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30MSC 2023 #12: Nikhilam Multiplication

30MSC 2023 #12: Nikhilam Multiplication

Nikhilam or base multiplication is commonly used when the multiplicands are composed of big digits. This type of multiplication is usually difficult when using the traditional method because the computations will involve many “carries”. But the bigger the digits of the multiplicands are, the nearer the numbers are to the base. This means that their complements are smaller and thus easier to deal with.

Nikhilam multiplication uses the base and ten’s complements of the multiplicands in the computation. Note that in Vedic Math, a perfect base is a power of 10.

The multiplicands can be represented by (x – a) and (x – b) where x is the base and a and b are their respectivedeficiency orten’s complements. Their product,

(x – a)(x – b)= x^2 – ax – bx + ab = x(x – a – b) + ab.

From the result, we can easily see that the product is obtained by

  1. Subtracting the deficiency, b, of one multiplicand from the other multiplicand (x – a),
  2. Multiplying the difference by the base, x
  3. And adding the product of the deficiencies.

This question from for the Intermediate age group of the 3rd International Vedic Mathematics Olympiad (IVMO 2023)  shows how easy base multiplication is, compared to ordinary multiplication.

               7946 x 9992 =

Using Nikhilam (All from 9 and the last from 10), we can quickly determine their deficiencies from  their base, 10,000: 2054 for 7946 and 8 for 9992.

  1. Subtraction 8 from 7946 gives 7938.
  2. Multiplying the difference by 10,000 would result into 79,380,000
  3. The product of their deficiencies 2054 and 8 can be mentally computed as 16, 432 which when added to 79,380,000 will produce a total of 79,396,432

A simplified two-part Base multiplication technique as well as other base multiplication procedures are discussed in Chapter 2 of the forthcoming 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 #13: Vertically and Crosswise Multiplication

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