# Japanese Long Multiplication

By | 2012-09-09

This YouTube video makes it pretty clear how to do it; but I thought Iâ€™d run one without the graphical aidsâ€¦

``593 x 472``

The trick is to arrange the nine necessary sub-multiplications in a standardised pattern. You can think of this as a square matrix rotated through 45 degreesâ€¦

``````                                c
5|a b c                      b     f
9|d e f                   a     e     i
3|g h i         --->   5     d     h     2
+-----                   9     g     7
4 7 2                      3     4``````

Then we calculate each element as the multiplication of the single digits labelling that elements row and column. Iâ€™ve drawn the orthogonal version again to make it clear which numbers are being multiplied, but you would do it on paper in the rotated grid â€¦

``````                                10
5|20 35 10                   35    18
9|36 63 18                20    63    6
3|12 21  6      --->   5     36    21    2
+--------                9     12    7
4  7  2                   3     4``````

The gridded lines method in the video doesnâ€™t multiply to make these numbers, it counts crossings. Itâ€™s pretty easy to see that you can save yourself the trouble of counting with simple single digit multiplications.

Now, starting at the right of the rotated form (and ignoring the labels), we write down the digits in columns according to their horizontal position in this diamond:

``````10000s |1000s |100s |10s | 1s
|      |  10 |    |                 6
|   35 |     | 18 |                18
20 |      |  63 |    |  6   --->      21
|   36 |     | 21 |               10
|      |  12 |    |               63
|      |     |    |               12
|      |     |    |              35
|      |     |    |              36
|      |     |    |             20``````

We pad on the right with zeroes, and sum (and the sums are easy, as itâ€™s at most three digits in any one column):

``````      6
180
210
1000
6300
1200
35000
36000
200000
------
279896``````

If youâ€™ve ever done long multiplication the â€œBritishâ€ way (I wonder if itâ€™s even taught in schools any more?), youâ€™ll recognise this method as being incredibly simple, repeatable and easy. The video shows how simple it is to do with the crossed lines; I wanted to show it in full in case the video wasnâ€™t clear.

Maybe itâ€™s just me, but I think thatâ€™s pretty wonderful.