File systems are also gematria systems. Gematria is a type of numeration in which the letters of an alphabet are used to stand for numerical values. The Rank-and-File system is a primitive kind of place value system, in which each place value gets it's own kind of "digits". This reduces the clutter of pure denominational systems, though with some draw backs, namely, the memorization of several fold more symbols. There are two main variants of Rank-and-File systems. The main type, the Rank-specific type, has units which are always specific to a denomination. While general expressions become significantly shorter, the draw back is that symbols are exhausted much more quickly, resulting in a reduced range, often only to thousands. There is also a more rare variant which solves this problem, the Unit-Rank Compound Type. In this, only one or two type of unit symbols exist. To extend further, a series of "rank" symbols are defined which can be appended to the unit symbols to provide "new" unit symbols. This system acts as an advanced denomination system. None of the systems so far discussed however have any way to mark a empty denomination, other than the lack of a symbol. In other words, there is no zero digit in any of these systems. This leads us to the final development...
Place Value Systems, are marked by only having one set of unit symbols, and no rank symbols. Instead rank is represented by position, and position is recorded by the number of unit symbols to the left or right. An empty rank is represented by a "naught" symbol of some variety, which can also double as the symbol for the number "zero". Our modern decimal notation is of this type. There are also some variants here, mainly in how the unit symbols are created, and how the unit symbols are grouped and separated. A simple place value system involves a set of unit symbols and a rule of primacy. For example, in modern decimal, left has primacy over right. A modified place value system, may include several enhancements, such as commas to separate groups of unit symbols, and a decimal point to separate whole units from fractional parts. The unit symbols can each be unique, or can themselves be compound symbols developed using any of the previous systems. For example, mayans have a place value system in which their unit symbols are actually compound symbols in a simple denominational system for writing numbers from 0 to 19. We can also consider things like scientific notation as advanced place value systems. There is also a variant on place value systems, which fit into none of the above categories. I'll call these combinatorial systems. In a combinatorial system, a finite set of symbols are used, but they represent neither units, nor ranks. Instead one simply goes through all the possible combinations of a fixed length before moving on to all the possible combinations of the next fixed length. The result is a system which requires no "zero" symbol but has the same strength as a place value system. Arithmetic can be developed within these kinds of systems, though it is somewhat awkward.
So in summary we have the following possible phases and sub-phases
>Tally Systems
>Qualitative
>Quantitative
>Grouping
>Denominational Systems
>Simple
>Modified
>Unit-Denomination Systems
>Rank-Specific
>Unit-Rank Compound
>Place Value Systems
>Simple
>Modified
>Combinatorial
We will be exploring all of these various types of systems and seeing what advantages and disadvantages they offer. The main question we will be interested in this search is: what is the limit of the strength of numeration systems? We'll discover that there is a fundamental built in limitation, caused by our very desire to express "every number". To progress further we'll eventually have to abandon this requirement, and allow us to "jump" from number to number. We'll be looking at this development in Section III and beyond, but first let's see how numeration systems can enhance our raw number sense with the capability to handle large numbers quite handally. We'll discover that we can easily juggle the kinds of numbers we are likely to need in this world ...
NEXT>> 1.2.2 - Carving the Notch : Dawn of Mathematics