Traditional Abacus and Bead Arithmetic/Addition and subtraction/Use of the 5th lower bead

Introduction
It is a mystery why traditional Chinese and Japanese abacuses had five beads in their lower deck as only four are required from the point of view of decimal numbers representation. As no extant ancient document seems to explain it, this mystery will probably last forever and we are limited to conjectures to try to understand its origin. In this line, we could think that, when they first appeared, fixed beads abacuses were conceived in image and likeness of counting rods, from which they were called to inherit every algorithm. With counting rods, the use of five rods to represent number five was compulsory in order to avoid the ambiguity between one and five, at least initially, when neither a representation for zero nor a checkerboard a la Japanese sangi were used. Furnishing the abacus with five lower beads allowed a parallel or similar manipulations of beads and rods, bringing some kind of hardware and software compatibility to fixed beads abacuses, in fact, the first Chinese books on suanpan also dealt with counting rods, so that both instruments were learned at the same time. We could also invoke a certain desire for compatibility between the abacus and rod numerals that, in one way or another, have been in use until modern times. So, for instance, one would like to change all 5’s to be represented by the five lower beads before writing down a result using rods numerals, in order to avoid very silly and catastrophic transcription mistakes.

Counting rods, by the way the most versatile and powerful abacus ever, had a flaw: it is extremely slow to manipulate. It is not a surprise that ancient Chinese mathematicians invented the multiplication table to speed up multiplication and that they also discovered the use of this multiplication table to also speed up division. Nor is it a surprise that they also discovered that, by using the abacus fifth bead, addition and subtraction operations could be somewhat simplified. They really had to be very sensitive to slowness.

Yifu Chen, as part of his doctoral thesis, has systematically analyzed 16 classic works on the abacus: twelve Chinese books from the late Ming and Qing dynasties and four Japanese books from the Edo period in which addition and subtraction are studied. As a result, Chen finds four different modes of using the fifth bead in addition and two in subtraction. These modes range from intensive or systematic use of the fifth bead to sporadic use or no use at all. From all those texts, only one Chinese book from the late Ming dynasty make full use of the fifth bead, belonging to Chen’s Mode 1 in both addition and subtraction: Computational Methods with the Beads in a Tray (Pánzhū Suànfǎ 盤珠算法) by Xú Xīnlǔ 徐心魯 (1573), by the way, the oldest extant book entirely devoted to the abacus. This fact should not lead us to the erroneous conclusion that the use of the fifth bead was a rarity of old times, since the books analyzed are neither treatises nor compendia on state-of-the-art abacus computation, but introductory manuals or textbooks for learning its use. Rather, it seems that the use or not of the fifth bead in these books corresponds more precisely to the didactic objective pursued by the authors, and that only Xu Xinlu considered it interesting to demonstrate it thoroughly from the beginning and included it in the syllabus of his course. It is mainly thanks to this work that we can rescue the traditional use of the fifth bead to simplify operations.

In what follows a small set of rules for the use of the fifth bead is presented along with their rationale and scope of use. These rules are not explicitly stated in any of the classical works, but can be inferred from the addition and subtraction demonstrations present in them, (especially in the Panzhu Suanfa)  as is done in Chen's thesis

Some terms and notation
In what follows we will use these concepts and notation in reference to the use (or not) of the lower fifth bead.




 * F: to denote a lower five (five lower beads activated) as opposed to:
 * 5: upper five (one upper bead activated).
 * T: ten on a rod (one upper bead and five lower beads activated). On 5+2 type abacus, it is also a lower ten as opposed to t an upper ten (two upper beads activated).
 * Q: lower fifteen on a rod (two upper beads and five lower beads activated) as opposed to q upper fifteen (suspended upper bead on the 5+2, three upper beads activated on the 5+3).
 * carry: this represents number 1 when it is to be added to a column as a carry from the right (addition).

Rules for addition

 * a1 Never use the 5th bead in addition except in the two cases that follow.
 * a2  4 + carry = F
 * a3  9 + carry = T

That is to say, when adding 1 to a rod you act as usual, for instance:

and

but when adding one as result of a carry you use the fifth lower bead:

text

You can see the above addition rules mentioned in a slightly different way in *.

