MSC # 29: Using the Ekadhika: Recurring Decimals

Our 29^th Math Special for Christmas is about a “fun, fast and easy” way to convert a fraction with a prime number denominator into a recurring decimal.

In our example here, instead of using the conventional method of dividing 3 by 19, we will divide by 2, its Ekadhika, instead. 2 is “one more than the one before” 9 in 19.

So we start with 0 and a decimal point  3/19 = 0.

Next, we divide 3 by 2 getting a quotient of 1 and a remainder of 1. We place the remainder before the answer figure: 3/19 = 0. ₁1…

We then divide our next dividend, 11 by 2 to get the next answer figure, 5 remainder 1. As before, we will always place the remainder before the answer digit.  3/19 = 0. ₁1 ₁5…

Then we divide 15 by 2 getting an answer of 7 and remainder of 1: 3/19 = 0. ₁1 ₁5 ₁7 …

Repeat the steps, 1)dividing the last answer figure (including the prefixed remainder) by the Ekadhika; 2) putting down the result as the next answer figure; and 3) placing any remainder before the answer figure, until we get an answer figure of 3.

We then stop, since we started with a numerator of 3 and the sequence of the answer figure will just repeat in a cycle.

3/19  = 0. ₁1  ₁5  ₁7  ₁8  9  ₁4  7  ₁3  ₁6  8  4  2  1  ₁0  5  ₁2  6  3…

We can also start computing from the right. Notice that the last answer figure is 3. Now instead of dividing by two, we will multiply by 2. From 3, we have 6, and ₁2, The next answer figure will be (2 x 2) + 1 or 5. Then we have 5 x 2 =10, (0 x 2) + 1 = 1 and so on. Getting the answer figures from the right is not possible using the conventional method.

There will be 18 digits in the recurring decimal. This is logical because when dividing by 19 there are only 18 possible remainders. To generalize, the recurring decimal equivalent of 1/n will have at most (n – 1) digits.

Once we have completed computing the recurring decimal equivalent of 3/19, we can now use the result to determine the decimal form of any fraction with 19 as the denominator. For example, if we want to know what the value of 12/19 is, we just look at the result of 3/19 and we see that the third from the last digit is ₁2 which is actually 12 and the answer is composed of the digits following it and cycling over to the beginning of the recurring decimal : 0.631578947368421052.  

Now we have purposely divided the answer in our illustration into two parts of 9 digits each. This is to show that there is a further short cut to this method. Take the difference between the denominator and the numerator, 19 – 3= 16.

Once we see a ₁6 in the answer line, which is the 9^th answer digit, we can stop computing and just use “all from 9” to get the next answer figure. You can see that the corresponding figures in the first and second lines add up to 9 and if we consider the remainders the total is 19.

This is easily explained by the fact that 3/19 +16/19 = 1 and their decimal equivalents will add up to 0.9999…