# 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

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

This question from for the Intermediate age group of the 3^{rd} 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**.

- Subtraction
**8**from**7946**gives**7938.** - Multiplying the difference by 10,000 would result into
**79,380,000** - 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**