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30MSC 2023 #14: Squaring

30MSC 2023 #14: Squaring

Squaring any number or polynomial can be done using the traditional method of multiplication. However, in Vedic Maths, there are several techniques, some specific and 1 general, available to easily perform squaring. The general method of squaring is called the duplex method, which is based on the Vertically and Crosswise sutra.

The duplex, denoted as D of numbers are as follows:

  1. D(a) = a^2
  2. D(ab) = 2 x (a x b)
  3. D(abc) = 2 x (a x c) + b^2
  4. D(abcd) = 2 x (a x d + b x c)
  5. D(abcde) = 2 x (a x e + b x d) + c^2

In general, for numbers with an even number of digits, their duplex is twice the sum of the products of the digits equidistant from the center. For numbers with an odd number of digits, their duplex, is twice the sum of the products of the digits equidistant from the center plus the square of the middle digit.

The square of a number is the sum of its duplexes.

In determining the duplexes of polynomials, we simply replace the digits by the terms of polynomials in the formulae.

In our featured example which was taken in the 1st Math2Shine International Vedic Mathematics Competition(open division) the square of the polynomial (2x^2 – 3x + 4)  is computed as:

  1. D(2x^2) = (2x^2)^2 = 4x^4
  2. D(2x^2 – 3x) = 2 x (2x^2)(– 3x) = – 12 x^3
  3. D(2x^2 – 3x + 4) = 2 x (2x^2)(4) + (3x)^2 = 16x^2 + 9x^2 = 25x^2
  4. D( – 3x + 4) = 2 x (-3x)(4) – 24x
  5. D(4) = (4)^2 = 16

(2x^2 – 3x + 4)^2 = 4x^4 – 12x^3 +  25x^2 – 24x + 16

 Other squaring techniques are discussed in Chapter 4 of our forthcoming book 30 Master Strategies in Computing, Vol II.

Follow us on our Facebook page, MATH-Inic Philippines at https://www.facebook.com/MATHInicPhils to see tomorrow’s 30MSC 2023 #15: Base Division

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