Using an Abacus/Addition and subtraction

Introduction
As has already been stated in the introduction to this book, addition and subtraction are the only two operations that can be carried out on the abacus; everything else must be reduced to a sequence of addition and subtraction, so learning these two operations is the most fundamental step in the study of the abacus.

Learning to add and subtract with the abacus is another case of psychomotor learning, similar to learning to dance, ride a bicycle, drive or learn a musical instrument.
 * In a first phase you will need a continuous cognitive effort trying to determine what is the next movement you have to do.
 * Later, you will notice that progressively you have to think less while the movements arise in an increasingly automatic way.
 * Finally, the movements will emerge spontaneously from you, you will have them definitely hardwired in your motor cortex and you will not have to think about them again. Although you will have a lifetime to perfect them.

Yes, it is like learning a musical instrument, but learning the abacus is much easier and faster than learning to play the viola and you will be sensitive to your progress from day to day.

In what follows we will deal with addition and subtraction together; It would be very difficult to separate the learning of one of these operations from the other since, as we will see, when we are adding we spend half the time subtracting complementary numbers and vice versa, when we are subtracting we spend half the time adding complementary numbers.

It has also been anticipated in the introduction of this book that it is not necessary to know how to add and subtract to use an abacus, only to know how to manipulate the beads or counters. In fact, for centuries the abacus was taught to people who had no previous knowledge of arithmetic and that the only knowledge they would have of it throughout their lives was going to be the use of the abacus itself. They learned to add and subtract by memorizing a long series of verses, rhymes, or rules intended to be sung as they were used. For example, taken from Xú Xīnlǔ's Pánzhū Suànfǎ, the first book entirely devoted to the abacus published in 1573 (late Ming Dynasty), and liberally translated from Chinese:

Which, obviously, inform us of which beads we have to move in order to add or subtract a digit. For example, the third line of the addition table contains three rules to try to add a 3: which apply, for example, to the following cases:
 * 3 activate 3, i.e. just activate three lower beads.
 * 3 activate 5 deactivate 2, i.e. activate one upper bead and deactivate two lower ones.
 * 3 subtract 7 carry 1, i.e. subtract 7 and add 1 to the left column.

You will understand these rules better later on, but don't worry anyway, you won't have to follow these 48 rules as you will go an easier path by memorizing just six rules that can be summed up into just three.

1-digit Addition and subtraction
The first step in learning addition and subtraction with an abacus is learning to add or subtract one of the 9 digits 1, 2, ..., 9 to / from any other 0, 1, 2, ..., 9; in total 180 cases that we will go through in our daily practice until we have them integrated into our motor memory. After this, adding or subtracting multi-digit numbers will be as simple as iterating this process in an orderly fashion for all the digits of the addend or subtrahend.

What do you need to know
To deal with the 180 cases mentioned above without memorizing the 48 rules of Panzhu Suanfa we need to memorize some almost trivial data:
 * the beads necessary to form a digit.
 * the complements to 5 of the digits 1, 2, 3, 4 and 5
 * the complements to 10 of the digits 1, 2, ..., 10

Beads needed to form a digit
Remember what was said in the Introduction of this book: "Addition is simulated by gathering the sets of counters representing the two addends, while subtraction is simulated by removing from the set of counters representing the minuend a set of counters representing the subtrahend". So we need to know the beads that make up each digit to be able to add or subtract them, but we already know this from the figure:

or in table form:

Complementary numbers
We also need to memorize two types of digit pairing: These are the digit pairs that together add up to 5 or 10. We can always find them mentally with our knowledge of addition and subtraction, but with practice they will end up solidly installed in our memory without the need to mentally "calculate" them. They are the basis of the mechanics of the abacus.
 * 5-complements
 * 10-complements

At a later stage, to deal with negative numbers, you will also need to handle 9-complements:

But for now you can safely forget about them.

