Games - Vedic mathematics

Home > Listing Index > Games > Vedic mathematics

	Games - Vedic mathematics

For the actual mathematics of the Vedic period, see the articles on Sulba Sutras, Ancient Vedic weights and measures and Indian mathematics. Nor should this article be confused with so-called Vedic Science.

Vedic mathematics is a system of mental calculation developed by Shri Bharati Krishna Tirthaji in the middle 20th century which he claimed he had based on a lost appendix of Atharvaveda, an ancient text of the Indian teachings called Veda. It has some similarities to the Trachtenberg system in that it speeds up some arithmetic calculations. It claims to have applications to more advanced mathematics, such as calculus and linear algebra. The system was first published in the book Vedic Mathematics ISBN 8120801644 in 1965. The system has since been developed further and there have been several other books released.

Criticism

Critics have questioned whether this subject deserves the name Vedic or indeed mathematics. They point to the lack of evidence of any sutras from the Vedic period consistent with the system, the inconsistency between the topics addressed by the system (such as decimal fractions) and the known mathematics of early India, the substantial extrapolations from a few words of a sutra to complex arithmetic, and the restriction of applications to convenient cases. They further say that such arithmetic as is speeded up by application of the sutras can be performed on a computer or calculator anyway, making their knowledge rather irrelevant in the modern world.

They are also worried that it deflects attention from genuine achievements of ancient and modern Indian mathematics and mathematicians, and that its promotion by Hindu nationalists may damage mathematics education in India. Actual Indian mathematicians, at leading institutions such as TIFR or IISc, largely share these opinions.

The system is based upon sixteen formulas and their corollaries, some of which are described below.

All from nine and the last from ten

Corollary 1: Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square of that deficiency.

For instance, in computing the square of 9 we go through the following steps:

#The nearest power of 10 to 9 is 10. Therefore, let us take 10 as our base. #Since 9 is 1 less than 10, decrease it still further to 8. This is the left side of our answer. #On the right hand side put the square of the deficiency, that is 1². Hence the answer is 81. #Similarly, 8² = 64, 7² = 49. #For numbers above 10, instead of looking at the deficit we look at the surplus. For example: :

11^2 = (11+1)\cdot 10+1^2 = 121.\,

12^2 = (12+2)\cdot 10+2^2 = 144.\,

14^2 = (14+4)\cdot 10+4^2 = 18\cdot10+16 = 196.\,

:and so on. This is based on the identities

(a+b)(a-b)=a^2-b^2

and

(a+b)^2=a^2+2ab+b^2

By one more than the one before

The proposition "by" means the operations this formula concerns are either multiplication or division. Thus this formula is used for either multiplication or division. It turns out that it is applicable in both operations.

An interesting application of this formula is in computing squares of numbers ending in five. Consider:

: 35 × 35 = (3 × (3 + 1)),25 = 12,25

The latter portion is multiplied by itself (5 by 5) and the previous portion is multiplied by one more than itself (3 by 4) resulting in the answer 1225.

This is a simple application of

(a+b)^2=a^2+2ab+b^2

when

a=10c

and

b=5

, i.e. :

(10c+5)^2=100c^2+100c+25=100c(c+1)+25

.

It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Consider:

: 37 × 33 = (3 × 4),7 × 3 = 12,21 : 29 × 21 = (2 × 3),9 × 1 = 6,09

This uses

(a+b)(a-b)=a^2-b^2

twice combined with the previous result to produce: :

(10c+5+d)(10c+5-d)=(10c+5)^2-d^2=100c(c+1)+25-d^2=100c(c+1)+(5+d)(5-d)

.

We illustrate this formula by its application to conversion of fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.

Method 1: Using Multiplication

1/19, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating decimal. (If the denominator contains only factors 2 and 5 is a purely non-circulating decimal, else it is a mixture of the two.)

So we start with the last digit

: 1

Multiply this by "one more", that is, 2 (this is the "key" digit from Ekadhikena)

: 21

Multiplying 2 by 2, followed by multiplying 4 by 2

: 421 → 8421

Now, multiplying 8 by 2, sixteen

: 68421 : 1 ← carry

[ Visit the complete Wikipedia entry for Vedic mathematics ]

Searches on eBay

vedic mathematics