[Math] Origin or author of ‘Japanese Multiplication Method’

arithmetic

What is the origin or author of the method¹ shown in the following image?

enter image description here

Notes

Also known as Japanese Multiplication Method for Kids.

Best Answer

According to Wikipedia, "It is not known where it arose first, nor whether it developed independently within more than one region of the world".

http://en.wikipedia.org/wiki/Lattice_multiplication

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I've been living in Japan for 15 years. In fact, that method is called "Indian multiplication method" in Japan. Though I don't know whether or not it actually came from India, many people believe so because they usually regard Indians as mathematically talented people. This method is not so common in Japan, yet some cramming schools teach this method for elementary school students. The following link shows another method of multiplication taught in some Japanese cramming schools.

http://www.youtube.com/watch?v=bnP1Ji1a-2w

[Math] How many number of multiplication/addition operations are there in a multiplication of two numbers of equal length

2) How did they find this "at-most $2n^2$" operations?

Lets consider the final addition part. For final addition, we are basically adding 4 numbers (results of multiplication by each digit). We can represent them like below (dashes and blank spaces replaced by 0's)

Now addition of $r_1$ and $r_2$ takes 7 basic opeartions.
Similarly addition of result of $(r_1+r_2)$ and $r_3$ takes 7 basic operations.
Similarly addition of result of $(r_1+r_2+r_3)$ and $r_4$ takes 7 basic operations.

A total of 21 operations for final addition. Here we are multiplying two small numbers. But if we would have multiplied $9999 \cdot 9999$ then final addition would be of 24 operations, i.e.
$$ r_1 - 00089991 + r_2 - 00899910 + r_3 - 08999100 + r_4 - 89991000 $$ 24 operations $= 2 n^2 - 2 n$ with $n = 4$ in our case.

(24 operations without considering carry overs. I am not sure if carry overs were also counted as additional opeartions by Tim Roughgarden)

Reference : https://people.mpi-inf.mpg.de/~mehlhorn/ftp/chapter2A-en.pdf

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[Math] Is there something similar for division as this Japanese multiplication method

[Math] How many number of multiplication/addition operations are there in a multiplication of two numbers of equal length

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