The rationale behind
Rule a1 goal is simply to always leave an unused lower bead at our disposal in case the current column has to accept a future carry from the right, while rules a2 and a3 specify the use of the 5th bead in such a situation. Then, we can expect to obtain:


 * a reduced number of finger movements because we avoid to deal with both upper and lower beads
 * to avoid skipping rods and to reduce the left-right hand displacement span
 * to avoid any “carry run” to the left (think of 99999+1=999T0 instead of 99999+1=100000)

The advantage
The above advantages are automatically realized by using rules a2 and a3, but rule a1 is of a different nature. Rule a1 is a provision for the future, it will simplify things if a future carry actually falls on the current column (which happens about 50% of the time on average), but it will simplify nothing otherwise. Rule a1 is so a kind of a bet (subtraction rules below are also of the same nature).

The scope of use
Rules a1 to a3 are for columns that can receive a carry, which excludes the rightmost column in normal (rightward) operation.

In inverse (leftward) operation, no column will receive a future carry from the right, so that rule a1 is out of scope and does not operate, but rules a2 and a3 should always be used. (This is mentioned because an ancient technique, now defunct, used leftward operation in alternation with normal operation to avoid long hand displacements. Not of general use but an extremely interesting exercise anyway).

Exceptionally, if you do know that some column will never receive a carry, you are also free of rule a1. (This seems a strange situation, but we need to introduce it to cope with the central part of the Test Drive below).

Rules for subtraction

 * s1 Always use lower fives (F) instead of upper fives (5). For instance: 7-2 = F


 * {| cellspacing="0" style="text-align:center"

!A !A !A
 * 25px
 * A - 2 =
 * 25px
 * not
 * 25px
 * 7
 * F
 * 5
 * }
 * F
 * 5
 * }
 * }


 * s2 Never leave a cleared rod (0) if you can borrow from the adjacent left rod (but not from a farther one!), leave a T instead, i.e. 27-7 = 1T


 * {| cellspacing="0" style="text-align:center"

!A !B !A !B
 * + Use this
 * 25px
 * 25px
 * B - 7 =
 * 25px
 * 25px
 * 2
 * 7
 * 1
 * T
 * }
 * 1
 * T
 * }

should be preferred to 27-7 = 20


 * {| cellspacing="0" style="text-align:center"

!A !B !A !B
 * + Do not use this
 * 25px
 * 25px
 * B - 7 =
 * 25px
 * 25px
 * 2
 * 7
 * 2
 * 0
 * }
 * 2
 * 0
 * }

.

Remark: These two rules do not apply on rods where you are borrowing from, i.e. 112-7 = 10F (not TF)


 * and 62-7 = 5F (not FF).

The rationale behind
Both rules tend to leave activated lower beads at our disposal for the case we need to borrow from them in the future (it is like always holding small change in our pocket just in case), saving us some movements and/or wider or more complex hand displacements, such as borrowing from non-adjacent columns or skipping rods.

The advantage
Is not automatically obtained, it is only fulfilled when we actually need to borrow from the present rod. This is similar to the case of addition rule a1.

The scope of use
Once more, the rightmost column is outside the scope of these rules as we will never borrow from it.

Also, In leftward or inverse operation we will never borrow from the current column, so these rules do not apply (which may be seen as an additional reason to prefer rightward operation in normal use).

Test drive
It was common in ancient books on the abacus to demonstrate addition and subtraction using the 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 exercise seems to have the Chinese name: Jiǔ pán qīng 九盤清, meaning something like clearing the nine trays). You can find the sequence of intermediate results of the Panzhu Suanfa in this 1982 article by Hisao Suzuki (鈴木 久男): Chuugoku ni okeru shuzan kagen-hou 中国における珠算加減法 (Abacus addition and subtraction methods in China). This is a Japanese text (spiced up with some classical Chinese) that deals with addition and subtraction methods as they appear in various Chinese books from the 16th century. In pages 12-17, the Panzhu Suanfa version of the 123456789 exercise is graphically displayed on the upper series of  1:5 diagrams. The short Chinese phrases below each bar specify how the current digit was obtained (Table 1 in the Appendix A below serves a similar purpose but in a different and more convenient way for us).

