# The Abacus:The Art of Calculating with Beads

## Luis Fernandes

##### Operate the abacus that appears below by click and releasing (not dragging) the mouse-pointer on the beads; the value for each column will update. Note: There was a problem with one of the images (nodiamond) which caused the abacus to malfunction (I bet the Chinese didn't have this problem way back when it was invented); it should be OK now.
###### Aug 1, 1995: Finally got some more time to work on the abacus...

Magic is fun, but it's always more fun to know how it's all done.

##### When complete, this will be a tutorial on using the abacus. Until then, enjoy what is here and look forward to what is yet to come.
WHAT'S NEW...
• overflow indicated by changing the display-color of the value to red and prefixing the value with an "*" (Aug 1, 1995);
TODO...

## 1.0 Introduction: The Anatomy of an Abacus

FIGURE 1:The Anatomy of an Abacus

The classic abacus has two decks. Each deck, separated by a beam, has several (normally 13) rods on which are mounted beads. Each rod on the top deck contains 2 beads, and each rod on the bottom deck contains 5 beads. Each bead on the upper deck has a value of five, while each bead on the lower deck has value of one. Beads are considered counted, when moved towards the beam that separates the two decks.

FIGURE 2:Numeric representation of the number: 87,654,321

##### Figure 2 is also a java-applet. It specifies the initial configuration using the VALUE resource.
The first column representing the number 8, is composed of 1 bead from the top-column (value 5) and 3 beads from the bottom-column (each with a value of 1, totaling 3); the sum of the column (5+3) is 8.

Similarily, the second column representing the number 7, is composed of 1 bead from the top-column (value 5) and 2 beads from the bottom-column (each with a value of 1, totaling 2); the sum of the column (5+2) is 7.

Addition on the abacus involves registering the numbers on the beads in the straight-forward left-to-right sequence they are written down in. As long as the digits are placed correctly, and the carry's noted properly, the answer to the operation immediately presents itself right on the abacus. There are 4 approaches to performing additions (or subtractions), each applied to particular situations. Each of these techniques is explained in tabular form in the sections that follow.

When performing the addition 6+2, one would move 1 bead from the upper deck down (value = 5) and one bead from the lower deck up (value = 1); this represents 6. Moving 2 beads from the lower deck (in the same column) up (value = 1 * 2 beads = 2) would complete the operation. The answer is then obtained by reading resultant bead positions.

### 2.2 Combined Adding-up And Taking Off

When the original number registered on a rod is smaller than 5, but will become greater than 5 after the addition, one bead from the upper-deck is moved down (added on to the beam) and one or more beads from the lower deck removed from the beam.

When a sum greater than 10 occurs on a certain rod, beads are removed from either or both the upper and lower decks and 1 bead is added to the rod directly to the left. Example: When adding 9 (10-1) to 8, one bead from the lower deck is removed (-1) and one bead from the lower deck on the row directly to the left is added (+10).

## 3.0 Subtraction

Subtraction is performed by first registering the minuend and then subtracting, starting from the left, by removing beads form either or both the lower or upper decks. The final bead-positions represent the answer.

### 3.1 Simple Taking-off

This is achieved by simply taking off one or more beads from the lower deck, or sometimes both. Example: When subtracting 7 (represented by -5-2= -7) from 9, remove 1 bead from the upper-deck (-5) and 2 beads from the lower deck (-2). The remaining 2 beads represent the result.

### 3.2 Combined Adding-up And Taking Off

When the number of beads in the lower deck is less than the subtracter (the number being subtracted), one or more beads are added in the lower deck and 1 bead is removed from the upper-deck.

Example: When subtracting 4 (+1-5 = -4) from 7 (represented by 1 bead in the upper-deck and 2 beads in the lower deck (less than 4, the subtracter), one bead is added to the lower deck (+1) and 1 bead is removed from the upper-deck (-5) leaving 3 beads, representing the result.

### 3.3 Taking-off From A Rod Of Higher Order And Adding-up

When a number on a specific rod is smaller than the subtrahend (4 is the subtrahend when performing 13 - 4; note that in the ones column, the 3 is less than the 4) one bead for the order of tens and one bead from the lower-deck has to be taken off, and one bead from the upper-deck is counted.

### 3.4 Combined Taking-off From A Rod Of Higher Order, Adding-up in the Upper-deck and Taking-off in the Lower-Deck

This technique is called for when a number on a specific rod is smaller than the supposed subtrahend (I have no idea what this means), but only in such cases as exemplified by 12 - 6.

If you've actually read this user's guide up to this point, hoping to learn how to use the abacus to its maximum potential, I should tell you that you'd be better off using a hand-held (or pop-up) digital calculator. (I hear you asking: Why did I bother going through all this if I was going to expound this heresy at the conclusion? Essentially, the exercise was designed to be didactic; from my programming stand-point, it was an interesting program to attempt and hopefully, the user would learn a thing or two (admit it, you now know how an abacus works). Unquestionably, though, it was unmitigated fun!).

Technically, the abacus is a hand-held digital calculator, but that the user must perform some sort of arithmetic manipulations before the solution is arrived at.

## 4.0 Acknowledgments

The original version of this this very same program is archived somewhere on the net as xabacus.

A great debt of gratitude goes to Agustine Lee, instructor at the Ryerson Electrical and Computer Engineering Department, who supplied a real, live abacus without which xabacus would not be possible, for supplying invaluable documentation, that was shamelessly plagiarized into the documentation you are now reading, for the Chinese characters, and for testing xabacus and providing helpful comments on improving it....

...thanks also to Nick Colonello, former sysadmin at EE and all-round technical-support person, for beta testing....

...and to Eva Dudova, who has expertise in unmercifully crashing applications (has a future as a beta-tester), and is cute too...

... and finally, thanks to those who have written X-applications and the demo Java applets, from whose code I have learned the art of X and Java.

The beads for the abacus, were generated using the Interactive WWW graphics generator. Check it out!

## 5.0 Do those Chinese characters really mean anything?

A few people have asked me this, so here's the translation:

The Abacus: The Art of Calculating with Beads/ Luis Fernandes elf@ee.ryerson.ca