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 :

(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.