Vedic Mathematics | Multiplication

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.

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