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MULTIPLICATION
We have already observed the application of Vedic sutras in multiplication.
Let us recall them.It enables us to have a comparative study of the applicability of these methods, to assess advantage of one method over the other method and so-on. Example (i) : Find the square of 195. The Conventional method : 1952 = 195 x 195 ______ 975 1755 195 _______ 38025 ¯¯¯¯¯¯¯ (ii) By Ekadhikena purvena, since the number ends up in 5 we write the answer split up into two parts. The right side part is 52 where as the left side part 19 X (19+1) (Ekhadhikena) Thus 1952 = 19 X 20/52 = 380/25 = 38025 (iii) By Nikhilam Navatascaramam Dasatah; as the number is far from base 100, we combine the sutra with the upa-sutra ‘anurupyena’ and proceed by taking working base 200. a) Working Base = 200 = 2 X 100. Now 1952 = 195 X 195  iv) By the sutras "yavadunam tavadunikritya vargamca yojayet" and "anurupyena" 1952, base 200 treated as 2 X 100 deficit is 5.  v) By ‘antyayor dasakepi’ and ‘Ekadhikena’ sutras Since in 195 x 195, 5 + 5 = 10 gives 1952 = 19 x 20 / 5 x 5 = 380 / 25 = 38025. vi) Now "urdhva-tiryagbhyam" gives  By the carryovers the answer is 38025 Example 2 : 98 X 92 i) ‘Nikhilam’ sutra 98 -2 x 92 -8 ______________ 90 / 16 = 9016 ii) ‘Antyayordasakepi’ and ‘Ekadhikena’ sutras. 98 X 92 Last digit sum= 8+2 =10 remaining digit (s) = 9 same sutras work.  98 X 92 = 9 X ( 9 + 1 ) / 8X2 =
90/16 = 9016. iii) urdhava-tiryak sutra 98 x 92 _______ 106 891 _______ 9016 vi) by vinculum method _ 98 = 100 – 2 = 102 _ 92 = 100 – 8 = 108 now _ 102 _ 108 ______ _ 10006 _ 1 1 _______ __ 11016 = 9016 Example 3: 493 X 497. 1) ‘Nikhilam’ Method and ‘Anurupyena’: a) Working base is 500, treated as 5 X 100  b) Working base is 500, treated as 1000 / 2 493 -7 497 -3 _________ 2) 490 / 021 _________ 245 / 021 = 245021 2) ‘Urdhva tiryak’ sutra.  3) Since end digits sum is 3+7 = 10 and remaining part 49 is same in both the numbers, ‘antyayordasakepi’ is applicable. Further Ekadhikena Sutra is also applicable. Thus 493 x 497 = 49 x 50 / 3x7 = 2450 / 21 = 245021 4) With the use of vinculum. _ 493 = 500 – 07 = 507 _ 497 = 500 – 03 = 503. _ _ Now 497 x 497 can be taken as 507 x 503 _ 507 _ x 503 ______ _ 50001 _ 252 _______ __ 255021 = 245021 Example 4: 99 X 99 1) Now by urdhva - tiryak sutra. 99 X 99 _______ 8121 168 _______ 9801 2) By vinculum method _ 99 = 100 - 1 = 101 Now 99 X 99 is _ 101 _ x 101 ______ _ 10201 = 9801 3) By Nikhilam method 99 -1 99 -1 _________ 98 / 01 = 9801. 4) ‘Yadunam’ sutra : 992 Base = 100 Deficiency is 1 : It indicates 992 = (99 – 1) / 12 = 98 / 01 = 9801. In the above examples we have observed how in more than one way problems can be solved and also the variety. You can have your own choice for doing multiplication. Not only that which method suits well for easier and quicker calculations. Thus the element of choice, divergent thinking, insight into properties and patterns in numbers, natural way of developing an idea, resourcefulness play major role in Vedic Mathematics methods. |
