Sunday, November 6, 2011

Japanese Method for Pulling Square Roots by Hand

Take two, you might say. An earlier post introduced this method (link), but while it is perfectly correct as far as it goes, it actually provides a German version of how to solve the problem of squaring. I learned of the Japanese method from Murai Takayuki. A second set of message from Murai finally enabled me to see how the Japanese actually do it. It is presented here. The example I shall use is to calculate the square root of 99—which presents problems when using the German mehod I described earlier.

The Japanese method involves a two-column approach. Calculations on the left side provide inputs for the division-like calculation on the right side. We also obtain successive digits of the actual answer on the left side. The following illustration sums up the method. Click to enlarge, press Esc to return:

As explained in the earlier post, the number is arrayed in digital pairs. The decimal point, if any, must fall on one of the spatial divisions. Therefore, the number 3.099, for example, must be divided 03 . 09 90 not 3.0 99.

The core of this process is the method by which the digits are determined. In the earlier post, showing the European formula instead of the Japanese (e.g. 18 *  ≤ 1800 explained above), two equations are used to calculate first the next digit of the Answer and the sum to be deducted from the last Remainder. Let us take the case, above where the first 4 is obtained. Using the Japanese method, we set up 198 *  ≤ 9900. We can start with the highest digit, 9. Therefore we get 1989 * 9 = 17901. That’s too big. Next we might try 5, therefore 1985 * 5 = 9925. Still too big. The next attempt, with 4, will succeed: 1984 * 4 = 7936. That’s quite simple. We get, at once, both the next digit of our answer, 4, and the sum to deduct, 7936.

The European method begins with a division. The last Remainder (9900) is divided by a number constructed by multiplying 2 * 10 * the already calculated Answer. In this case that number is 99, and 99 rather than 9.9 because the decimal is ignored. 2 * 10 * 99 = 1980. Then 9900 / 1980 = 5. Next we test that number. We take (1980 + 5) * 5 = 9925. But that number is too high. Therefore we reduce 5 by 1 to get 4. Next we apply the new number, thus (1980+4) * 4 = 7936.

As is evident, this process is much more complicated than the method Murai suggests. We have to engage in double-column bookkeeping, to be sure, but everything is clearer, and the procedure is much simpler.

Nice, handy method, readily used with a sheet of paper divided in two and a hand calculator.

For those able to read Japanese an excellent tutorial with live demos is available here from Google will translate the Japanese into English. The result is so-so but one can make out the sense and follow the numbers.

Thanks Murai. This has been a lot of fun.

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