Just as the complements of numbers below powers of ten are used in Nikhilam or base multiplication, their deficiencies from their bases are used in base division.
In cases where the divisor is a little more than the base, the Sutra Transpose and Apply is applied in base division.
Mastering base division is very important for young learners because the same method can be extended to Algebraic division even with trinomials and longer divisors.
Conventional division becomes hard when the divisor is composed of big digits. But big digits mean smaller complements. The technique in base division is not to use the divisor. Instead, we use its complement as multiplier.
The most simple example of these base divisions is division by 9. (See pp. 37 to 40 of our book, 25 Math Short Cuts which is included in our Christmas book bundles.)
Our featured example how this difficult division problem, 12,012 ÷ 89, can be solved easily by base division:
Here is the step-by-step solution:
- In the space for the divisor, place 89 and its complement, 11, underneath it. This complement is easily determined using the Sutra Nikhilam navatascaramam dasatah or All from 9 and the Last from 10”. We will use this complement as our multiplier.
- Separate the last two digits of the dividend by a remainder bar. The number of digits separated is equal to the number of zeroes in the base. In this case the base is 100.
- As in long division, we will consider the first four digits 120. Bring down the first digit of the dividend, 1, as the first digit of the quotient. This is because we have the same number of 89s as there are hundreds in the dividend.
- We then multiply this first quotient figure by 11 which is the remainder when 89 is taken from every hundred and place the product under the second and third digits of the dividend.
- The sum obtained in the second column (2 +1 = 3) will be the second figure in the answer.
- Write the product of this latest answer 3 and 11 which is 33 under the 3rd to the 4th columns of the dividend as shown.
- Add the figures in the 3rd column, 0 + 1 + 3 = 4 and write this as the 3rd (and last) answer figure.
- Again, we multiply this quotient figure by 11 and place the product 44 in the 4th and 5th columns as shown
- Add all the numbers to the right of the remainder bar – in the 4th and 5th columns. (1 + 3 + 4 = 8) and (4 + 4 = 8) to get the remainder of 88.
More examples of base division can be found on Chapter 4 of the Inspirational Maths from India: A Teacher’s Handbook.