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NIKHILAM NAVATAS?CARAMAM DASATAH

The formula simply means : ?all from 9 and the last from 10?

The formula can be very effectively applied in multiplication of numbers, which are nearer to bases like 10, 100, 1000 i.e., to the powers of 10 . The procedure of multiplication using the Nikhilam involves minimum number of steps, space, time saving and only mental calculation. The numbers taken can be either less or more than the base considered.

The difference between the number and the base is termed as deviation. Deviation may be positive or negative. Positive deviation is written without the positive sign and the negative deviation, is written using Rekhank (a bar on the number). Now observe the following table.

Number	Base	Number ? Base	Deviation

14	10	14 - 10	4
			_
8	10	8 - 10	-2 or 2
			__
97	100	97 - 100	-03 or 03

112	100	112 - 100	12
			___
993	1000	993 - 1000	-007 or 007

1011	1000	1011 - 1000	011

Some rules of the method (near to the base) in Multiplication

a) Since deviation is obtained by Nikhilam sutra we call the method as Nikhilam multiplication.

Eg : 94. Now deviation can be obtained by ?all from 9 and the last from 10? sutra i.e., the last digit 4 is from 10 and remaining digit 9 from 9 gives 06.

b) The two numbers under consideration are written one below the other. The deviations are written on the right hand side.

Eg : Multiply 7 by 8.

     Now the base is 10. Since it is near to both the numbers,
                                                                                            7
     we write the numbers one below the other.                          8
                                                                                          -----
      Take the deviations of both the numbers from
      the base and represent                                                        _
                                                                                          7   3
                                                                                               _
      Rekhank or the minus sign before the deviations               8    2
                                                                                         ------
                                                                                         ------

                                                                                     or 7 -3
                                                                                          8 -2
                                                                                        -------
                                                                                        -------

or remainders 3 and 2 implies that the numbers to be multiplied are both less than 10

c) The product or answer will have two parts, one on the left side and the other on the right. A vertical or a slant line i.e., a slash may be drawn for the demarcation of the two parts i.e.,

(or)

d) The R.H.S. of the answer is the product of the deviations of the numbers. It shall contain the number of digits equal to number of zeroes in the base.

                                  _
                    i.e., 7    3
                                  _
                            8    2
                    _____________
                                / (3x2) = 6

Since base is 10, 6 can be taken as it is.

e) L.H.S of the answer is the sum of one number with the deviation of the other. It can be arrived at in any one of the four ways.

i) Cross-subtract deviation 2 on the second row from the original number 7 in the first row i.e., 7-2 = 5.

ii) Cross?subtract deviation 3 on the first row from the original number8 in the second row (converse way of (i))
i.e., 8 - 3 = 5

iii) Subtract the base 10 from the sum of the given numbers.
i.e., (7 + 8) ? 10 = 5

iv) Subtract the sum of the two deviations from the base.
i.e., 10 ? ( 3 + 2) = 5

Hence 5 is left hand side of the answer.

                                              _
                            Thus     7    3
                                              _
                                        8    2
                                ¯¯¯¯¯¯¯¯¯¯¯¯
                                        5 /

            Now (d) and (e) together give the solution
                                  _
                            7    3                7
                                  _
                            8    2    i.e., X 8
                          ¯¯¯¯¯¯¯         ¯¯¯¯¯¯
                            5 / 6               56

f) If R.H.S. contains less number of digits than the number of zeros in the base, the remaining digits are filled up by giving zero or zeroes on the left side of the R.H.S. If the number of digits are more than the number of zeroes in the base, the excess digit or digits are to be added to L.H.S of the answer.

The general form of the multiplication under Nikhilam can be shown as follows :

Let N₁ and N₂ be two numbers near to a given base in powers of 10, and D₁ and D₂ are their respective deviations from the base. Then N₁ X N₂ can be represented as

Case (i) : Both the numbers are lower than the base. We have already considered the example 7 x 8 , with base 10.

Now let us solve some more examples by taking bases 100 and 1000 respectively.

