Vedic Mathematics - Suthra 01 - Finding the Recurring Decimal Expansions of Vulgar Fraction with Denominator ending with the digit '9'

 

Vedic Mathematics

Suthra 1     :        Ekadhikena Purvena

Application : Recurring Decimal Expansion of Vulgar Fractions with denominator ending with ‘9’.

Meaning    :        By One More than the Previous One

We can apply the same suthra to find out the decimal representation(Recurring Decimal)  of VULGAR fractions with denominator ending with the digit ‘9’.

First Method : Division Method

Let us use this suthra to find out the recurring decimal of

1/19

Here the number of decimal places before repetition is

“Difference of Numerator and Denominator”

i.e., 19 – 1 = 18 places

In this fraction, 1/19, “Purva”(before 9) is 1

Hence “Ekadhikena Purvena” means 1 + 1 =2

While applying “Ekadhikena Purvena”

We have

Step 1 : Divide numerator 1 by 20

            1/20 = 0.1/2 = ₁0 (0 times, 1 remainder)

Step 2 : Divide 10 by 2

             0.0₀5(5 times, o remainder)

Step 3 : Divide 5 by 2

             0.05₁2 (2 times, 1 reminder)

Step 4 : Divide ₁2 I.e., 12 by 2

             0.052₀6(6 times, 0 reminder)

Step 5 : Divide 6 by 2

             0.0526₀3(3 times, 0 reminder)

Step 6 : Divide 3 by 2

             0.05263₁1(1 times, 1 remainder)

Step 7 : Divide 11 by 2

             0.052631₁5(5 times, 1 reminder)

Step 8 : Divide 15 by 2

             0.0526315₁7 (7 times, 1 remainder)

Step 9 : Divide 17 by 2

             0.05263157₁8(8 times, 1 reminder)

Step 10 : Divide 18 by 2

             0.052631578₀9(9 times, 0 remainder)

Step 11 : Divide 9 by 2

            0.0526315789₁4(4 times, 1 remainder)

Step 12 : Divide 14 by 2

            0.05263157894₁7(7 times, 1 remainder)

Step 13 : Divide 7 by 2

            0.052631578947₁3(3 times, 1 reminder)

Step 14 : Divide 13 by 2

            0.0526315789473₁6(6 times, 1 reminder)

Step 15 : Divide 16 by 2(8 times, 0 remainder)

            0.05263157894736₀8

Step 16 : Divide 8 by 2(4 times, 0 remainder)

            0.052631578947368₀4

Step 17 : Divide 4 by 2(2 times, 0 remainder)

            0.0526315789473684₀2

Step 8 : Divide 2 by 2(1 time, 0 remainder)

            0.052631578947368421

Hence 1/19     = 0. 052631578947368421052631578947368421……….

                        =0. 052631578947368421