Japanese Long Multiplication

By | 2012-09-09

This YouTube video makes it pretty clear how to do it; but I thought I’d run one without the graphical aids…

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 degrees…

                                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. I’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 …

                                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 doesn’t multiply to make these numbers, it counts crossings. It’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 it’s at most three digits in any one column):

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

If you’ve ever done long multiplication the “British” way (I wonder if it’s even taught in schools any more?), you’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 wasn’t clear.

Maybe it’s just me, but I think that’s pretty wonderful.

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