Using the addition rules explained above, we should get the following sequence of results each time we complete the addition of 123456789 (see Table 1 for more details):

000000000, 123456789, 246913F78, 36T36T367, 4938271F6, 617283945, 74073T734, 864197F23, 9876F4312,   ...

at this point, adding 123456789 once more results in 1111111101, but this number appears in the Panzhu Suanfa as:

TTTTTTTT1

which cannot be obtained by the use of the above rules only. A similar situation occurs when repeating this exercise but starting with 999999999 instead of a cleared abacus (see Table 2), reaching 1TTTTTTTT0. This is why we introduced the last comment on the scope of addition rules above. It might be that, by inspection or intuition, we realize that using the 5th bead here does not generate any carry, so that we can overcome the a1 rule and proceed to this, somewhat theatrical, result ...

From here, by subtraction we should get:

TTTTTTTT1, 9876F4312, 864197523, 740740734, 61728394F, 493827156, 36T370367, 246913578, 123456789, 000000000

As it can be seen here, few F’s and T’s appear on the intermediate results, but a few more appear in the middle of calculation (Table 1), being immediately converted to 4’s and 9’s by borrowing, which is the purpose for which they were introduced. The F’s and T’s remaining on the intermediate results are only the unused ones.

Additional rules
Of course, the rules for addition can also be directly used in multiplication and the rules for subtraction in division, roots, etc.

Additionally, if using traditional division method (see chapter: Modern and traditional division; close relatives) on the 2:5 or 3:5 abacus, we can introduce an additional rule:


 * k1 Always use lower five’s, ten’s, and fifteen’s (F, T, Q) when adding to the remainder after application of the division rules.

This is so because, although we are adding to a rod, the next thing we will do is start subtracting from it (if the divisor has more than one digit). It is a kind of extension of the first rule for subtraction (s1). For instance, initiating 87÷98:

Just after application of the division rule 8/9>8+8 we should have:

By the way, you may sometimes find somewhat conflicting the use of the second rule for subtraction (s2) in Chinese division. For instance, 1167/32 = 36,46875

Now, which rule should be used here? 1/3->3+1 or 2/3->6+2 ? In fact, we can use any of them and revise up as needed, but it is faster to realize that the remainder is actually 3207 so that the second Chinese rule is the appropriate one, so, simply change columns EF to 62 and continue.

Finally, if you are using the traditional Chinese multiplication method or similar on the suanpan, you may face overflow on some columns, so that an additional rule:


 * m1 [14] + carry = Q

can also be considered.

About the advantage
It is clear that the use of the 5th bead may reduce the number of bead or finger movements required in some calculations (Think of 99999 + 1 = 999T0 vs. 99999 + 1 = 100000). An estimate based on the 123456789 exercise and some of its derivatives (see the next chapter) leads to a reduction of 10% on average (counting simultaneous movements of upper and lower beads separately). This is a modest reduction, but the advantage of the 5th bead goes beyond simply reducing the number of finger movements, as it also reduces the number and/or the extent of other hand gestures required in calculations (hand displacement, changes of direction, skipping rods,...). As already stated in the introduction to this book, each gesture:


 * as a physical process, takes a time to complete
 * as governed by our brains, requires our attention, consuming (mental or biochemical) energy
 * as done by humans (not machines), has a chance to be done in the wrong way, introducing mistakes

So, under this optic, we can expect that the use of the 5th bead will result in a somewhat faster, more relaxed and reliable calculation by reducing the total number of required gestures. It is not easy to measure this triple advantage using a single parameter.

