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.
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
493 -7
497 -3
_________
2) 490 / 021
_________
245 / 021 = 245021
2) ‘Urdhva tiryak’ sutra.
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.
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