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Dialog with an Uncertain Future / Excerp (Irregular updates)

The 59th InstallmentIndian mathematics –A driving force behind the IT superpower?

by Satoko Moriguchi,
Assistant Professor, Master Program of Information Systems Architecture

83×87, 76×74, 48×42...

While I was watching a variety television show, I saw a TV personality solve some double-digit multiplication problems in the blink of an eye to the surprise of everyone else on the show. The trick behind the instant solution was revealed to be the calculation method taught in the Indian education system, which has been well talked about in Japan and referred to as “Indian mathematics”.

The TV show introduced the formula for the multiplication of two double-digit numbers with identical tens digits, and ones digits that add up to ten. The formula worked simply by writing down, from the left, the figure resulting from multiplying the common tens digit with that same common digit plus one. Then, next to the first figure, the figure resulting from the multiplication of the two ones digits is written down. If I explain by using specific numbers it would look as follows for the problem, 83×87:
Using the common tens digit, "8", the first figure is 8×(8+1) = 8×9 = 72.
Multiply the ones digits, the second figure is 3×7 = 21.
Then simply arrange them from the left and you end up with the answer 7,221.
Therefore if the problem is 76×74, arrange the problem as (7×8) and (6×4) and you get the solution 5,624.
Therefore if the problem is 48×42, arrange the problem as (4×5) and (8×2) and you get the solution 2,016.
As you can see, you can calculate an instant solution to the problems. Needless to say, all the solutions are correct but if you have any doubt, you may check them yourself by manually calculating the solutions or using a calculator.

The TV show introduced the technique as a method that anyone can use and concluded with the comment, "Indian arithmetic and mathematics education is amazing!" Due to the nature of variety shows, I suppose it can't be helped that the segment ended without going very deep, but personally I was left feeling that something was missing, wondering if the show had missed conveying an important message. I felt uncomfortable that the TV show only superficially mentioned the method.

India has been producing many IT engineers. As far as I can remember, Indian mathematics originally drew attention in Japan because we only memorize the times table up to 9×9 (81 calculations) but in India they memorize up to 19×19, which is 361 calculations (in the past Indian students apparently had to memorize up to 30×30, which is 900 calculation). However, this is not the only noteworthy thing about arithmetic and mathematics education in India. The TV show I mentioned earlier did note that it was not the memorization of a large times table that characterized Indian mathematics, but rather it was that they teach students methods to calculate the answers quickly. However, the program did not discuss any justification for using the method, nor did they talk about the actual process behind the method. This can only be applied to very specific problems (multiplication of two double digit numbers with identical tens digits, and ones digits that add up to ten) so this is hardly a useful skill.

In fact, in addition to the method mentioned in the show, Indian arithmetic and mathematics education teaches a variety of methods that enable the solving of seemingly complicated calculations through the use of a combination of simple calculations. Additionally, the schools have children think thoroughly through the process of the calculation. They also get the students to think thoroughly about how they can justify the methods and the process used to originally develop these methods. The memorization of these methods alone is not much use. The more important benefit of learning these skills is the development competencies of a foundation for logical thinking and continued learning. It is only when these things are combined with the mathematical problem solving methods that you have developed truly useful skill.

The concept of zero originated in India and goes a long way to showing the historical superiority of mathematics in the region. It is understandable that many people attribute India's development of many outstanding IT engineers to the unique Indian mathematics methods. However, I believe that what we should note and learn is that it is a competency-based education system.

I would like to take this opportunity to conclude the column by proving at least the first formula. I said prove, but as you can see, this will only require a simple deformation, which should be at a junior high school level, so even those who think they are no good at mathematics shouldn't give up before having a look.

First, to use the two double-digit numbers? with identical tens digits, and ones digits that add up to ten? in a formula I will use "m" to represent both the identical tens digits (a whole number that is 1≦m≦9) and I will use "n" and "10-n" to represent the ones digits (whole numbers that are 1≦n≦9), then the two numbers to be multiplied together can be written as follows:
10m+n and 10m+(10-n)
For 83 and 87, m=8 and n=3 therefore the calculation is 10×8+3 and 10×8+(10-3). Let's multiply those two and breakdown the calculation:
(10m+n)(10m+(10-n))
=10m×10m+10m(10-n)+10mn+n(10-n)
=10m(10m+10)-10mn+10mn+n(10-n)
=100m(m+1)+n(10-n)
The solution to the above is to first multiply m, the original common tens digit, and (m+1). That figure is then multiplied by 100 and added to the figure resulting from the multiplication of the two original ones digits. Therefore, the final solution to the problem can be obtained, as mentioned earlier, by simply arranging the results of the two multiplications from the left.

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