**The rules**

These chinese short division rules are directly taken from a chinese book on abacus (suan pan) division:

```
1/1 = forward 1
```1/2 = 5

2/2 = forward 1

1/3 = 3 plus 1

2/3 = 6 plus 2

3/3 = forward 1

1/4 = 2 plus 2

2/4 = 5

3/4 = 7 plus 2

4/4 = forward 1

1/5 = 2

2/5 = 4

3/5 = 6

4/5 = 8

5/5 = forward 1

1/6 = 1 plus 4

2/6 = 3 plus 2

3/6 = 5

4/6 = 6 plus 4

5/6 = 8 plus 2

6/6 = forward 1

1/7 = 1 plus 3

2/7 = 2 plus 6

3/7 = 4 plus 2

4/7 = 5 plus 5

5/7 = 7 plus 1

6/7 = 8 plus 4

7/7 = forward 1

1/8 = 1 plus 2

2/8 = 2 plus 4

3/8 = 3 plus 6

4/8 = 5

5/8 = 6 plus 2

6/8 = 7 plus 4

7/8 = 8 plus 6

8/8 = forward 1

1/9 = 1 plus 1

2/9 = 2 plus 2

3/9 = 3 plus 3

4/9 = 4 plus 4

5/9 = 5 plus 5

6/9 = 6 plus 6

7/9 = 7 plus 7

8/9 = 8 plus 8

9/9 = forward 1

The following table shows the same rules in table form (instead of 'forward' I write '<' here):

```
- divisor -
| 1 2 3 4 5 6 7 8 9
---------------------------------------------
1 | <1 5 1+3 2+2 2 1+4 1+3 1+2 1+1
d 2 | ... <1 6+2 5 4 3+2 2+6 2+4 2+2
i 3 | ... ... <1 7+2 6 5 4+2 3+6 3+3
v 4 | ... ... ... <1 8 6+4 5+5 5 4+4
i 5 | ... ... ... ... <1 8+2 7+1 6+2 5+5
d 6 | ... ... ... ... ... <1 8+4 7+4 6+6
e 7 | ... ... ... ... ... ... <1 8+6 7+7
n 8 | ... ... ... ... ... ... ... <1 8+8
d 9 | ... ... ... ... ... ... ... ... <1
```

You will be able to see many patterns in this table, looking a diagonals or in horizontal or vertical directions. This will help memorizing the rules. E.g. in the 7-table, you will recognize the fraction 1/ = 0.142857..., just in the right order with increasing quotient figures as the dividend figure increases. Also watch how beautiful the 9-table is!

**Motivation**

What is the motivation to use these rules instead of division as taught in school? Well, in school we learn a multiplication table by heart, from 1 x 1 to 9 x 9 (all multiplications with one digit). We learn that these are the building blocks of multiplication and will be needed for larger multiplications later and for division. If we do not know the table by heart, we will struggle to do larger exercises. We also learn an addition table, sometimes explicitly as a table, sometimes just by experience, ding lots of additions. After a while, we know that 8+6 = 14, we do not actually need to calculate this anymore. When we know the addition table, we also know the subtraction table, more or less (often less well).

**The school method**

But how about division? Who did ever learn a division table and used it? Typically, in school, we either use the multiplication table in reverse, or we do trial division. An example:

`234/8 =`

We start by looking at the 2, see that 8 doesn't go into 2, revise by using 23, see that 8 goes into 23 and try to figure out how many times. We might remember from the multiplication table that 3*8=24, so we can try with 2 as the first quotient figure. 2*8=16, 23-16=7:

```
234/8 = 2
16
------
07
```

And now, bringing down the next figure, we go on with 74/8 ... and here many people get stuck since they cannot figure out immediately how many times 8 fits into 74. After a while we find that 9 will work:

```
074/8 = 29
072
------
002
```

and we are back at 2 again. But as 2 is too small, we need a zero and look for 20/8 ... and so on.

