Ekadhikena Purvena (एकाधिकेन पूर्वेण)
The Sutra means: “By one more than the previous one”.
पहले से एक अधिक के द्वारा।
1. This sutra is useful to the ‘squaring of numbers ending in 5’.
Example: 252.
For the number 25, the last digit is 5 and the 'previous' digit is 2. According to formula 'One more than the previous one', that is, 2+1=3. The Sutra gives the procedure 'to multiply the previous digit 2 (by one more than itself) by 3. It becomes the L.H.S (left hand side) of the result, that is, 2 X 3 = 6. The R.H.S (right hand side) of the result is 52, that is, 25.
252 = 2 X 3 / 25 = 6/25=625.
152 = 1 X (1+1) /25 =225;
952 = 9 X 10/25 = 9025;
1352 = 13 X 14/25 = 18225;
2. Vulgar fractions whose denominators are numbers ending in Nine
We take examples of 1 / a9, where a = 1, 2………… In the conversion of vulgar fractions into recurring decimals, Ekadhika process can be effectively used both in division and multiplication.
Division Method: Value of 1 / 19.
The numbers of decimal places before repetition is the difference of numerator and denominator, 19 -1=18 places. For the denominator 19, the purva (previous) is 1. Hence Ekadhikena purva (one more than the previous) is 1 + 1 = 2.
1. 0.10 (Divide numerator 1 by 20, 0 times, 1 remainder)
2. 0.005 (Divide 10 by 2, 5 times, No remainder)
3. 0.0512 (Divide 5 by 2, 2 times, 1 remainder)
4. 0.0526 (Divide 12 or12 by 2, 6 times, No remainder)
5. 0.05263 (Divide 6 by 2, 3 times, No remainder)
6. 0.0526311(Divide 3 by 2, 1 time, 1 remainder)
7. 0.05263115 (Divide 11 or 11 by 25 times, 1 remainder)
8. 0.052631517 (Divide 15 or 15 by 2, 7 times, 1 remainder)
9. 0.0526315718 (Divide 17 or 17 by 2, 8 times, 1 remainder)
10. 0.0526315789 (Divide 18 or 18 by 2, 9 times, No remainder)
11. 0.052631578914 (Divide 9 by 2, 4 times, 1 remainder)
12. 0.052631578947 (Divide 14 or 14 by 2, 7 times, No remainder)
13. 0.05263157894713 (Divide 7 by 2, 3 times, 1 remainder)
14. 0.052631578947316 (Divide 13 or 13 by 2, 6 times, 1 remainder)
15. 0.052631578947368 (Divide 16 or 16 by 2, 8 times, No remainder)
16. 0.0526315789473684 (Divide 8 by 2, 4 times, No remainder)
17. 0.05263157894736842 (Divide 4 by 2, 2 times, No remainder)
18. 0.052631578947368421 (Divide 2 by 2, 1 time, No remainder)
0
Multiplication Method: Value of 1 / 19
For 1 / 19, 'previous' of 19 is 1. And one more than of it, is 1 + 1 = 2. Therefore 2 is the multiplier for the conversion. We write the last digit in the numerator (अंश) as 1 and follow the steps leftwards.
1. 1
2. 21(multiply 1 by 2, put to left)
3. 421(multiply 2 by 2, put to left)
4. 8421(multiply 4 by 2, put to left)
5. 168421 (multiply 8 by 2=16, 1 carried over, 6 put to left)
6. 1368421 (6 X 2 =12, +1 = 13, 1 carried over, 3 put to left)
7. 7368421 (3 X 2, = 6 +1 = 7, put to left)
8. 147368421 (7 X 2 =14, 1 carried over, 4 put to left)
9. 947368421 (4 X 2, = 8, +1 = 9, put to left)
10. 18947368421(9 X 2 =18, 1 carried over, 8 put to left)
11. 178947368421(8 X 2 =16,+1=17, 1 carried over, 7 put to left)
12. 1578947368421(7 X 2 =14,+1=15, 1 carried over, 5 put to left)
13. 11578947368421(5 X 2 =10,+1=11,1 carried over, 1 put to left)
14. 31578947368421(1 X 2 =2,+1=3, 3 put to left)
15. 631578947368421(3 X 2 =6, 6 put to left)
16. 12631578947368421(6 X 2 =12, 1 carried over, 2 put to left)
17. 52631578947368421(2 X 2 =4,+1=5, 5 put to left)
18. 1052631578947368421(5 X 2 =10, 1 carried over, 0 put to left)
Now from step 18 onwards the same numbers and order towards left continue.
Thus 1 / 19 = 0.052631578947368421
Example: Value of 1 / 7.
1/7 में हर के अंक को 9 बनाने के लिए हर और अंश में 7 से गुणा करते हैं।
(1/7 = 7/49); हर के 49 का पूर्वेण हैं, 4; जिसका एकाधिक 5 हैं। भिन्न के आवृति दशमलव स्वरूप का अन्तिम अंक 7 होगा तथा 7-1 = 6 अंकों के पश्चात् दशमलव अंकों की पुनरावृति होगी।
1. 7
2. 357 (multiply 7 by 5 =35, 3 carried over, 5 put to left)
3. 2857 (5 X 5 =25, +3 = 28, 2 carried over, 8 put to left)
4. 42857 (8 X 5 =40, +2 = 42, 4 carried over, 2 put to left)
5. 142857 (2 X 5 =10, +4 = 14, 1 carried over, 4 put to left)
6. 2142857 (4 X 5 =20, +1 = 21, 2 carried over, 1 put to left)
अतः
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