Traditional Abacus and Bead Arithmetic/Division/Learning the division table

Memorization of the division table.

The division table contains 45 rules, including the 9 diagonal elements for multi-digit divisors.

Division Table (八算, Hassan, Bā suàn)

1/9>1+1	2/9>2+2	3/9>3+3	4/9>4+4	5/9>5+5	6/9>6+6	7/9>7+7	8/9>8+8	9/9>9+9
1/8>1+2	2/8>2+4	3/8>3+6	4/8>5+0	5/8>6+2	6/8>7+4	7/8>8+6	8/8>9+8
1/7>1+3	2/7>2+6	3/7>4+2	4/7>5+5	5/7>7+1	6/7>8+4	7/7>9+7
1/6>1+4	2/6>3+2	3/6>5+0	4/6>6+4	5/6>8+2	6/6>9+6
1/5>2+0	2/5>4+0	3/5>6+0	4/5>8+0	5/5>9+5
1/4>2+2	2/4>5+0	3/4>7+2	4/4>9+4
1/3>3+1	2/3>6+2	3/3>9+3
1/2>5+0	2/2>9+2
1/1>9+1

The same number of independent elements that we find in the multiplication table (given the commutativity of this operation) whose memorization was one of the feats of our childhood in school. Memorizing the division table is therefore a similar task to learning the multiplication table.

These rules:

• From an operational point of view, these rules should be read or interpreted slightly differently depending on whether we use the traditional (TDA) or the modern (MDA) division arrangement.
  • when using MDA, the rule a/b>q+r must be read: “write q as interim quotient digit to the left, clear a and add r to the right”
  • When using TDA, the rule a/b>q+r must be read: “change a into q as interim quotient digit and add r to the right”
• From a theoretical point of view, each rule expresses the result of a Euclidean division: ${\textstyle (10\times a)/b=q,r}$ ($q$: quotient, $r$: remainder, $a,b$ digits from 1 to 9) or, equivalently ${\textstyle 10a=q\cdot b+r}$

If we think about this last point, in fact there is no need to memorize the division rules since we can obtain them in situ, when we need them, by a simple mental process. But then we would be making a mental effort similar to that required with the modern method of division and we would be moving away from the philosophy of the traditional method. There is no doubt, the efficiency and goodness of the traditional method is only achieved by memorizing the rules and we should only resort to the aforementioned mental process during the learning phase, when some rule resists coming to memory.

Fortunately, a series of patterns that appear in the division table come to our aid making it easier for us to learn it, leaving only 14 hard rules out of a total of 45.

Easy rules

In the chapter: Guide to traditional division (帰 除法) we already mentioned that the division rules by 9, 5 and 2, as well as the diagonal rules, have a particularly simple structure that allows almost immediate memorization.

Easy rules

Diagonal	Divide by 9	Divide by 5	Divide by 2
1/1>9+1	1/9>1+1	1/5>2+0	1/2>5+0
2/2>9+2	2/9>2+2	2/5>4+0
3/3>9+3	3/9>3+3	3/5>6+0
4/4>9+4	4/9>4+4	4/5>8+0
5/5>9+5	5/9>5+5
6/6>9+6	6/9>6+6
7/7>9+7	7/9>7+7
8/8>9+8	8/9>8+8
9/9>9+9

For this reason, the examples presented in that chapter only made use of divisors starting with 2,5 and 9. If you practice several examples with such divisors, it will not be difficult for you to memorize these 22 rules (almost half of the total!); which is a drastic reduction in the work to be done and not the only one.

Division by 8

Of the remaining rules, the division by 8 series is the longest but not the most difficult, since it has an internal structure:

Division by 8 rules

1/8>1+2	5/8>6+2
2/8>2+4	6/8>7+4
3/8>3+6	7/8>8+6
4/8>5+0

Leaving aside 4/8>5+0 (think of this as 8x5 = 40), the two sub-series 1, 2, 3 and 5, 6, 7 have the same remainders and the quotients are as simple as 1, 2, 3 and 6, 7, 8; so, without a doubt, this will not be the series that will be the most difficult for you to learn.

Subdiagonal rules

Finally, as a last resort for learning, note the following series of terms adjacent to the diagonal of the table.

Subdiagonal rules

4/5>8+0

5/6>8+2

6/7>8+4

7/8>8+6

8/9>8+8

There are really only two new rules here, but grasping the structure of the table above will also help you memorize the rules for divisors 5, 8, and 9.

