Applications of an Alternate Number System

                       ROBERT R. FORSLUND
                          Nov.30 1996

Thunder Bay, Ontario
Canada
email: forslund@tbaytel.net

ABSTRACT.  The intent of this article is to illustrate further
applications of a recently developed alternate positional number
system for the set of positive integers.  This system eliminates
the need for the digit zero, and hence all digits behave the
same.  Examples of counting, arithmetic operations, computing and
number theory are discussed and re-examined with respect to this
alternate system.


Keywords and Phrases. positional number system, zero,
placeholder, abacus.

                         INTRODUCTION

   In a previous article [4], an Alternate Number System (ANS)
was introduced that eliminates the need for the zero placeholder
currently required in the Existing Number System (ENS).  I am
intentionally focusing only on the set of positive integers in
order to simplify preliminary explorations of ANS.  The previous
article gave some elementary examples in number theory, and
focused on the use of ANS to reinterpret apparent errors in
ancient archaeological documents.  Before examining additional
applications, I will review ANS.

   In a selected base (b), ANS is represented by the following
mathematical notation : 

N = SUM (a(j)*b^(j-1)) + a(1)  where 1 <= a(j) <= b,  b >= 1.

  The SUM is over j with 2 <= j <= n, where a(j) (including a(1))
are the n digits of the positive integer N in base b.  The
difference between this representation and ENS, is that base 1 is
now valid, and is equivalent to a tally system.  In addition, the
allowable digits range from 1 to b inclusive in ANS rather than 0
to (b-1), which is the case in ENS.  

The above number system notation traditionally is : sum (a(j)*b^j)
where the sum is over j and 0 <= j <= (n-1) is used.   Since ANS
has no need for zero, I have altered the value of j to agree with
ANS.  This eliminates the need for b^0 = 1 and includes digit
symbols a(1) to a(n) for the n digits rather than a(0) to a(n-1). 
Also, the notation is split to illustrate that two types of
number systems are embedded in the traditional notation.  The
first system involves a symbolic system represented by the single
digits (a(1)). In this system, unique symbols (1,2,3,4,5,6,7,
...) are selected for each number (N) from one to the base.  When
N becomes greater than the base, either another symbol is
selected, or a multi-digit number system (such as ENS or ANS) is
implemented.  The second system is therefore a multi-digit system
which involves combining the symbolic digits with the base using
implied operators.  The implied operators in the case of both ENS
and ANS are addition, multiplication and powers of the base. 
Since the power values change for each digit, therefore ANS (as
well as ENS) are positional systems which implies that digit
order is necessary.

   If we let the symbol "A" represent the digit ten, then an
example of a number in ANS (base A) is :

A1A3 = A*A^3 + 1*A^2 + A*A^1 + 3

   In ENS, the same number in base 10 is 

10203 = 1*10^4 + 0*10^3 + 2*10^2 + 0*10^1 + 3

   Notice that the ANS notation has one less digit compared to
the ENS version of the same number.  Since the original reason
for introducing number systems was to create a means to represent
larger numbers more concisely, therefore ANS provides a slight
improvement over ENS.

   The purpose of this article, is to illustrate additional
applications of ANS.  For familiarity, I will usually focus on
base ten in the following discussions, however the extension to
any other positive integer base will equally apply.


                        COUNTING IN ANS

   In base ten, a tenth digit (A) is introduced in ANS, and
counting begins at one and proceeds as follows :

1,2,3,4,5,6,7,8,9,A,11,12, ... ,18,19,1A,21,22, ... ,28,29,2A,
... ,98,99,9A,A1,A2,A3,A4,A5,A6,A7,A8,A9,AA,111,112, ...

   I define a "turning point" as a number where either the number
of digits increases by 1, or where the most significant digit
increases by 1.  I will omit the single digit symbols themselves
as turning points.  Therefore, in ENS in base 10, we refer to the
numbers :

10, 20, 30, ... , 90, 100, 200, 300, ... , 900, 1000, 2000, ...

as the familiar turning points.  Similarly, the turning points
for ANS in base A, are the numbers :

11, 21, 31, ... , 91, A1, 111, 211, ... , A11, 1111, 2111 ...

   An interesting aspect of applying ANS to years, is that
turning points such as decades, centuries and millennia, take on
a different relevance.  The "magic" of the digits in the year
2000 in ENS is replaced with the year 199A in ANS, which is
followed by 19A1, and so on up to the year 1AAA.  Therefore, if
ANS is used, the next turning point does not occur until we reach
the year 2111.  This idea may be relevant to the millenium
problem where computers will have trouble performing calculations
when we reach the year 2000.

