			       BACK TO BASICS
			       --------------

			     John Olwoch (#706)


1) INTRODUCTION

   Amidst  all  the  latest  hi-tech  gadgets  and gizmos around today, there
exists fundamental tools that are still a joy to use.   One  such  tool,  the
abacus,  is  the  subject  of  this  article.   The  abacus is a flat, oblong
calculating instrument with a number of counting beads that  slide  back  and
forth  along rods or in grooves.  A crosswise bar divides the abacus into two
parts.  In the Chinese version the upper part contains two  rows  of  5-value
counting  beads and the lower contains five rows of 1-value beads.  There are
normally thirteen columns, but this number varies.  The column with the  most
significant  value is on the left hand side.  For most calculations, only one
bead in the upper part and four beads in the lower part are used, making  the
value nine.  If you use all the beads in a column they add up to fifteen, and
you  have  a hex calculator.  The values of the beads are determined by their
position.  They obtain value only  when  they  are  moved  towards  the  bar.
Figure  1  shows  an  abacus  with  the  value  123,456,789.   The  beads are
represented by the letter 'O'.


		 -------------------------------------------
		 |  O  O  O  O  O  O  O  O  O  O  O  O  O  |
		 |  O  O  O  O  O  O  O  O  |  |  |  |  |  |    Upper part
		 |  |  |  |  |  |  |  |  |  O  O  O  O  O  |
		 |-----------------------------------------|    Bar
		 |  |  |  |  |  O  O  O  O  |  O  O  O  O  |
		 |  O  O  O  O  |  O  O  O  O  |  O  O  O  |
		 |  O  O  O  O  O  |  O  O  O  O  |  O  O  |    Lower part
		 |  O  O  O  O  O  O  |  O  O  O  O  |  O  |
		 |  O  O  O  O  O  O  O  |  O  O  O  O  |  |
		 |  O  O  O  O  O  O  O  O  O  O  O  O  O  |
		 -------------------------------------------

	       Figure 1: Abacus showing the value 123,456,789.


2) EXAMPLES

2.1) ADDITION

   Addition  and  subtraction  are  the  bases  of  all   calculation.    The
calculations  proceed  from  left  to  right.  Suppose we wish to perform the
addition, 87+52.  Choose any three columns (hundreds,  tens  and  units)  for
your calculation and do the following:

1)  In  the  'tens'  column,  set  one  5-value  bead and three 1-value beads
    against the bar.  In the 'units' column, set one  5-value  bead  and  two
    1-value beads against the bar.  This gives the value 87 (Fig 2a).

2)  Add 5 to the 8 in the 'tens' column.  Since 8+5>9 , move the 5-value bead
    in  the  'tens' column away from the bar and move one 1-value bead in the
    'hundreds' column against the bar.  This is equivalent to  adding  50  to
    87.  The result at this stage is 137 (Fig 2b).

3)  Add  2  to  the  7  in  the  'units' column by moving three 1-value beads
    against the bar.  The final result is 139 (Fig 2c).


		-----------      -----------      -----------
		  O  O  O          O  O  O          O  O  O
		  O  |  |          O  O  |          O  O  |
		  |  O  O          |  |  O          |  |  O
		-----------      -----------      -----------
		  |  O  O          O  O  O          O  O  O
		  O  O  O          |  O  O          |  O  O
		  O  O  |          O  O  |          O  O  O
		  O  |  O          O  |  O          O  |  O
		  O  O  O          O  O  O          O  O  |
		  O  O  O          O  O  O          O  O  O
		-----------      -----------      -----------
		    (a)              (b)              (c)

			Figure 2: Example 87+52=139.


2.2) SUBTRACTION

   The following steps will subtract 49 from 238.

1)  Set 238 (Fig 3a).

2)  Subtract  4  from  3  in the 'tens' column.  Since 4>3, borrow 1 from the
    'hundreds' column by moving one 1-value bead away from the bar, and add 6
    to the 'tens' column.  This is equivalent  to  adding  -100+60  (-40)  to
    238.  The result at this stage is 198 (Fig 3b).

3)  Subtract  9  from  8 in the 'units' column.  Since 9>8, borrow 1 from the
    'tens' column and add 1 to the 'units' column.   This  is  equivalent  to
    adding -10+1 (-9) to 198.  The result is 189 (Fig 3c).


		-----------      -----------      -----------
		  O  O  O          O  O  O          O  O  O
		  O  O  |          O  |  |          O  |  |
		  |  |  O          |  O  O          |  O  O
		-----------      -----------      -----------
		  O  O  O          O  O  O          O  O  O
		  O  O  O          |  O  O          |  O  O
		  |  O  O          O  O  O          O  O  O
		  O  |  |          O  O  |          O  |  O
		  O  O  O          O  |  O          O  O  |
		  O  O  O          O  O  O          O  O  O
		-----------      -----------      -----------
		    (a)              (b)              (c)

			Figure 3: Example 238-49=189.


3) VIRTUAL ABACUS

   I  have  written a program to simulate a five column Chinese abacus on the
'48 SX/GX.  The complete source in system RPL, is  available  in  Jazz  ready
format.   For  those  who  don't  have  that  excellent  development tool, an
executable version is also available.  The program and all  documentation  is
included on the conference disk.  The keys used are listed in Table 1.


       --------------------------------------------------------------
       |           KEY             |           FUNCTION             |
       |------------------------------------------------------------|
       | Up Arrow                  | Move lower bead to the bar     |
       | Down Arrrow               | Move upper bead to the bar     |
       | Left Shift Up Arrow       | Move upper bead away from bar  |
       | Left Shift Down Arrow     | Move lower bead away from bar  |
       | Left Shift DEL            | Reset abacus to zero           |
       | ON                        | Exit Program                   |
       | Right Shift ON            | Switch off 48                  |
       --------------------------------------------------------------

		       Table 1: Program key functions.


4) BACK TO BASICS

   The  abacus  is  a  great  tool for teaching basic numerical concepts.  In
manipulating the beads one is able to follow the process of calculation.  For
example, to do the sum '7+8=15', one can see  how  a  bead  from  the  'ones'
column  has  to  be  carried  over to the 'tens' column.  Experts can perform
complicated, multi-digit calculations at very high speeds.  Real afficionados
can simply visualize the abacus to perform  any  calculations  mentally.   In
this  age  of falling maths standards in schools, perhaps this is the tool we
should turn to.

   For a good quality abacus, check out http://www.soroban.com for a Japanese
version, or pop down to your local Chinatown.  Even if you don't use  it,  it
makes  a  great ornament, or gift.  Whatever you decide to do with it, do try
it out, and be amazed.  It really is worth going back to basics.


				   -------
