30MSC 2023 #16: Transpose and Apply
Nikhilam division is used when the divisor is near but below a power of 10. When the divisor is slightly greater than a power of 10, the Vedic Sutra, “Transpose and Apply” is used instead. The process is similar to that of Nikhilam division where the complement of the divisor or its deficiency from the base is used as multiplier. In transpose and apply, the excess of the divisor over the base is transposed or converted into its additive inverse.
This method is important because it can be applied in polynomial division, even for quadratic or higher order divisors.
We will explain this technique using a problem given in the 1^{st} International Vedic Mathematics Olympiad (IVMO 2021), Senior group:
What are the first three terms in the series expansion for (2+ 3x –x^2)/(1+ x)(2 − 3x) ?
The common solution to this type of problem is separate the expression into partial fractions, expand the fractions, and later add the expansions together. Using the transpose and apply sutra will provide a quicker and easier solution:
 Multiply out the factors of the divisor – (1 + x)(2 – 3x) = (2 – x – 3x^2)
 Transpose all the terms of the divisor except the first one (note that we can use the coefficients only but for this example we would also show the x terms):
2 – x – 3x^2  2 + 3x – x^2
x + 3x^2 

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 Divide the first term of the dividend by the first term of the divisor and bring down the result as the first term of the quotient ( 2 ÷ 2 = 1). Then multiply it by the transposed figures and place the result below the dividend starting at the second column.
2 – x – 3x^2  2 + 3x – x^2
x + 3x^2  x +3x^2

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1
 Add the figures in the second column and divide the sum by 2 which is the first term of the divisor. 3x + x = 4x; 4x ÷ 2 = 2x. Bring down 2x as the second term of the quotient the multiply it by the transposed term. Write the product on the 3^{rd} row starting at the 3^{rd} column.
2 – x – 3x^2  2 + 3x – x^2
x + 3x^2  x +3x^2
 2x^2 + 6x^3
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 + 2x
 Add the figures on the 3^{rd} column and divide the sum by 2. ( – x^2 + 3x^2 + 2x^2) ÷ 2 = 2x^2.
The 1^{st} three figures of the quotient are 1 + 2x + 2x^2
More examples of transpose and apply division are shown on Chapter 6 of our forthcoming book, 30 Master Strategies in Computing Vol II.
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