The rules to use
The mechanics of addition and subtraction are based on three rules to be tried in sequence with the following protocol:
 * Only if a rule fails (because we do not have the necessary beads to complete the operation) we proceed to try the next rule.
 * The second of the rules only works for digits 1, 2, 3 and 4.
 * The third rule decomposes the operation into two other "simpler" ones: a carry to the column directly to the left or a borrow from that column plus an operation of the opposite type (i.e. a subtraction if we are adding or a sum if we are subtracting). This case raises the following points to take into account:
 * The first operation (carry or borrow) is trivial most of the time.
 * The second operation (the opposite of the starting one) is guaranteed to culminate using rules 1 or 2 (never 3) of the opposite operation.
 * We need to decide in what order we do these two operations.

Addition rules
These are the rules for the addition of a digit:

Subtraction rules
And these are the rules for subtraction:

Joint rules for addition and subtraction
The above rules for addition and subtraction are of identical structure so we can merge them into:

and we have only three rules to memorize!

Order of operation
Before moving on to some preliminary examples we have to decide what order of operation to use in case we reach rule 3, which will happen half the time. This rule leads us to a split of the original problem into two hopefully simpler ones: a carry or borrow and a operation of the opposite type to the one we are performing. What do we do first?...(sakidama, atodama...)

The Japanese standard method currently taught since the end of the 19th century proposes to first carry out the borrow and then the addition of the complementary number in the case of subtraction (sakidama 先珠), while in the case of addition, the subtraction of the complementary number is done first and then carry to the left column (atodama 後珠). This seems inspired by the structure of Chinese rules / verses / rhymes used for teaching the abacus since ancient times, but there does not seem to be any compelling logical reason to do so and not everyone agrees.

As we will see, with the abacus one works from left to right during the addition and subtraction of multi-digit numbers, so it seems natural to try to respect this movement from left to right of the hand without disturbing it with continuous comings and goings to the column of the left. Always using sakidama (carries and borrows first) seems the most natural thing to do.

It goes without saying that if you have a teacher or coach you should scrupulously follow their directions, but if you are self-taught feel free to experiment until you find your way.

By the way, in some Asian countries it is taught to use the left hand for carries and borrows.

Some preliminary examples
Ex: Enter 1 to a column of your abacus and add 3 to it:
 * 1) Do we have at our disposal (inactive) the necessary beads (3 lower ones) to add them to the 1 in our abacus? Yes!
 * 2) *then we activate them and we have completed the operation with the first rule.

Ex: Enter 3 to a column of your abacus and add 3 to it:
 * 1) Do we have at our disposal (inactive) the necessary beads (3 lower ones) to add them to the 3 in our abacus? No!
 * 2) * then we go to the second rule.
 * 3) As the addend 3 is less than 5 we can try the second rule: Do we have at our disposal (inactive) an upper bead? Yes!
 * 4) * then we apply the second rule: we activate the upper bead and retire two lower beads (the 5-complement of the addend 3).

Ex: Enter 9 to a column of your abacus and add 3 to it:
 * 1) Do we have at our disposal (inactive) the necessary beads (3 lower ones) to add them to the 3 in our abacus? No!
 * 2) * then we go to the second rule.
 * 3) As the addend 3 is less than 5 we can try the second rule: Do we have at our disposal (inactive) an upper bead? no!
 * 4) * the we proceed to the third rule:
 * 5) Carry one to A and subtract 7 (the 10-complement of 3) from B
 * 6) Do we have at our disposal (active, we are subtracting now!) the necessary beads (one upper bead and 2 lower ones) to retire them from the 9 on our abacus? Yes!
 * 7) *then retire them and we have completed this part of the operation with the first rule.

As you can see, the rules used here are the same that appeared in Xu Xinlu's Panzhu Suanfa, but condensed into only three rules thanks to the concept of complementary numbers!

Let us see now the reverse mouvements for subtraction:

Ex: Enter 4 to a column of your abacus and subtract 3 from it:
 * 1) Do we have at our disposal (active) the necessary beads (3 lower ones) to retire them from the 4 in our abacus? Yes!
 * 2) *then we deactivate them and we have completed the operation with the first rule.

Ex: Enter 6 to a column of your abacus and subtract 3 from it:
 * 1) Do we have at our disposal (active) the necessary beads (3 lower ones) to subtract them from the 6 in our abacus? No!
 * 2) * then we go to the second rule.
 * 3) As the subtrahend 3 is less than 5 we can try the second rule: Do we have at our disposal (active) an upper bead? Yes!
 * 4) * then we apply the second rule: we deactivate the upper bead and add two lower beads (the 5-complement of the subtrahend 3).

