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.