This is a very tedious procedure and doing this mentally is just far too may figures to remember and juggle in your head. This is where the (true) division table come in handy.

**The chinese method**

Here it is:

1/8 = 1 + 2 2/8 = 2 + 4 3/8 = 3 + 6 4/8 = 5 5/8 = 6 + 2 6/8 = 7 + 4 7/8 = 8 + 6 8/8 = forward 1

You read this table in a special way. When doing division, you work from left to right, as usual, and do it figure by figure. The figure, you are currently working with is called the working figure of the dividend. A quotient figure replaces the working figure, and when it says '+ something', you add that amount to the next dividend figure on the right side of your working figure. If it says 'forward something', you add the number to the figure left of your working figure. All this means that the actual quotient fomrs in the place of the dividend. There is no need to spell out the quotient after the '=' sign (especially when done mentally), but let me do it here for clarity. Like so:

`234/8 = `

The rule is: 2/8=2+4. 234 changes into 274 (the 2 is replaced by 2, the 7 is 3+4).

The left-most 2 in 274 is the first figure of the final quotient, our initial result. Now the 7 is the next working figure:

`274/8 = 2`

The rule is: 7/8=8+6. 274 changes into 28[10] (the 7 is replaced by 8, the 4 turns into 4+6=10). Now we can immediately see that our next working figure is 10, which is larger than the divisor:

`28[10]/8 = 28`

So we can apply the rule '8/8 = forward 1' and just remove 8 from the 10 and add 1 to the quotient figure just obtained (the 8). So, 28[10] changes to 292. The rightmost 2 (which was 10 before) is still our working figure since there is a reminder left (we actually only dealt with 8 out of 10 and had 2 left over):

`292/8 = 29`

The rule is again: 2/8=2+4. 292 changes to 2924. Next working figure is 4:

`2924/8 = 292`

The rule is: 4/8=5. 2924 changed to 2925.

Now, there is no more working figure (everything to the right of the 5 is zero), so we are finished. The result is:

`234/8 = 29.25`

We can always make use of shortcuts here, as soon as we have understood and learnt the principle of the division table. Seeing a 2/8, we know this will turn into .25, so we can make this one step instead of two (remember that 2/8 = 1/4 = 0.25):

292 changes into 2925 when the rightmost figure in 292 is our working figure.

Also, when having 274 with 7 as the working figure, we can already see (since we know the rule 7/8=8+6) that the figure to the right will grow above 8 (it is 7 already and we have a quite large dividend figure). So, instead of saying 7/8 = 8+6 we can also say

7/8=9-2

which is equivalent. So, 274 changes immediately into 292, which immediately changes to 2925.

So, if you get fluent using the table, your calculation just involves a few steps without any guesswork or trial division and without the additional step of finding the remainder and subtracting it from the remaining part of the dividend:

234/8 > 274 > 292 > 2925 = 29.25

How do you know where the decimal point goes? It will always be one position to the left of the place in the original number/figure!

**Anatomy of the division table**

How is the division table constructed? Instead of e.g. 7/8, view this as 70/8. 8 goes into 70 8 times, making it 64 (8*8) and leaving 6 as a remainder. This is what the division table says: 7/8 = 8+6, i.e. 70/8 = 8 with a remainder of 6. There is nothing more to the table than that. Actually, looking at the table, you see that for every step in the 8-table you add 1 to the first figure and 2 to the second figure (after the + sign). When this last figure grows larger that the divisor 8, you add another 1 to the left figure and remove 8 from the second. This principle can be seen in all tables.

The 9-table is especially easy to remember, and can make a great speedup:

123/9 > 133 > 136 > 1366 > 13666 ... = 13.66666....

And the 5-table is elegant, since it has no remainders to deal with:

713/5 > 1213 (forward 1) > 1413 > 1423 > 1426 = 142.6

**Extensions**

This method can also be used when the divisor is larger than only one figure. I can write about this in another post.