A Logical Alternative to the Existing Positional Number System
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Electronic Journal: Southwest Journal of Pure and Applied
Mathematics
Internet: http://rattler.cameron.edu/swjpam/swjpam.html
ISSN 1083-0464
Vol. 01, 1995, pp. 27-29
Submitted: July 19, 1995. Published: September 18, 1995.
Copyright 1995 Cameron University
A LOGICAL ALTERNATIVE TO THE
EXISTING POSITIONAL NUMBER SYSTEM
Robert R. Forslund
Ministry of Natural Resources
Centre for Northern Forest Ecosystem Research
c/o Lakehead University, 955 Oliver Road
Thunder Bay, Ontario
Canada P7B 5E1
email:forslund@tbaytel.net
ABSTRACT. This article introduces an alternative positional number
system. The advantages of this alternative system over the existing
one are discussed, and an illustration of the use of the system to
re-interpret apparent errors in ancient archaeological documents is
presented.
(1991) A.M.S. (MOS) Subject Classification Codes. 11,01.
Key Words and Phrases. positional number system, archaeology, zero,
placeholder.
INTRODUCTION
To simplify this article, the history of positional number
systems will not be discussed, and interested readers can find this
information in the references listed. In addition, the discussion
is restricted to the set of positive integers; however, the
extension to decimal and negative values, if necessary, is easily
accomplished.
The existing positive integer number system in a selected base
(b), is represented by the following mathematical notation :
N = SUM( a(j)*b^j ) where 0 <= a(j) <= (b-1), b > 1,
and the SUM is over j with 0 <= j <= (n-1), and a(j) are the n
digits of the positive integer N in base b.
For example:
10203 = 1*10^4 + 0*10^3 + 2*10^2 + 0*10^1 + 3
Traditionally we have selected base : b = 10, as in the example,
and commonly use other bases such as binary (b = 2), octal (b = 8)
and hexadecimal (b = 16). Ancient civilizations introduced other
bases such as 60 used by the Babylonians [1], and 18 and 20 used by
the Mayans [1].
THE ALTERNATIVE SYSTEM
There is a problem with the existing number system. Base 1 is
illogical since the digits are equal to zero, and therefore we must
select base, b > 1 when defining the system. In addition, the
digit integers starting at one are concrete concepts, while the
concept of zero (nothingness) is abstract. Therefore, I sought a
system that did not require the use of zero as a placeholder. As
you will see shortly, there is such a system, and the concept of a
zero placeholder adds unnecessary complexity.
As a result, I reasoned as follows. First of all, and most
importantly, why not use digits from 1 to b rather than 0 to (b-1)?
In base 10, this simply means introducing a 10th symbol for the
digit "ten". Any symbol will do, however the symbol "A" borrowed
from the hexadecimal system provides some familiarity. This is the
only change to the notation used in the introduction for N : viz:
1 <= a(j) <= b and b >= 1. This alternative system has no need for
a placeholder (zero) as I illustrate below.
In the alternative system (base A), the number 10 is now
represented by the digit A. The number 20 now becomes :
1A = 1*A + A, and 90 is 8A. The number 100 now becomes :
9A = 9*A + A, and 1000 = 99A = 9*A^2 + 9*A + A = 900 + 90 + 10, and
so on. Notice that, with the exception of integers that have
digits equal to zero in the existing system, all other numbers
remain the same as in the alternative system as long as the digits
lie between 1 and (b-1) inclusive. Therefore the smaller the base,
the more the differences will be between the two systems. Base 2
for example will remain the same as the existing system only when
all digits are 1. An example in base 2 is : 21 = 10101 in the
existing system, and 1221 in the alternative system. In
comparison, the number 31 in both systems remains at 11111 in base
2.