Hard rules

In summary, of the 45 rules included in the division table, 31 fall within one of the previous patterns (grayed)

Division Table (八算, Hassan, Bā suàn)

1/9>1+1	2/9>2+2	3/9>3+3	4/9>4+4	5/9>5+5	6/9>6+6	7/9>7+7	8/9>8+8	9/9>9+9
1/8>1+2	2/8>2+4	3/8>3+6	4/8>5+0	5/8>6+2	6/8>7+4	7/8>8+6	8/8>9+8
1/7>1+3	2/7>2+6	3/7>4+2	4/7>5+5	5/7>7+1	6/7>8+4	7/7>9+7
1/6>1+4	2/6>3+2	3/6>5+0	4/6>6+4	5/6>8+2	6/6>9+6
1/5>2+0	2/5>4+0	3/5>6+0	4/5>8+0	5/5>9+5
1/4>2+2	2/4>5+0	3/4>7+2	4/4>9+4
1/3>3+1	2/3>6+2	3/3>9+3
1/2>5+0	2/2>9+2
1/1>9+1

and we are left with only 14 "hard" rules to memorize with no other help. This is no longer a huge job. Cheer up and don't give up! with some effort and practice, the greatest of the arcane mysteries of Traditional Bead Arithmetic will be yours!

The combined multiplication-division table

What follows is a simple historical note with little or no practical relevance.

The multiplication table in the English language contains all the 81 two-digit products in any order; that is, it includes both 8x9 = 72 and 9x8 = 72, which is unnecessary given the commutativity of the multiplication. On the contrary, in Chinese it only contained one of the terms of these pairs 8x9 = 72; always with the first factor less than or equal to the second. On the other hand, the division rules were enunciated by giving first the divisor that is always greater than the dividend, with the exception of the rules that we have called diagonals in which it is equal. This allows a combined multiplication-division table to be conceived that covers the entire "space" of pairs of digits as operands:

Combined table of multiplication and division rules

9✕9 81	9\8 8+8	9\7 7+7	9\6 6+6	9\5 5+5	9\4 4+4	9\3 3+3	9\2 2+2	9\1 1+1
8✕9 72	8✕8 64	8\7 8+6	8\6 7+4	8\5 6+2	8\4 5+0	8\3 3+6	8\2 2+4	8\1 1+2
7✕9 63	7✕8 56	7✕7 49	7\6 8+4	7\5 7+1	7\4 5+5	7\3 4+2	7\2 2+6	7\1 1+3
6✕9 54	6✕8 48	6✕7 42	6✕6 36	6\5 8+2	6\4 6+4	6\3 5+0	6\2 3+2	6\1 1+4
5✕9 45	5✕8 40	5✕7 35	5✕6 30	5✕5 25	5\4 8+0	5\3 6+0	5\2 4+0	5\1 2+0
4✕9 36	4✕8 32	4✕7 28	4✕6 24	4✕5 20	4✕4 16	4\3 7+2	4\2 5+0	4\1 2+2
3✕9 27	3✕8 24	3✕7 21	3✕6 18	3✕5 15	3✕4 12	3✕3  9	3\2 2+6	3\1 3+1
2✕9 18	2✕8 16	2✕7 14	2✕6 12	2✕5 10	2✕4  8	2✕3  6	2✕2  4	2\1 5+0
1✕9  9	1✕8  8	1✕7  7	1✕6  6	1✕5  5	1✕4  4	1✕3  3	1✕2  2	1✕1  1

Where we have altered the writing of our division rules to adapt them to the order of arguments used in Chinese. To highlight this fact we have replaced "/" by "\", so that the division rules as they appear in the above table must be interpreted in the form: Read a\b c+d:  as: a divide into b0 c times leaving d as remainder.

The combined table has 81 elements or rules, to which we must add the diagonal rules

Diagonal

1/1>9+1

2/2>9+2

3/3>9+3

4/4>9+4

5/5>9+5

6/6>9+6

7/7>9+7

8/8>9+8

9/9>9+9

and the rules for revising down given in the previous chapter.

Rules to revise down (two-digit divisors)

While dividing by	Revise q to	Add to remainder
1	q-1	+1
2	q-1	+2
3	q-1	+3
4	q-1	+4
5	q-1	+5
6	q-1	+6
7	q-1	+7
8	q-1	+8
9	q-1	+9

that were studied separately. This adds up to a total of 99 rules to which we can add the approximately 50 addition and subtraction rules. The traditional learning of the abacus consisted fundamentally of the memorization and practice of these 150 rules.

Statistical rules

What follows is a matter that arises from practice, not from any book in the past. The diagonal rules for divisors 1 and 2

Diagonal rules for 1 and 2

2/2>9+2

1/1>9+1

are excessive in the sense that we are often forced to revise up the divisor several times. In practice the following two statistical rules (to give them a name) behave better allowing a faster calculation.

Statistical rules

2/2>7+6

1/1>7+3

Please try them sometime during your practice!