Nikhilam or base multiplication is a very fast and easy way to get the product of two numbers near powers of 10. It got its name from the sutra Nikhilam Navatascaraman Dasatah meaning All from 9 and the Last from 10 because it uses the ten’s complement of the multiplicands in computing.
From our experience, the bigger the digits are in the multiplicands, the harder it is to perform multiplication when using the traditional right to left method. But then bigger digits implies smaller complements.
There are many cases where base multiplication can be applied. Here we will consider when one multiplicand is above and one is below the base.
It is hard to imagine trying to solve the example given, 1324 x 997, mentally but it is very easy with base multiplication.
The nearest base to the multiplicands is 1,000 and it is very easy to get the excess of 1324 over 1000 and the deficiency of 997 from 1000 which are 324 and 3 respectively.
We then place the multiplicands one on top of the other and their excess/deficiency together with their corresponding signs on the right side.
The answer will be in two parts:
The first part is obtained by cross addition: add the excess (or deficiency) to the other multiplicand. 324 + 997 or – 003 + 1324 will give the same result which is 1 321 .
The second part is the product of the excess 324 and the deficiency – 3 which is – 972. This is expressed in bar form in our illustration.
The final answer is the difference between 1,321,000 and 972 which is 1,320, 028.
While some may dismiss this technique as a “trick”, it can be easily proved using algebra or geometry. Please refer to page 68 of 25 Math Short Cuts for the algebraic proof.
Base multiplication with using powers of 10 as bases will be discussed by Ms. Heather Banagui, of Cordillera Regional Science High School on Dec 4 while Multiplication using different bases will be explained by Ms. Angel Prudente, center director of MATH-Inic San Pablo City on Dec 5 in the Inspirational Maths from India webinars. Both Ms. Banagui and Ms. Prudente are Advanced Diploma in Vedic Mathematics holders and life members of the IAVM.
An example of Nikhilam multiplication when both numbers below the base given below
8 ,7 8 9 x 9, 9 9 6 = ?
8, 7 8 9 – 1 2 1 1
9, 9 9 6 – 0 0 0 4
8 7 8 5 | 4 8 4 4 = 87, 854, 844
Here the deficiencies of the multiplicand from their base of 10,000 is easily determined using Nikhilam (All from 9 and the Last from 10). The first part of the answer is obtained by cross subtraction 8789 – 4 = 8785 and the second part by multiplication of the deficiencies 1211 x 4 = 4844. Combining the two parts will give an answer of 87, 854, 844.