Skipping columns, as Yifu Chen comments in his two works mentioned above, seems to have traditionally been viewed as something to be avoided as a possible source of errors. Without this concept the subtraction rule (s2) cannot be understood since it does not always lead to a reduction in the number of finger movements, but it always reduces the range of hand movement and the need to skip rods. Have you ever felt insecure with divisors or roots that contain embedded zeros? They force us to skip columns.

In any case, the advantage of using the fifth bead, although not negligible, is only modest, and each one must decide whether it is worth using it or not. After getting used to and becoming fluent in using the 5th bead, there is no better test of its efficiency than using a 4+1 abacus again and being sensitive to the amount of additional work required to complete tasks on it.

Addition


ABCDEFGHI      ABCDEFGHI       ABCDEFGHI       ABCDEFGHI       ABCDEFGHI -      -       -       -       -  000000000       123456789       246913F78       36T36T367       4938271F6 100000000 A+1  223456789  A+1  346913F78  A+1  46T36T367  A+1  5938271F6  A+1 120000000 B+2  243456789  B+2  366913F78  B+2  48T36T367  B+2  6138271F6  B+2 123000000 C+3  246456789  C+3  369913F78  C+3  49336T367  C+3  6168271F6  C+3 123400000 D+4  246856789  D+4  36T313F78  D+4  49376T367  D+4  6172271F6  D+4 123450000 E+5  246906789  E+5  36T363F78  E+5  49381T367  E+5  6172771F6  E+5 123456000 F+6  246912789  F+6  36T369F78  F+6  493826367  F+6  6172831F6  F+6 123456700 G+7  246913489  G+7  36T36T278  G+7  493827067  G+7  6172838F6  G+7 123456780 H+8  246913F69  H+8  36T36T358  H+8  493827147  H+8  617283936  H+8 123456789 I+9  246913F78  I+9  36T36T367  I+9  4938271F6  I+9  617283945  I+9 ABCDEFGHI      ABCDEFGHI       ABCDEFGHI       ABCDEFGHI -      -       -       -      617283945       74073T734       864197F23       9876F4312 717283945 A+1  84073T734  A+1  964197F23  A+1  T876F4312  A+1 737283945 B+2  86073T734  B+2  984197F23  B+2  TT76F4312  B+2 740283945 C+3  86373T734  C+3  987197F23  C+3  TTT6F4312  C+3 740683945 D+4  86413T734  D+4  987597F23  D+4  TTTTF4312  D+4 740733945 E+5  86418T734  E+5  987647F23  E+5  TTTTT4312  E+5 740739945 F+6  864196734  F+6  9876F3F23  F+6  TTTTTT312  F+6 74073T645 G+7  864197434  G+7  9876F4223  G+7  TTTTTTT12  G+7 74073T725 H+8  864197F14  H+8  9876F4303  H+8  TTTTTTT92  H+8 74073T734 I+9  864197F23  I+9  9876F4312  I+9  TTTTTTTT1  I+9

Subtraction
ABCDEFGHI      ABCDEFGHI       ABCDEFGHI       ABCDEFGHI       ABCDEFGHI -      -       -       -       -  TTTTTTTT1       9876F4312       864197523       740740734       61728394F 9TTTTTTT1 A-1  8876F4312  A-1  764197523  A-1  640740734  A-1  F1728394F  A-1 98TTTTTT1 B-2  8676F4312  B-2  744197523  B-2  620740734  B-2  49728394F  B-2 987TTTTT1 C-3  8646F4312  C-3  741197523  C-3  617740734  C-3  49428394F  C-3 9876TTTT1 D-4  8642F4312  D-4  740797523  D-4  617340734  D-4  49388394F  D-4 9876FTTT1 E-5  8641T4312  E-5  740747523  E-5  617290734  E-5  49383394F  E-5 9876F4TT1 F-6  864198312  F-6  740741523  F-6  617284734  F-6  49382794F  F-6 9876F43T1 G-7  864197612  G-7  740740823  G-7  617283T34  G-7  49382724F  G-7 9876F4321 H-8  864197532  H-8  740740743  H-8  6172839F4  H-8  49382716F  H-8 9876F4312 I-9  864197523  I-9  740740734  I-9  61728394F  I-9  493827156  I-9 ABCDEFGHI      ABCDEFGHI       ABCDEFGHI       ABCDEFGHI -      -       -       -      493827156       36T370367       246913578       123456789 393827156 A-1  26T370367  A-1  146913578  A-1  023456789  A-1 373827156 B-2  24T370367  B-2  126913578  B-2  003456789  B-2 36T827156 C-3  247370367  C-3  123913578  C-3  000456789  C-3 36T427156 D-4  246970367  D-4  123F13578  D-4  000056789  D-4 36T377156 E-5  246920367  E-5  123463578  E-5  000006789  E-5 36T371156 F-6  246914367  F-6  123457578  F-6  000000789  F-6 36T370456 G-7  246913667  G-7  123456878  G-7  000000089  G-7 36T370376 H-8  246913587  H-8  123456798  H-8  000000009  H-8 36T370367 I-9  246913578  I-9  123456789  I-9  000000000  I-9