Ex. 1: Find 97 X 94. Here base is 100. Now following the rules, the working is as follows:

Ex. 2: 98 X 97 Base is 100.

Ex. 3: 75X95. Base is 100.

Ex. 4: 986 X 989. Base is 1000.

Ex. 5: 994X988. Base is 1000.

Ex. 6: 750X995.

Case ( ii) : Both the numbers are higher than the base.

The method and rules follow as they are. The only difference is the positive deviation. Instead of cross ? subtract, we follow cross ? add.

Ex. 7: 13X12. Base is 10

Ex. 8: 18X14. Base is 10.

Ex. 9: 104X102. Base is 100.

                               104    04
                               102    02
                            ¯¯¯¯¯¯¯¯¯¯¯¯
                               106 / 4x2    =    10608      ( rule - f )
                            ¯¯¯¯¯¯¯¯¯¯¯¯

Ex. 10: 1275X1004. Base is 1000.

                              1275   275
                              1004   004
                            ¯¯¯¯¯¯¯¯¯¯¯¯
                            1279 / 275x4   =    1279 / ₁100     ( rule - f )
                            ____________    =    1280100

Case ( iii ): One number is more and the other is less than the base.

In this situation one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore have to be subtracted. To have a clear representation and understanding a vinculum is used. It proceeds into normalization.

Ex.11: 13X7. Base is 10

Note : Conversion of common number into vinculum number and vice versa.

Eg :
                                            __
                      9 = 10 ? 1    = 11
                                              _
                  98 = 100 ? 2 = 102
                                              _
                196 = 200 ? 4 = 204
                    _
                  32 = 30 ? 2    = 28
                    _
                145 = 140 ? 5 = 135
                    _
                322 = 300 ? 22 = 278. etc

The procedure can be explained in detail using Nikhilam Navatascaram Dasatah, Ekadhikena purvena, Ekanyunena purvena in the foregoing pages of this book.]

Ex. 12: 108 X 94. Base is 100.

Ex. 13: 998 X 1025. Base is 1000.

Algebraic Proof:

Case ( i ):

Let the two numbers N1 and N2 be less than the selected base say x.

N₁ = (x-a), N₂ = (x-b). Here a and b are the corresponding deviations of the numbers N₁ and N₂ from the base x. Observe that x is a multiple of 10.

Now N₁ X N₂ = (x-a) (x-b) = x.x ? x.b ? a.x + ab

= x (x ? a ? b ) + ab. [rule ? e(iv), d ]

= x [(x ? a) ? b] + ab = x (N₁?b) + ab[rule?e(i),d]

or = x [(x ? b) ? a] = x (N₂ ? a) + ab. [rule ?e (ii),d]

x (x ? a ? b) + ab can also be written as

x[(x ? a) + (x ? b) ? x] + ab = x[N₁+N₂ ? x] + ab [rule ? e(iii),d].

A difficult can be faced, if the vertical multiplication of the deficit digits or deviations i.e., a.b yields a product consisting of more than the required digits. Then rule-f will enable us to surmount the difficulty.

Case ( ii ) :

When both the numbers exceed the selected base, we have N₁ = x + a, N₂ = x + b, x being the base. Now the identity (x+a) (x+b) = x(x+a+b) + a.b holds good, of course with relevant details mentioned in case (i).

Case ( iii ) :

When one number is less and another is more than the base, we can use (x-a)(x+b) = x(x?a+ b)?ab. and the procedure is evident from the examples given.

          Find the following products by Nikhilam formula.

          1) 7 X 4                 2) 93 X 85             3) 875 X 994

         4) 1234 X 1002     5) 1003 X 997        6) 11112 X 9998

         7) 1234 X 1002     8) 118 X 105

Nikhilam in Division

Consider some two digit numbers (dividends) and same divisor 9. Observe the following example.

i) 13 ÷ 9 The quotient (Q) is 1, Remainder (R) is 4.

                            since 9 ) 13 ( 1

                                                 9
                                              ____
                                                 4

ii) 34 ÷ 9, Q is 3, R is 7.

iii) 60 ÷ 9, Q is 6, R is 6.

iv) 80 ÷ 9, Q is 8, R is 8.

