Chinese short division

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#1 30 August, 2017 - 08:05
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Chinese short division


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 38=24, so we can try with 2 as the first quotient figure. 28=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.

30 August, 2017 - 19:28
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Joined: 5 months 3 weeks ago

Thanks for writing that up Torstenberg, I am curious about how to dvide by 2 digit numbers as I find that is the most common division I need to undertake.

3 September, 2017 - 16:52
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Joined: 5 years 9 months ago

I have been playing with this method and I like it.

The system, as you present it, is a little more suited to the Chinese Suan Pan with 2 heaven beads and 5 earth. In contrast to the soroban, on the Suan Pan you can put numbers 0-15 in on row, so you can add a remainder to a row, where the result is bigger than 9, like in your example of 234/8.
Do you know how the Japanese do this?

I also tried this on a soroban.
In the first couple of divisions I did, in order to learn exactly what is going on, I split the answer part from the remainder part. I had a couple of rows in between them. This has the advantage of just using one of the empty rows - on the left of the remainder part - for remainders bigger than 9. It has the disadvantage that the answer is more tot he left now. Will get back to that in a second.

I was playing around om the soroban using this app:
https://play.google.com/store/apps/details?id=br.net.btco.soroban

In the beginning I was just playing around with the beads. First I was making mistakes. Always I had to transpose my answers from left to right because the app only approves the answer if it is in the right most columns.

Then, all of a sudden I was setting new best times!
So I am having lots of fun with this method!

9 September, 2017 - 15:30
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Joined: 2 months 4 weeks ago

The japanese do not use this division method. They typically learn the traditional way of dividing, we also use when doing such exercises written on paper. But you are right, the fact that the suan pan has two heaven beads and 5 earth beads, it allows to display numbers up to 15 and thus to hold larger numbers which can come up using this method.

9 September, 2017 - 15:37
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This method can also be used when the divisor is larger than only one figure. There are two ways to do this.

Firstly, you can just establish a division table for the divisor. This is, however, only practical with divisors that are not too large since there will be as many rules as the divisor is large. Larger divisors will result in a longer list of rules, meaning to memorize many rules.

A good practical example is the number 12. Division by 12 is a typical division exercise when dividing items into monthly parts given some yearly value. Also, dividing into 12 hours or even 24 hours (which is just 2 x 12) can be done with this table. So, how do we determine the rules?

Let us start with 1/12. We always need to expand the dividend to a 2-digit number, as done for the divisors 1–9. So 1 becomes 10. 12 goes into 10 zero times with a reminder of 10. Peculiar enough, this is the first rule:

1/12 = 0 plus 10

2/12 is easier. We expand 2 to 20 and find 20/8 = 1 with a remainder of 8.

From this, the following rules are built by always adding 1 to the fraction and subtracting 2 from the remainder:

1/12 = 0 plus 10
2/12 = 1 plus 8
3/12 = 2 plus 6
4/12 = 3 plus 4
5/12 = 4 plus 2
6/12 = 5 (plus 0)
7/12 = 5 plus 10 (the same as 6 plus -2)
8/12 = 6 plus 8
9/12 = 7 plus 6
10/12 = 8 plus 4
11/12 = 9 plus 2

Let’s try an example:

121/12 = 10 with remainder 1 = 10.083

We follow the same procedure as for the 1-digit divisors:

121 (the original number)
0[12]1 (1/12 = 0+10; note here that we are now allowed to carry the 1 from 12, since the working figure 12 is equal to our divisor 12)
101 (did the carry, now the right-most 1 is the next working figure)
100[10] (no carry allowed here, since 10 is lower than the divisor 12)
10084 (10/12 = 8+4)
100834 (4/12 = 3+4)
1008334 (4/12 = 3+4)

From here, you can easily see this will continue forever with 3. So, 121/12 = 10.08333... Again, the decimal point has shifted one figure to the left. Another example:

1873/12 =
0[18]73
1673 (carry here. With a little practice you will be able to see this coming and make this one single step)
1573
155[13]
1561 (again a carry)
1560[10]
156084
1560834

And so on with 3. Already in the line with the working figure 1 as the right-most figure, you will soon see that 1 (as it is 1/12) always will develop to .08333...

Another way to use the short division table for divisors larger than 9 is by using a modified traditional long division. I will show this in one of the next posts.

10 September, 2017 - 20:00
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Joined: 5 months 3 weeks ago

Thanks for the write-up, food for thought.

11 September, 2017 - 05:08
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Joined: 5 years 9 months ago

Quote:

1873/12 =

These kinds of calculations are difficult for me to do exactly according to the Chinese short division method.
When confronted with 1873 and division by 12, I almost immediately see: 1800 + 72 + 1 .

So this makes me curious as to how to divide by for example 42.
I suspect that we do a division by 4 or 40 first and then make a correction.

11 September, 2017 - 14:17
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Joined: 2 months 4 weeks ago

Of course, you can mit the division table with what you can achieve without it. If you immediately see that 1800 goes into 12, then just use this partial result of 150 and continue with 73/12:

73
5[13]
61
60[10]
6084
60833….

So the result is 150+6.08333 = 156.08333….

If you also see immediately that 72 goes into 12, use it right away. For me, this chinese method is a good helper, which I can use when I do not see the answer right away or get stuck at some point. Then, the rules will help me further quickly and I do not need to guess the (partial) answer. I can, so to say, switch this helper on and off as I wish.

What I do is that I start with 18/12 instead of 1/12 and see that the partial result is 1 with a remainder of 6 and since 6/12 is 0.5 I can start writing „15“ as partial result and then continue with the remaining „73“.

A divisor of 42 will result in a quite long set of rules, and I would not recommend memorizing it unless you know this will make you faster somehow. Dividing by 4, shifting the decimal point and then doing a correction, is probably the way to go. Or, you use the short division „4-rules“ and apply them to traditional long division.

13 September, 2017 - 02:57
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Joined: 5 years 9 months ago

Quote:

For me, this chinese method is a good helper, which I can use when I do not see the answer right away or get stuck at some point. Then, the rules will help me further quickly and I do not need to guess the (partial) answer. I can, so to say, switch this helper on and off as I wish.

Indeed. The automation process is what I like about the Chinese way of doing. No need to think. There is only doing.

Quote:

A divisor of 42 will result in a quite long set of rules, and I would not recommend memorizing it unless you know this will make you faster somehow. Dividing by 4, shifting the decimal point and then doing a correction, is probably the way to go. Or, you use the short division „4-rules“ and apply them to traditional long division

This is how I mentally divide by 42.
Let's take 1890.
I guess that 42 fits 4 times into 189.
I first subtract 160, then 8:
189 => 29 => 21.
210. I realize that 21 is half 42, so 210 has 5 times 42 in it.
Answer: 45

If we would modify the Chinese short division system, it would go something like this I think:

1890.
Focus on the '1'.
The 4-rule dictates:
1/4 = 2 plus 2:
2[10]90
Carry:
4290
Now do the correction:
4*2=8
Subtract 8 from the 9:
4210

Focus on the '2'.
The 4-rule dictates:
2/4 = 5:
4510
Now do the correction:
5*2=10
Subtract 10 from the last 2 digits:
4500

Answer: 45

The difficulty lies, imho, that you need to know on what rod(s) to do the correction.
Btw, for that same reason I originally moved the answer part a little more to the left.
A bit of practice will take care of this, so this is not a big issue for me.

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