Nikhilam navatascaramam 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).


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.

Example :- 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.

Example :- 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 fromthe 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 linei.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 number7 in
the first row i.e., 7-2 = 5.
ii) Cross–subtract deviation 3 on the first row from the original number 8 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
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