Table 2: The 123456789 exercise over a background
(See also the next chapter)

0         1           2           3           4 000000000  0111111111  0222222222  0333333333  0444444444 123456789  02345678T0  0345678T11  045678T122  05678T1233 246913F78 0357T24689  046913F7T0  057T246911  0691357T22 36T36T367 0481481478  0592592F89  06T36T36T0  0814814811 4938271F6 0604938267  0715T49378  082715T489  09392715T0 617283945 0728394TF6  08394T6167  09F0617278  1061738389 74073T734 08F18F1845  09629629F6  1074073T67  118F18F178 864197F23 097F308634  1086419745  1197F2T8F6  1308641967 9876F4312 109876F423  1209876F34  1320987645  14320987F6 TTTTTTTT1 1222222212  1333333323  1444444434  1555FFFF45 9876F4312 1098765423  1209876534  132098764F  1432098756 864197523 097F308634  108641974F  1197F30856  1308641967 740740734 08F18F184F  0962962956  0T74074067  118F18F178 61728394F 072839F056  0839F06167  09F0617278  0T61728389 493827156 05T4938267  0716049378  0827160489  093827159T 36T370367 0481481478  0592592589  06T370369T  0814814811 246913578 0357T24689  046913579T  0F7T246911  0691358022 123456789 023456789T  0345678T11  04F678T122  0F678T1233 000000000 0111111111  0222222222  0333333333  0444444444    5          6           7           8           9 0555555555  0666666666  0777777777  0888888888  0999999999 0678T12344  078T1234F5  08T1234F66  0T1234F677  11234F6788 07T2469133 091357T244  0T246913F5  11357T2466  1246913F77 0925925922 1036T36T33  1148148144  12592592F5  136T36T366 1049382711 115T493822  12715T4933  1382715T44  14938271F5 11728394T0 128394T611  1394T61722  1F06172833  1617283944 1296296289 14073T73T0  1F18F18F11  1629629622  174073T733 14197F2T78 1530864189  164197F2T0  17F3086411  1864197F22 1543209867 1654320978  176F431T89  1876F431T0  19876F4311 16666666F6 1777777767  1888888878  1999999989  1TTTTTTTT0 1F43209867 16F4320978  176F432089  1876F4319T  19876F4311 14197F3078 1F30864189  164197529T  17F3086411  1864197522 1296296289 140740739T  1F18F18F11  1629629622  1740740733 117283949T 12839F0611  139F061722  14T6172833  1617283944 0T49382711 115T493822  1271604933  1382716044  149382715F 0925925922 0T36T37033  1148148144  125925925F  136T370366 07T2469133 0913580244  0T2469135F  11357T2466  1246913577 0678T12344 078T12345F  08T1234566  0T12345677  1123456788 0FFF55555F 0666666666  0777777777  0888888888  0999999999

External resources
You can practice using the fifth bead online with Soroban Trainer (see chapter: Introduction) using this file 123456789-5bead.sbk that you should download to your computer and then submit it to Soroban Trainer (It is a text file that you can inspect with any text editor and that you can safely download to your computer).