Now we have another type of representation for the above examples as given hereunder:

i) Split each dividend into a left hand part for the Quotient and right - hand part for the remainder by a slant line or slash.

Eg. 13 as 1 / 3, 34 as 3 / 4 , 80 as 8 / 0.

ii) Leave some space below such representation, draw a horizontal line.

Eg. 1 / 3 3 / 4 8 / 0

______ , ______ , ______

iii) Put the first digit of the dividend as it is under the horizontal line. Put the same digit under the right hand part for the remainder, add the two and place the sum i.e., sum of the digits of the numbers as the remainder.

Eg.
                      1 / 3         3 / 4         8 / 0
                           1              3              8
                     ______ ,   ______ ,  ______
                      1 / 4         3 / 7         8 / 8

Now the problem is over. i.e.,

13 ÷ 9 gives Q = 1, R = 4

34 ÷ 9 gives Q = 3, R = 7

80 ÷ 9 gives Q = 8, R = 8

Proceeding for some more of the two digit number division by 9, we get

             a) 21 ÷ 9  as
                                9) 2 / 1     i.e     Q=2, R=3
                                            2
                                     ¯¯¯¯¯¯
                                     2 / 3

             b) 43 ÷ 9  as
                                9) 4 / 3     i.e     Q = 4, R = 7.
                                            4
                                     ¯¯¯¯¯¯
                                     4 / 7

The examples given so far convey that in the division of two digit numbers by 9, we can mechanically take the first digit down for the quotient ? column and that, by adding the quotient to the second digit, we can get the remainder.

Now in the case of 3 digit numbers, let us proceed as follows.

i)
                    9 ) 104 ( 11                 9 ) 10   / 4
                             99                                1 / 1
                         ¯¯¯¯¯¯            as            ¯¯¯¯¯¯¯
                             5                                 11 / 5

ii)
                    9 ) 212 ( 23                 9 ) 21 / 2
                           207                               2 / 3
                          ¯¯¯¯¯              as            ¯¯¯¯¯¯¯
                              5                                23 / 5

iii)
                    9 ) 401 ( 44                 9 ) 40 / 1
                           396                                 4 / 4
                         ¯¯¯¯¯             as            ¯¯¯¯¯¯¯¯
                             5                                 44 / 5

Note that the remainder is the sum of the digits of the dividend. The first digit of the dividend from left is added mechanically to the second digit of the dividend to obtain the second digit of the quotient. This digit added to the third digit sets the remainder. The first digit of the dividend remains as the first digit of the quotient.

Consider 511 ÷ 9

Add the first digit 5 to second digit 1 getting 5 + 1 = 6. Hence Quotient is 56. Now second digit of 56 i.e., 6 is added to third digit 1 of dividend to get the remainder i.e., 1 + 6 = 7

Thus
                    9 )     51 / 1
                               5 / 6
                            ¯¯¯¯¯¯¯
                            56 / 7

Q is 56, R is 7.

Extending the same principle even to bigger numbers of still more digits, we can get the results.

Eg : 1204 ÷ 9

i) Add first digit 1 to the second digit 2. 1 + 2 = 3

ii) Add the second digit of quotient 13. i.e., 3 to third digit ?0? and obtain the Quotient. 3 + 0 = 3, 133

iii) Add the third digit of Quotient 133 i.e.,3 to last digit ?4? of the dividend and write the final Quotient and Remainder. R = 3 + 4 = 7, Q = 133

   In symbolic form         9 ) 120 / 4
                                                13 / 3
                                             ¯¯¯¯¯¯¯¯
                                           133 / 7

Another example.
                                    9 ) 13210   / 1          132101 ÷ 9
                gives
                                          1467   / 7          Q = 14677, R = 8
                                         ¯¯¯¯¯¯¯¯¯¯
                                          14677 / 8

In all the cases mentioned above, the remainder is less than the divisor. What about the case when the remainder is equal or greater than the divisor?