   Another interesting topic with respect to counting, is the use
of the abacus.  Traditionally, ENS is the system employed on an
abacus, and the absence of beads at a position indicates the
digit zero.  With a little practice, ANS can also be implemented
on an abacus.  The only difference is that in ANS, each position
contains at least one bead, and ten beads are allowed at each
position (in base A).  In other words, there are no empty
positions within the number itself.  Of course the absence of
digits to the left of the number, will still be represented by
the absence of beads.  A good exercise to become familiar with
ANS, is to use the abacus to count in base ten.


                   ARITHMETIC OPERATIONS IN ANS

   For illustration purposes, I will consider only the positive
operators of addition, multiplication and powers respectively. 
Performing operations on numbers in ANS, is just as simple as in
ENS, but takes some practice to retrain ourselves to eliminate
the idea of zero as a placeholder.

   Addition in ANS (in base A), operates as follows : 1 + 9 = A,
1 + A = 11, A + A = 1A.  The logic of A + A = 1A is no different
from 9 + 9 = 18 or 8 + 8 = 16.  Similar to ENS, there is a
maximum carry digit of 1, and calculations such as 7 + 0 = 7
disappear.

   Like addition, multiplication in ANS is similar to ENS.  Using
base A again : 9 x A = 8A, A x A = 9A,  and so on.  Here too, the
logic is no different from 8 x 9 = 72 and 9 x 9 = 81, except that
a maximum carry digit of 9 can occur rather than 8 in ENS and
calculations such as 7 x 0 = 0 disappear.

   Powers take on a different character in ANS.  It seems natural
for us to use numbers such as 1000, 100000, and so on, and we
think of them as 10^3 and 10^5 respectively.  In ENS, these are
the focus of our number system.  In ANS however, they are no
longer the turning points, and therefore they have the same
number of digits as the preceding number in all bases > 1.  The
number 999 is followed by 99A in ANS (1000 in ENS), which in turn
is followed by 9A1 and so on.  Similar to ENS, a specific pattern
to the digits of powers in ANS makes them distinct from other
numbers.  Therefore, in any base, b>1, a number such as :

(b-1);(b-1); ... ;(b-1);b

represents a power of b, where the semicolons are used for digit
separators, and the power is equal to the number of digits.  For
example, the number 9999A = A^5 in base A,  7778 = 8^4 in base 8,
1111112 = 2^7 in base 2, and so on.


                        COMPUTING AND ANS

   The binary number system is the basis of computing in ENS. 
ANS in base 2 should therefore be applicable to computing.  A
slight modification is necessary for an ANS computer to function
properly.  Three bit states are necessary, rather than only two
presently required for an ENS computer, since a computer (as well
as an abacus) uses a fixed number of bits (positions) to
represent a piece of information.  Therefore an 8 bit byte such
as 00010101 in ENS, would require the bit pattern 00001221 in
ANS, where the 0 symbols in this case, represent the absence of
digits, and three conditions : 0, 1 and 2 are required.  The use
of the three states in ANS does not imply the use of a base 3
number system in this situation.  In ENS, the 0 symbol has a dual
purpose and is used to represent not only digits, but also the
absence of digits.  For comparison in ANS, the zero bit clearly
represents only the absence of digits.  It is possible, in
theory, to create such an ANS computer with three states rather
than two, however its advantages remain to be seen, other than a
slight improvement on the conciseness of binary numbers.


                 NUMBER THEORY WITH RESPECT TO ANS

   As briefly illustrated in the previous article, ANS can be
used to re-examine digit patterns.  All concepts of number theory
related to digits remain the same in both ENS and ANS, as long as
the digits lie between 1 and (b-1) inclusive, and the base, b is
greater than 1.  For example, Mersenne numbers = 2^n - 1 are
simply the digit patterns 1...1 (base 2) with n digits all equal
to 1.  Since the digits are all 1, therefore the concept of
Mersenne numbers (hence Mersenne primes) remains the same in both
ENS and ANS in base 2.  Mersenne numbers are the only numbers
that remain the same between ANS and ENS in base 2.


   Generalizing this idea to other bases, any number of the form:

(b^n - 1)/(b - 1)

is 1...1 (base b) with n digits equal to 1.  So any results with
respect to such numbers will remain consistent in both ANS and
ENS, as long as a base conversion is not performed.  For example,
1111111 (base 5) = (5^7 - 1)/4, however if this is converted to
base 7, then the number becomes 77641 in ANS whereas it is 110641
in ENS.  So any manipulation or calculation performed on the
digits will differ between the two systems when the digit 0
occurs in ENS.
  
   Another number theory example is the palindrome which exhibits
central digit symmetry.  For example, 21 (base 10), is 10101
(base 2) in ENS and 1221 in ANS.  In this case, both are
palindromes, however this is not the general case, and conversion
of palindromes between systems does not guarantee that they will
both be palindromes.  For example, the palindrome 20302 in ENS
(base 7), is 17272 in ANS and the palindrome 27372 in ANS (base
7), is 30402 in ENS.  Therefore, the identification of
palindromes will change between the two systems.