Ex: Enter 12 to a pair of columns of your abacus (AB) and subtract 3 from B:
 * 1) Do we have at our disposal (active) the necessary beads (3 lower ones) to retire them from B? No!
 * 2) * then we go to the second rule.
 * 3) As the subtrahend 3 is less than 5 we can try the second rule: Do we have at our disposal (active) an upper bead? no!
 * 4) * the we proceed to the third rule:
 * 5) Borrow 1 from A and ADD 7 (the 10-complement of 3) to B
 * 6) Do we have at our disposal (inactive, we are adding now!) the necessary beads (one upper bead and 2 lower ones) to add them to the 2 on B? Yes!
 * 7) *then activate them and we have completed this part of the operation with the first rule.

Types of one-digit addition and subtraction
The following table shows for each of the 180 elementary operations of addition and subtraction which rule solves the problem. It can be useful during your first practice, to choose which digits to add or subtract.

As you can see, the tables for addition and subtraction are mirror images of each other. Also note how half the cases correspond to rule three, that is, they require carries and borrows and from them, those marked in bold, end with an opposite type 2 operation. Also check how rule 2 only affects the addition / subtraction of digits less than 5.

Examples
Ex: Enter 5 to a column of your abacus and add 7 to it:

In this example, which requires a carry, the subtraction of the complementary number in turn requires the use of rule 2, affecting the upper bead.

Ex: Enter 95 into your abacus and add 7 to it:

Now the carry leads to another type 3 operation, requiring in turn a new carry. Three columns of the abacus are affected in this operation.

Ex: Enter 999995 into your abacus and add 7 to it:

This is an extreme situation, extrapolation from the previous case, which you should study carefully. The carry spreads or runs through the columns to the left until it finds a hole to lodge!

Note that, if we had at our disposal a lower fifth bead, in the case of the traditional abacus, we could have avoided this "carry run" at least temporarily.

For details on the use of the lower fifth ball you can consult the book: Traditional Abacus and Bead Arithmetic once you have acquired practice in addition and subtraction.

Ex: Enter 50 into your abacus and subtract 3 from it it: In this case of a type 3 operation, the borrow in turn requires a type 2 operation (affecting the upper bead).

Ex: Enter 10006 into your abacus and subtract 7 from it it:

Finally, this is a case of "borrow run" where we have to travel far to the left to find something to subtract from! Study this case carefully as well.

Two advices
So far our theoretical or intellectual explanations about the abacus. Now you know what the eastern abacus "is" and you are on your way. This intellectual knowledge will be your guide during your first steps, but with practice the movements of the beads will become second nature to you and you will never think about all these rules again (at least, until you write your first book on the abacus). To achieve this you will need to practice and practice and we offer you a couple of important tips to help you complete the path you are taking now.


 * Never read intermediate results. This is a bad habit that does not lead to anything, only to wasting time and wasting mental energy, and what you want is to acquire speed and comfort in the use of the abacus. Your abacus is there, and you have paid for it, just to keep your numbers safe for you without you having to worry about them. You only have to "react" to the arrangement of the beads without having to be aware of what number they represent.
 * Forget the addition and subtraction tables, except what we have extracted from them in the form of numbers complementary to five and ten. In particular, never think: "I have to add 7 + 8, this gives 15, then a fifteen has to appear on the abacus." If you do this, you will be "thinking" while adding and subtracting and that will tire you out and slow you down. If you have to think about something, think about the rules of movement of beads and not numbers, until you are able to add and subtract mechanically while thinking about anything else.

If you do not follow these tips, you will develop a bad habit that can be very difficult to correct later, as with the bad habits that are acquired when studying a musical instrument.

And now the practice
Your first exercises should be as simple as possible and nothing seems easier than randomly choosing two digits, for example: 6 and 8, and trying to add or subtract them (perhaps adding a one in front of the first digit if when subtracting you will need to borrow). You can use the table of types of operations explained earlier in this chapter to know in advance the type of operations to be carried out.