Eg.
                    9 ) 3 / 6              9)   24 / 6
                                3                      2 /   6
                         ¯¯¯¯¯¯        or        ¯¯¯¯¯¯¯¯
                        3 / 9 (equal)           26 / 12 (greater).

We proceed by re-dividing the remainder by 9, carrying over this Quotient to the quotient side and retaining the final remainder in the remainder side.

              9 ) 3 / 6                  9 ) 24 / 6
                            / 3                         2 / 6
                        ¯¯¯¯¯¯¯                   ¯¯¯¯¯¯¯¯
                        3 / 9                       26 / 12
                        ¯¯¯¯¯¯¯                   ¯¯¯¯¯¯¯¯
                        4 / 0                       27 /   3

Q = 4, R = 0 Q = 27, R = 3.

When the remainder is greater than divisor, it can also be represented as

                            9 ) 24 / 6
                                        2 / 6
                                    ¯¯¯¯¯¯¯¯
                                    26 /1 / 2
                                              / 1
                                     ¯¯¯¯¯¯¯¯
                                        1    / 3
                                     ¯¯¯¯¯¯¯¯
                                      27    / 3

Now consider the divisors of two or more digits whose last digit is 9,when divisor is 89.

We Know 113 = 1 X 89 + 24, Q =1, R = 24

10015 = 112 X 89 + 47, Q = 112, R = 47.

Representing in the previous form of procedure, we have

                89 ) 1 / 13                 89 ) 100 / 15
                                /   11                        12 / 32
                            ¯¯¯¯¯¯¯                    ¯¯¯¯¯¯¯¯¯¯
                            1 / 24                        112 / 47

But how to get these? What is the procedure?

Now Nikhilam rule comes to rescue us. The nikhilam states ?all from 9 and the last from 10?. Now if you want to find 113 ÷ 89, 10015 ÷ 89, you have to apply nikhilam formula on 89 and get the complement 11.Further while carrying the added numbers to the place below the next digit, we have to multiply by this 11.

           89 ) 1 / 13         89 ) 100 / 15
                                           ¯¯
                           / 11         11      11 /            first digit 1 x 11
                       ¯¯¯¯¯¯¯¯
                        1 / 24                    1 / 1      total second is 1x11
                                                             22    total of 3^rd digit is 2 x 11
                                                 ¯¯¯¯¯¯¯¯¯¯
                                                  112 / 47

What is 10015 ÷ 98 ? Apply Nikhilam and get 100 ? 98 = 02. Set off the 2 digits from the right as the remainder consists of 2 digits. While carrying the added numbers to the place below the next digit, multiply by 02.

Thus
                    98 )   100   / 15
                    ¯¯
                    02        02 /                 i.e., 10015 ÷ 98 gives
                                 0 /   0              Q = 102,     R = 19
                                    /    04
                             ¯¯¯¯¯¯¯¯¯¯
                             102   / 19

In the same way
                            897 ) 11 / 422
                            ¯¯¯
                            103      1 / 03
                                          /   206
                                     ¯¯¯¯¯¯¯¯¯
                                      12 / 658

gives 11,422 ÷ 897, Q = 12, R=658.

In this way we have to multiply the quotient by 2 in the case of 8, by 3 in the case of 7, by 4 in the case of 6 and so on. i.e., multiply the Quotient digit by the divisors complement from 10. In case of more digited numbers we apply Nikhilam and proceed. Any how, this method is highly useful and effective for division when the numbers are near to bases of 10.

* Guess the logic in the process of division by 9.

* Obtain the Quotient and Remainder for the following problems.

          1) 311 ÷ 9               2) 120012 ÷ 9            3) 1135 ÷ 97

         4) 2342 ÷ 98           5) 113401 ÷ 997

         6) 11199171 ÷ 99979

Observe that by nikhilam process of division, even lengthier divisions involve no division or no subtraction but only a few multiplications of single digits with small numbers and a simple addition. But we know fairly well that only a special type of cases are being dealt and hence many questions about various other types of problems arise. The answer lies in Vedic Methods.

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