# VMO Tip: Nikhilam (base) Multiplication: Numbers above the base

Our first Vedic Mathematics Olympiad (VMO) tip is about NIkhilam or base Multiplication, a fun fast and easy way to do multiplication of numbers close to bases.

In VM powers of 10 are often used as base for easy computations. The difference of a number from the next higher power of 10 is called its **10’s complement** or **deficiency** while its difference from the lower power of 10 is called its **excess**.

The product of two numbers that are just above a common base is is composed of two parts. For our featured example 107 x 104,

- To get the first part,
**add the excess of one factor from the base to the other factor.**This could be either (107 + 4) or (104 + 7) which both will give 111. - To get the second part,
**multiply the excesses of the two factors from the base.**The number of digits allotted for the second part is equal to the number of zeroes of the base. Here it is 7 x 4 = 28. The product is**11,128.**

The algebraic proof for is method is as follows:

Let **x** = base or power of 10

**a, b** = excess from the base

**(x + a)** and **(x + b)** are the factors or multiplicands

**( x + a) ( x + b) = x2 + ax + bx + ab**

** = (x + a + b)x + ab**

** = [(x + a) + b]x + ab**

In the case of 102 x 104, we have for the first part (102 + 4) or (104 + 2) = 106 and for the second part, 4 x 2 = 8. But the base, 100 has two zeroes. So the second part must be expressed as 08 instead of 8. Thus 102 x 104 = 10,608.

Most of us have learned by heart the multiplication up to 10 x 10. With base multiplication we will be able to easily extend our multiplication power up to 20 x 20.

For example, 14 x 12 can be simply computed as 14+2| 4 x 2 = 16|8 or 168.

17 x 19 involves a “carry” operation but still very simple. 17 + 9 | 7 x 9 = 26 | 63. Here the base is 10 and has only 1 zero but the second part has 2 digits. We must “carry” the 6 over to the first part to get 323.

Exercise P-1 Find the following products mentally:

- 12 x 13; 2) 14 x 17; 3) 15 x 18; 4) 108 x 101; 5) 116 x 102; 6)108 x 112 ; 7) 112 x 114; 8) 123 x 106; 9) 1021 x 1006; 10) 1432 x 1002

(See answers in the next post)

- From Prudente, Virgilio,
*25 Math Short Cuts*, pp 61 -63.

Suggested readings:

Glover, James, *The Curious Hats of Magical Maths*, Book 1, pp 1 – 7

Williams, Kenneth, *Vedic Mathematics Teacher’s Manual (Intermediate Level)*p 33

IAVM, Inspirational Maths from India, pp 26-28.