{"id":477,"date":"2012-09-09T01:00:00","date_gmt":"2012-09-08T23:00:00","guid":{"rendered":"https:\/\/www.fussylogic.co.uk\/blog\/?p=477"},"modified":"2012-09-11T20:20:05","modified_gmt":"2012-09-11T19:20:05","slug":"japanese-long-multiplication","status":"publish","type":"post","link":"https:\/\/www.fussylogic.co.uk\/blog\/?p=477","title":{"rendered":"Japanese Long Multiplication"},"content":{"rendered":"<p>This <a href=\"http:\/\/www.youtube.com\/watch?v=e-P5RGdjICo\">YouTube video<\/a> makes it pretty clear how to do it; but I thought I\u00e2\u20ac\u2122d run one without the graphical aids\u00e2\u20ac\u00a6<\/p>\n<pre><code>593 x 472<\/code><\/pre>\n<p>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\u00e2\u20ac\u00a6<\/p>\n<pre><code>                                c\n5|a b c                      b     f\n9|d e f                   a     e     i\n3|g h i         ---&gt;   5     d     h     2\n +-----                   9     g     7\n  4 7 2                      3     4<\/code><\/pre>\n<p>Then we calculate each element as the multiplication of the single digits labelling that elements row and column. I\u00e2\u20ac\u2122ve 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 \u00e2\u20ac\u00a6<\/p>\n<pre><code>                                10\n5|20 35 10                   35    18\n9|36 63 18                20    63    6\n3|12 21  6      ---&gt;   5     36    21    2\n +--------                9     12    7\n   4  7  2                   3     4<\/code><\/pre>\n<p>The gridded lines method in the video doesn\u00e2\u20ac\u2122t multiply to make these numbers, it counts crossings. It\u00e2\u20ac\u2122s pretty easy to see that you can save yourself the trouble of counting with simple single digit multiplications.<\/p>\n<p>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:<\/p>\n<pre><code>10000s |1000s |100s |10s | 1s\n       |      |  10 |    |                 6\n       |   35 |     | 18 |                18\n    20 |      |  63 |    |  6   ---&gt;      21\n       |   36 |     | 21 |               10\n       |      |  12 |    |               63\n       |      |     |    |               12\n       |      |     |    |              35\n       |      |     |    |              36\n       |      |     |    |             20<\/code><\/pre>\n<p>We pad on the right with zeroes, and sum (and the sums are easy, as it\u00e2\u20ac\u2122s at most three digits in any one column):<\/p>\n<pre><code>      6\n    180\n    210\n   1000\n   6300\n   1200\n  35000\n  36000\n 200000\n ------\n 279896<\/code><\/pre>\n<p>If you\u00e2\u20ac\u2122ve ever done long multiplication the \u00e2\u20ac\u0153British\u00e2\u20ac\u009d way (I wonder if it\u00e2\u20ac\u2122s even taught in schools any more?), you\u00e2\u20ac\u2122ll 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\u00e2\u20ac\u2122t clear.<\/p>\n<p>Maybe it\u00e2\u20ac\u2122s just me, but I think that\u00e2\u20ac\u2122s pretty wonderful.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This YouTube video makes it pretty clear how to do it; but I thought I\u00e2\u20ac\u2122d run one without the graphical aids\u00e2\u20ac\u00a6 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\u00e2\u20ac\u00a6 c 5|a b c b\u2026 <span class=\"read-more\"><a href=\"https:\/\/www.fussylogic.co.uk\/blog\/?p=477\">Read More &raquo;<\/a><\/span><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[35],"_links":{"self":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/477"}],"collection":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=477"}],"version-history":[{"count":6,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/477\/revisions"}],"predecessor-version":[{"id":734,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=\/wp\/v2\/posts\/477\/revisions\/734"}],"wp:attachment":[{"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=477"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=477"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fussylogic.co.uk\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=477"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}