Subsequently, you should proceed to a systematic practice of all 180 cases of addition and subtraction of a single digit, for which the following exercise is proposed, which will also serve as an introduction to the addition and subtraction of multi-digit numbers.

Start with the abacus in the following state and add the same digit to each of the nine columns B-J proceeding from left to right

For example, to add 1 to each digit of 123456789 follow the steps indicated in the following table

and you should arrive at the result: 234567900; that is, 123456789+111111111. The following table shows the results of adding 111111111, 222222222, ... 999999999 to 1234568789.

For subtraction, add an additional 1 to column A for future borrows:

and proceed similarly

and the result is 1012345678 = 1123456789-111111111. For the rest of digits, the following table shows the results of subtracting 111111111, 222222222, ... 999999999 from 1123456789.

During a time you should practice these exercises daily until you notice that little by little you are replacing your intellectual work (thinking about the rules to use) by an instinctive mechanical response. Then you can say that you have started to learn the abacus.

Always work from left to right
In English, numbers are stated starting with the highest power of ten; 327 is "three hundred twenty-seven" and not "seven-twenty-three hundred". This is the case for many other languages, including Chinese and Japanese, but not for others such as those of the Semitic family. This is the main reason why in the abacus the sum of multi-digit numbers is worked from left to right; everything will be much easier for you, whether you have to read the numbers on a list or if someone else is dictating them to you.

For instance, let us obtain 44+78. Start with a cleared abacus and introduce the first addend 44 anywhere on the abacus (aligned to a unit rod mark if you wish, this is convenient but not essential)

then enter 4 (40) in B then enter 4 in C now add 7 (70) to B and, finally, add 8 to C The result: 122 appears on ABC.

In a more synthetic form:

Another example. Suppose we have to obtain the total of these amounts in euros: Start by clearing your abacus and enter the first number (from left to right). Align it with some of the unit dots markers if you wish.

With exercise sheets
You should start your practice by adding and subtracting short series of small integers; for example, 3 to 5 numbers of 2 or 3 digits. For instance:

Progressively increase the size of these series until you reach 10 numbers and, from here, progressively increase the size of the numbers to be added / subtracted to 5 or 6 digits. For instance:

You will therefore need collections of problems of this type that you can generate with some free utilities on the internet .

With this type of practice you will develop two different skills The latter is essential to, for example, make use of the abacus in accounting.
 * efficiently add and subtract with the abacus.
 * read numbers at a glance and keep them in memory long enough to work on the abacus.

The 123456789 exercise
It was common in ancient books on the abacus to demonstrate addition and subtraction using a well-known exercise that consists of adding the number 123456789 nine times to a cleared abacus until the number 1111111101 is reached, and then erase it again by subtracting the same number nine times. This is a convenient exercise because it uses many of the 1-digit addition and subtraction cases (but not all) and allows you to practice addition and subtraction without a paper worksheet, but it is not an elementary exercise given its length. You will need some practice time to complete it without error; consider it rather as a test of your proficiency in addition and subtraction.

Throughout this exercise the following partial results are obtained:

For more details, please refer to the chapter: Extending the 123456789 exercise of the book: Traditional Abacus and Bead Arithmetic.

Addition and subtraction with other abacuses
Everything you learn about the eastern abacus will work well with other types of abacus or at least simplify your learning. Remember that the basic operations of the abacus are addition and subtraction and everything else must be reduced to a sequence of these two operations and to the problem of how to organize such a sequence of operations on the abacus.

Counting rods
Counting rods are another example of a bi-quinary abacus, so the same three rules of addition and subtraction studied here apply to it. You just have to bear in mind that the concepts of activating / deactivating beads translate to placing / removing rods from the table and that "having at our disposal" rods to add does not refer to the pile of rods ready to be used that we have in the box, but to "fit" within the limits of representation of numbers (one 5-rod and 5 1-rods maximum).

Russian abacus
The Russian abacus (Schoty) is not a bi-quinary abacus; so the second of the addition and subtraction rules given here does not work, everything is solved with the help of the first and third rules only.