   Another logical feature of ANS, is that the permutation of the
digits, including reversing and rotating digits, results in a
number with the same number of digits.  Digit permutation in ENS
often presents a problem, since for example, the reversal of
12300 becomes 321, which "loses" two zero digits in ENS.  This
situation does not occur in ANS, and is therefore an advantage of
ANS over ENS with respect to digit rearrangements and
manipulations.

  Digit reversal further illustrates that the zero digit behaves
differently from the other digits.  ANS, on the otherhand, uses
digits that all behave in the same way.  This distinct behaviour
of zero is reflected as well in the operations performed with
zero.  For example N x 0 = 0 and N + 0 = N.  Therefore, the
elimination of the digit zero by introducing ANS, results in
consistent digit behaviour in the number system and the
operations performed on the digits.

   Kaprekar's constant is an interesting concept for exploration
and also involves the manipulation of digits.  Kaprekar's
constant is the number that results when the digits of a number
are arranged in descending and ascending order, subtracted from
one another, and then the same procedure applied to the result
until a constant number (or a cycle of numbers) occurs.  In ENS,
four digit numbers in base 10 usually result in the constant
6174.  In ANS, an interesting thing happens.  6174 does occur,
however the number 8082 = 7A82 also occurs when the operation is
performed in ANS since A872 - 278A = 7A82 in ANS.  Finally,
notice that in ANS, numbers will retain all of their digits,
whereas some numbers in ENS (which include zero digits) will
"lose" digits when the digits are arranged in ascending order
from left to right.  Once again the consistent digit behaviour in
ANS is beneficial.


   Digit manipulation is the essence of the previous examples. 
Of course if all digits of a number lie between 1 and (b-1)
inclusive, then the number, and hence any patterns associated
with the digits, will remain the same in both systems.  The
smaller the base, the greater the dissimilarity between the
digits of the two systems.  Since base 1 is invalid in ENS,
therefore the binary system will contain the greatest differences
between the digits of numbers in ENS versus ANS.

                    SUMMARY AND CONCLUSIONS

   The Alternate Number System provides a logical simplification
of the Existing Number System.  By eliminating the zero digit
concept, digit behaviour is now consistent.  This consistency
simplifies the results when permutations of digits, and
operations on digits are performed.

   As illustrated in these few examples, the Alternate Number
System contains many possible paths to explore.  My purpose in
this follow-up article, is to attempt to motivate others to
continue exploring additional applications of this alternate
number system, especially in the theory of numbers.  I have
purposely focused on the set of positive integers and the
positive operations between them since I believe we must first
fully understand their nature in order to begin to comprehend the
many extensions that we have superimposed on them.  It is my hope
that ANS will help us in this understanding.


   There are many excellent books available.  The accompanying
bibliography includes the more recent books that I have
encountered.  It is revealing to re-examine the many methods,
conjectures and theorems contained in these books, from this
alternate perspective, to gain new insight into the set of the
positive integers.
                            REFERENCES

1. Boyer, C.B. and Merzbach, U.T. : A History of Mathematics
(Second Edition), John Wiley & Sons Inc., New York, 1991.

2. Dunham, W. : The Mathematical Universe, John Wiley & Sons,
Inc., New York, 1994.

3. Dunham, W. : Journey through Genius - The Great Theorems of
Mathematics, John Wiley & Sons, Inc., New York, 1990.

4. Forslund, R.R. : A Logical Alternative to the Existing
Positional Number System, Southwest Journal of Pure and Applied
Mathematics, Vol.1 27-29, 1995.

5. Halmos, P.R. : Problems for Mathematicians Young and Old, The
Mathematical Association of America, 1991.

6. 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.

7. Ifrah, G. : From One to Zero - A Universal History of Numbers,
Viking Penguin Inc. New York, 1985.

8. McLeish, J. : The Story of Numbers - How Mathematics Has
Shaped Civilization, Ballantine Books, New York, 1992.

9. Pappus, T. : The Joy of Mathematics - Discovering Mathematics
All Around You. Wide World Publishing/Tetra, San Carlos,
California 1991.

10. Pappus, T. : More Joy of Mathematics - Exploring Mathematics
All Around You. Wide World Publishing/Tetra, San Carlos,
California, 1991.

11. Richards, S.P. : A Number for Your Thoughts, S. P. Richards,
New Providence, New Jersey, 1982.

12. Swetz, F.J. : From Five Fingers to Infinity, Open Court,
Chicago and La Salle Illinois, 1995.


                         ACKNOWLEDGEMENTS

I wish to thank Dr. Mark Johnston, Mr. Luke Dalla Bona and Dr.
Robert Rempel of the Centre for Northern Forest Ecosystem
Research (Ministry of Natural Resources in Thunder Bay, Ontario)
for their help in publishing this over the Internet.

Last Modified Aug.8 1997