Traditional Abacus and Bead Arithmetic/Addition and subtraction/Use of the 5th lower bead
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| T | T | T | T | T | T | T | T | 1 |
Introduction[edit | edit source]
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[1], 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)[2], 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[edit | edit source]
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[edit | edit source]
- 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:
| A | A | |
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A + 1 = | 
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| 4 | 5 |
and
| A | B | A | B | |
|---|---|---|---|---|
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B + 1 = |
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| 0 | 9 | 1 | 0 |
but when adding one as result of a carry you use the fifth lower bead:
| A | B | A | B | |
|---|---|---|---|---|
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B + 5 = | 
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| 4 | 6 | F | 1 |
text
| A | B | A | B | |
|---|---|---|---|---|
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B + 5 = |
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| 9 | 6 | T | 1 |
You can see the above addition rules mentioned in a slightly different way in *Chen, Yifu (2018), "The Education of Abacus Addition in China and Japan Prior to the Early 20th Century", Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators, Springer Publishing, ISBN 978-3-319-73396-8 {{citation}}: Unknown parameter |editor1first= ignored (|editor-first1= suggested) (help); Unknown parameter |editor1last= ignored (|editor-last1= suggested) (help); Unknown parameter |editor2first= ignored (|editor-first2= suggested) (help); Unknown parameter |editor2last= ignored (|editor-last2= suggested) (help).
The rationale behind[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
- s1 Always use lower fives (F) instead of upper fives (5). For instance: 7-2 = F
A A A 
A - 2 = 
not 
7 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
Use this A B A B 
B - 7 = 
2 7 1 T
should be preferred to 27-7 = 20
Do not use this A B A B
B - 7 =
2 7 2 0
.
Remark: These two rules do not apply on rods where you are borrowing from, i.e. 112-7 = 10F (not TF)
| A | B | C | A | B | C | A | B | C | ||
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ABC - 7 = |
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not |
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| 1 | 1 | 2 | 1 | 0 | F | 0 | T | F |
- and 62-7 = 5F (not FF).
| A | B | A | B | A | B | ||
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AB - 7 = |
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not |
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| 6 | 2 | 5 | F | F | F |
The rationale behind[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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[edit | edit source]
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)[3]. 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[edit | edit source]
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:
| Abacus | Comment |
|---|---|
| ABCDEFG | |
| 87 98 | Dividend AB, divisor FG |
| 8Q 98 | A: Rule 8/9>8+8 |
| -64 | |
| 886 98 | etc. |
Just after application of the division rule 8/9>8+8 we should have:
| A | B | C | D | E | F | G | ||
|---|---|---|---|---|---|---|---|---|
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| 8 | Q | 0 | 0 | 0 | 9 | 8 |
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
| Abacus | Comment |
|---|---|
| ABCDEFG | |
| 32 1167 | 1/3->3+1 rule |
| 32 3267 | -3*2=-6 in f, use 2nd subtraction rule |
| -6 | |
| 32 31T7 |
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.
| Abacus | Comment |
|---|---|
| ABCDEFG | |
| 32 3627 | |
| ... |
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[edit | edit source]
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.
Table 1: The 123456789 exercise step by step[edit | edit source]
Addition[edit | edit source]
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[edit | edit source]
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[edit | edit source]
(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
References[edit | edit source]
- ↑ Chen, Yifu. "L'étude des différents modes de déplacement des boules du boulier et de l'invention de la méthode de multiplication Kongpan Qianchengfa et son lien avec le calcul mental". theses.fr. Retrieved 13 July 2021.
- ↑ Xú Xīnlǔ (徐心魯) (1993) [1573]. Pánzhū Suànfǎ (盤珠算法) (in Chinese). Zhōngguó kēxué jìshù diǎnjí tōng huì (中國科學技術典籍通彙).
{{cite book}}: Unknown parameter|trans_title=ignored (|trans-title=suggested) (help) - ↑ Suzuki, Hisao (1982). "Zhusuan addition and subtraction methods in China". Kokushikan University School of Political Science and Economics (in Japanese). 57 – via Kokushikan.
Further readings[edit | edit source]
- Heffelfinger, Totton; Hinkka, Hannu (2011). "The 5 Earth Bead Advantage". 算盤 Abacus: Mystery of the Bead. Archived from the original on August 1, 2021.
{{cite web}}: Unknown parameter|accesdate=ignored (|access-date=suggested) (help)
External resources[edit | edit source]
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).