Vedic Mathematics/Sutras/Nikhilam Navatashcaramam Dashatah

Nikhilam Sutra stipulates subtraction of a number from the nearest power of 10 ie 10, 100, 1000, etc. The powers of 10 from which the difference is calculated is called the Base. These numbers are considered to be references to find out whether given number is less or more than the Base.

If the given number is 104, the nearest power of 10 is 100 and is the base. Hence the difference between the base and the number is 4, which is Positive and it is called NIKHILAM. The value of Nikhilam may be reference base, the Nikhilam of 87 is -13 and that of 113 is +13 respectively.

Nikhilam Sutra in Vedic Mathematics can be used as shortcuts to multiply numbers, divide numbers in faster approach. In English it is translated as All from 9 and last from 10. i.e. subtract last digit from 10 and rest of digits from 9.

Multiplication using Nikhilam Sutra is used when numbers are closer to power of 10 i.e. 10, 100, 1000, etc.

Case 1
Digits of multiplicand and multiplier (number with 9’s) are same. Columns of answer:
 * 1st column: 1 less of multiplicand
 * 2nd column: complement of above column or (Multiplier – previous column).

Case 2
Multiplier (number with 9’s) has digits less than multiplicand. Columns of answer:
 * 1st Column: If multiplicand starts with 1, subtract 2 from it. If multiplicand starts with 2, subtract 3 from it and so on.
 * 2nd Column: Take compliment of last digit of multiplicand.

Case 3
Multiplier (number with 9’s) has digits more than multiplicand. Columns of answer:
 * 1st Column: 1 less than Multiplicand
 * 2nd column: all 9’s
 * 3rd Column: 10’s compliment of multiplicand or (9’s complement of 1st columns). column: all 9’s