Vedic mathematics and the division by 109
In my previous article Vedic mathematics and fast calculus, I talked about the curiosity aroused by early readings on Vedic mathematics, which originates from the Hindu sacred texts, the Vedas. I was amazed when I learned that the four elementary operations (addition, subtraction, multiplication, division) can be performed differently from what we learned in primary school. This is just one of the special features of Vedic mathematics, which contains many “alternative” calculus techniques that are excellent exercises in mental gymnastics.
Let us now focus on division. It rarely happens now, but if we were to do division with pen and paper we would do the classic “column” division, in which we calculate quotients and remainders in sequence. “Our” division is done vertically; in Vedic mathematics it is done horizontally.
About 25 years ago I was watching an episode of a well-known Saturday night TV program in which contestants competed in skill tests in which they had to complete a given task in a defined time. That evening there was a sprightly octogenarian who bet that he could calculate in 2 minutes the first 30 decimal digits of the division of a number less than 100 by 109. The test began, they chose a number at random in a sealed envelope. I don’t remember it, but suppose it was 67.
67 : 109 = 0.614678……
When the stopwatch was started, the contestant began listing the digits of the decimal part of the result in sequence, roughly one every 2-3 seconds:
6 … 1 … 4 … 6 … 7 … 8 …
and managed to complete the series of 30 decimal numbers within the two-minute time limit without error.
Given my natural curiosity about numbers and mathematics, I was fascinated and watched that proof carefully. The contestant was doing calculations in his head, I could clearly see it from the movements of his mouth, but I was sure that he was not performing classical column division. I thought he was using another method.
After about 20 years, when I learned the “one-line” division procedure of Vedic mathematics1, I remembered that episode.
I applied the method to division by 109, and realized that that sprightly competitor had used a shortcut that depends precisely on having chosen 109 as the divisor and a number less than 109 as the dividend. The algorithm is greatly simplified, and it is not impossible to perform it in your head. The decimal digits are calculated by the following iterative procedure:
67 divided by 11 = 6 remainder 1
The quotient (in this case 6) is the decimal digit of the result. Then you construct the number 1*10 + 6 = 16 (remainder * 10 + quotient), and proceed with 16 divided by 11, and then proceed in the same way.
16 divided 11 = 1 remainder 5
51 divided 11 = 4 remainder 7
74 divided 11 = 6 remainder 8
86 divided 11 = 7 remainder 9
97 divided 11 = 8 remainder 9
So the number is 0.614678….
When we approach other cultures with an open mind and without prejudice, we often learn something.
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