Japanese Multiplication – How Does it Work?

I have come across some fun math problems (yes, math CAN be fun!) through the internet and friends.  Here is a video of Japanese Multiplication.  First watch the video, and then see if you can figure out how they are multiplying  before you read my explanation below!

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Well, did you figure it out?  Let’s take a closer look at the first example:

Now take it apart:

  • The first set of lines, the green ones in my image above, represent 2 sets of 10 , or 20
  • The second region with one line, which is orange in my image, represents 1 set of 1
  • Together these sets of lines, read top to bottom,  represent 2×10 + 1×1  which is expanded notation for 21

Let’s look at the next set of lines that were drawn perpendicular to these lines:

  • The first region that contains the 1 blue line represents 1 set of 10 or 1×10
  • The second set of lines that were drawn, the three red lines, represent 3 sets of 1
  • Together these 4 lines, read left to right, represent 1×10+3×1 = 13 in expanded notation.

Now for the tricky part!

Those of you who have ever FOIL-ed in Algebra will recognize the process of distributing the values by “First, Outer, Inner, Last”

Here is a quick Algebra example to remind you
(x+3)(2x+5) =

First = x * 2x = 2x^2
Outer= x*5 = 5x
Inner = 3*2x = 6x
Last = 3*5 = 15

Then,  2x^2 + 5x +6x +15 = 2x^2 + 11x +15 (the Outer and Inner were “like” terms, so could be added together)

Now back to the arithmetic.   If you look at the product 21×13 by separating out each factor by its place values, you have:

(20 + 1)(10 + 3)  and now you can FOIL out the values, just like in the Algebra problem!

First = 20×10 = 200
Outer =  20×3 = 60
Inner = 1×10 = 10
Last =  1×3 = 3

The 200 is represented by the 2 sets of crossing lines circled in yellow on the image above- that location on the paper represents the hundreds place value, so having a 2 in the hundreds location represents 2×100 = 200.  In the video a 2 is placed as the first digit of the product, which will be the hundreds place.

Next:

The 60 is represented by the 6 sets of crossing lines in green on the top right
The 10 is represented by the 1 set of crossing lines in green on the bottom left

Together the 60+10 gives 70.  In the video, the areas circled in green on the image above both represent the tens place value, so they are adding up the 6 crossed marks and the 1 crossed mark to get 7 sets in the tens place, or 7×10=70.    They then place a 7 to the right of the 2 in the product (placing it in the tens place)

Finally:

The 3 is represented by the 3 crossed marks in the lower right (circled in red on the image above).   This area of the paper represents the ones place, so we have 3×1 = 3.   They then place a 3 to the right of the 7 in the product, placing the 3 in the ones place.

This gives the final product of 200+60+10+3 or 200+70+3 = 273

~Now look at the second product in the video and see if you can figure out how it works!

 

9 comments

  1. Adriaan van der Lugt says:

    Very interesting indeed. I wonder if anybody has any idea how old this technique is. I could not find any clue to this.

  2. Invisible says:

    Though interesting that it really works, it is far to labour intensive to be practical. I took a stopwatch and solved a random 596 x 482 the classic pen and paper way and that took 25 seconds. A time the drawing method will never achieve I guess

    • Samantha says:

      It took me a minute to figure out how to multiply by hand again since it’s been so long. Also if you accidentally multiply wrong it can be a pain. It’s hard to count wrong… Just my opinion.

  3. invisbile2 says:

    you’d still probably be faster if you did the single digit multiplication and then added the diagonals (e.g. did the drawing in your mind):
    (5×4 =) 20 (5×8)= 40 10
    36 72 18
    24 48 12 (=6×2)

  4. abby says:

    this method is really fun. but what about multiplying it with zero? (ex 302, 350… and the like)

  5. Bart Mathias says:

    I still don’t quite see how to distinguish 101×101 from 11×11. Does one measure the distance from the 10**2 lines to the 10**0 lines?

    • Paul Nishop says:

      It’s probably best demonstrated using large-squared graph paper. A three-digit by three-digit multiplication would take up a 3×3 section of squares. For the zeros leave the appropriate columns blank or use a dotted line for zero. Do not count anything where there is no line crossing nor intersections that include a dotted linelines that coss the dotted lines. Those would be zeros in the later addition.

  6. Moe says:

    I suppose it could be beneficial to kids (and adults) who have learning barriers such as those who learn and understand only through tangible visual aids.

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