ADVANTAGES OF THE ALTERNATIVE SYSTEM
Base 1 is now permissible and completely logical and is simply
a tally system. For example, 7 = 1111111 (base 1) in the
alternative system. Secondly, for number theorists, the re-
arrangement of digits in the existing system causes digits to be
lost. For example the reversal of 12300 in the existing system is
321 and two zero digits are "lost". In the alternative system this
does not occur. For example, 12300 = 1229A, which is A9221 upon
reversal. As this simple example illustrates, this alternative
system lends itself to further research in number theory. In
addition, in the existing system, powers of the base are identified
by the number of zeros in the number - for example 10000 = 10^4.
Powers are similar in the alternative system and are identified as
in this example by 999A = A^4. More generally, in base b, powers
= ...(b-1)(b-1)...(b-1)b. Notice that fewer digits are often
needed to represent the numbers - this too is an advantage, and one
of the main reasons positional notation was first introduced.
Also, in base 10 for example, instead of having 9 numbers with one
digit, 90 numbers with 2 digits, and so on, the alternative system
now has the first 10 numbers having a single digit, the next 100
numbers having 2 digits, and so on. This too is simpler since the
number of integers with n digits in base b, is b^n rather than
b^n - b^(n-1) in the existing system.
APPLICATION OF THE ALTERNATIVE SYSTEM TO ARCHAEOLOGY
It appears that this alternative system would have been a
slightly more logical first step in the development of a number
system in ancient times. One possible use of this system might be
in the re- interpretation of ancient calculations found in
archaeological documents. History tends to identify the
introduction of the zero placeholder into the number system, as an
advancement in mathematics. However, the alternative system, with
its inherent simplicity, illustrates that the introduction of zero
may have unnecessarily complicated the use of integers, and numbers
in general.
Perhaps by re-interpreting ancient mathematical documents using
this alternative system, mathematical archaeologists will reveal
that some of the ancient calculations, previously thought to be in
error, really were correct. I will illustrate using the commonly
used digit symbols in base ten, realizing that ancient documents
used other symbols for digits. The ancients may have used 10
symbols (if base 10 was used); however, the two symbols "10" may
really have meant 1A - which is twenty (not ten). In other words,
it may be that the zero symbol represented the digit ten - not
zero. To detect such discrepancies, ancient documents must contain
calculations such as addition. For example : 10 + 11 = 31 - this
appears to be incorrect. However, if the symbol "0" actually
represents the digit ten (A) and not zero, then the equation is
correct and equivalent to 1A + 11 = 31 = 20 + 11 in the existing
system. Another example : 101 + 9 = 100. Once again, this appears
incorrect, however, if "0" symbol is re-interpreted as "A", then
this is: 1A1 + 9 = 1AA, which is 201 + 9 = 210 in the existing
system. In base 10, these apparent errors would only exist when
the symbol for ten (and interpreted as zero) occurred in the digits
of numbers. Larger bases such as 60 used by the Mesopotamians [1],
would be even more void of such "errors", and therefore it would be
harder to determine if the alternative number system was used. I
leave it to archaeologists specializing in the interpretation of
these ancient documents to examine this usage of the alternative
positional number system.
In summary, the alternative system appears to be a slightly more
logical system relative to the existing one. As illustrated, it
has immediate application to the field of archaeology. Readers
should have little problem finding other fields for the application
of this system especially in the theory of numbers. If this
article motivates others to explore the possibilities of this
alternative system, then it has succeeded.
REFERENCES
1. Boyer, C.B. and Merzbach, U.T. : A History of Mathematics
(Second Edition), John Wiley & Sons Inc., New York, 1991.
2. Hellemans, A. and Bunch, B. : The Timetables of Science - A
Chronology of the Most Important People and Events in the History
of Science, Simon & Schuster Inc., New York, 1991.
3. Swetz, F.J. : From Five Fingers to Infinity, Open Court, Chicago
and La Salle Illinois, 1995.
ACKNOWLEDGEMENTS
I wish to thank Dr. Mark Johnston and Mr. Luke Dalla Bona at
the Centre for Northern Forest Ecosystem Research for providing me
with access to the use of electronic communication via the Internet
and for their encouragement and comments regarding this paper.
email:forslund@tbaytel.net
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