Easier Solution for Complex Divisions

Now let us have a look at a sutra - "Ekadhiken Purven" out of a total of sixteen such sutras of Vedic Mathematics. Though this deals with multiplication and division both. Will talk about multiplication using this later, lets concentrate on division for the time being.

So first things first, when this sutra would apply

Values like 1/x9 (i.e. 1/19, 1/29, 1/39, etc.)

There are two methods by which we can approach this sutra I shall explain both in detail:

Prob: 1/19
First Conclusion:

19 is not a factor of either 2 or 5 which means that the result of this division is a purely circulating decimal.

Therefore, we must find out what are going to be the number of digits in this circulating decimal and the answer is 18 (divisor -1).

These 18 digits form two equal parts of 9 each and they complement each other to sum up to 9 as shown below: (1 denotes the left most digit of the result and 18 denotes the right most or last digit of the circulating sequence we shall receive out of this result)

1     2    3     4     5    6    7    8    9
10  11  12  13  14  15  16  17  18

This shows that the sum of digits at 1st place and 10th place in the circulating sequence should be 9 and so with sum of 2nd and 11th place and for all the rest of the pairs.

Thus, we effectively need to find out either 1st to 9th or 10th to 18th digits complement them respectively with 9 to get the other part of the answer.

There are two method in which the sutra can be used as shown below:

Method 1: Multiplication - To find out digits at places 10th to 18th
Method 2: Division - To find out digits at the places 1st to 9th

1. Multiplication (Deriving digits at places 10th to 18th of the answer)

This method gives answer from the right most digit (i.e. 18th) onwards to the left most digit (i.e. 10th)

18th Digit: Simply put the dividend (or numerator of the problem 1/19) here i.e. 1

17th to 10th Digits:

Just look at the problem as 1/x9 which will give us value of x=1, now add one to x which will make it 2 and thats our multiplier for the deriving the rest of the answers. so

Place  Digit
18th      1
17th      2 (i.e. 2 multiplier as explained above is multiplied with 1 i.e. 18th digit)
16th      4 (i.e. 2 multiplied with 17th digit)
15th      8 (i.e. 2 multiplied with 16th digit)
14th      6 (i.e. 2 multiplied with 15th digit, carry over 1)
13th      3 (i.e. 2 multiplied with 14th digit + carry over 1 = 13, so 1 carried over)
12th      7 (i.e. 2 multiplied with 13th digit)
11th      4 (i.e. 2 multiplied with 12th digit, carry over 1)
10th      9 (i.e. 2 multiplied with 16th digit + carry over 1 = 9)

So the set of digits from 10th to 18th place looks like this

9 4 7 3 6 8 4 2 1

Now lets subtract all of them from 9 to get 1st to 9th digits respectively;

9 4 7 3 6 8 4 2 1
0 5 2 6 3 1 5 7 8

So the final answer is:

1/19 = 0.052631578947368421 (circulating sequence)

2. Division (Deriving digits at places 1st to 9th of the answer)

This method gives answer from the left most digit (i.e. 1st) onwards to the right most digit (i.e. 9th)

Again, let us just look at the problem as 1/x9 which will give us value of x=1, now add one to x which will make it 2 and thats our magic divisor for the deriving the answer.

1st Place: a / b where a = dividend of the problem i.e. 1 and b = the magic divisor = 2

So, 1 / 2 is 0 with a remainder of 1. Hence, 1st place = 0

2nd place onwards: (in terms of a/b for each place)

Thus our 1st to 9th digits are

0 5 2 6 3 1 5 7 8

Now lets subtract all of them from 9 to get 1st to 9th digits respectively;

0 5 2 6 3 1 5 7 8

9 4 7 3 6 8 4 2 1

So the final answer is:

1/19 = 0.052631578947368421 (circulating sequence)

So with blessings of Vedic Mathematic, a little practice on this and you shall be quick like a gun!!

Will come back with application of this Vedic Maths sutra for multiplication which is even more interesting till then stay tuned and visit GyanCircle.com