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RULES OF VEDIC MATHEMATICS

COROLLARY
: Anurupyena
Meaning
: By one more than the previous one

2. Nikhilam Navatashcaramam Dashatah
COROLLARY
: Sisyate Sesasamjnah
Meaning: All from 9 and the last from 10

3. Urdhva-Tiryagbyham
COROLLARY
Meaning: Vertically and crosswise

4. Paraavartya Yojayet
COROLLARY
: Kevalaih Saptakam Gunyat

5. Shunyam Saamyasamuccaye
COROLLARY
: Vestanam
Meaning: When the sum is the same that sum is zero.

6. (Anurupye) Shunyamanyat
COROLLARY
: Vestanam
Meaning
: If one is in ratio, the other is zero

7. Sankalana-vyavakalanabhyam
COROLLARY
Meaning: By addition and by subtraction

8. Puranapuranabyham
COROLLARY
: Antyayordashake'pi
Meaning
: By the completion or non-completion

9. Chalana-Kalanabyham
COROLLARY
: Antyayoreva
Meaning: Differences and Similarities

COROLLARY
: Samuccayagunitah
Meaning
: Whatever the extent of its deficiency

11.Vyashtisamanstih
COROLLARY
: Lopanasthapanabhyam
Meaning: Part and Whole
12. Shesanyankena Charamena
COROLLARY
: Vilokanam
Meaning: The remainders by the last digit

COROLLARY:
Gunitasamuccayah Samuccayagunitah
Meaning: The ultimate and twice the penultimate

14. Ekanyunena Purvena
COROLLARY: Dhvajanka
Meaning: By one less than the previous one

15. Gunitasamuchyah
COROLLARY: Dwandwa Yoga
Meaning: The product of the sum is equal to the sum of the product

16. Gunakasamuchyah
Meaning
: The factors of the sum is equal to the sum of the factors

Few of the rules in detail

EKĀDHIKENA PŪRVEŅA

The Sutra (formula) means: “By one more than the previous one”.

Now let us apply this sutra to the ‘squaring of numbers ending in 5’.

Consider the example of 252.

Here the number is 25. We have to find out the square of the number. For the number 25, the last digit is 5 and the 'previous' digit is 2. Hence, 'one more than the previous one', that is, 2+1=3. The Sutra, in this context, gives the procedure 'to multiply the previous digit 2 by one more than itself, that is, by 3. It becomes the L.H.S (left hand side) of the result, that is,
2 X 3 = 6. The R.H.S (right hand side) of the result is 52, that is, 25.

NIKHILAM NAVATAS’CHARAMAM DASATAH

The formula simply means : “all from 9 and the last from 10
The formula can be very effectively applied in multiplication of numbers, which are nearer to bases like 10, 100, 1000 i.e., to the powers of 10(eg: 96 x 98 or 102 x 104). The procedure of multiplication using the Nikhilam involves minimum number of steps, space, time saving and only mental calculation. The numbers taken can be either less or more than the base considered.

URDHVA TIRYAGBHYAM

Urdhva – tiryagbhyam is the general formula applicable to all cases of multiplication and also in the division of a large number by another large number. It means “Vertically and cross wise.”

Now let us apply this sutra to “Multiplication of two 2 digit numbers”.

Ex. Find the product 14 X 12

SOL: Steps:

i) 4 X 6 = 24 : 2, the carried over digit is placed below the second digit.
ii) (3 X 6) + (4 x 1) = 18 + 4 = 22 ; 2, the carried over digit is placed below third digit.
iii) (2 X 6) + (3 X 1) + (4 X 3) = 12 + 3 + 12 = 27 ; 2, the carried over digit is placed below fourth digit.
iv) (2 X 1) + ( 3 X 3) = 2 + 9 = 11; 1, the carried over digit is placed below fifth digit.
v) ( 2 X 3 ) = 6.

This sutra means whatever the extent of its surplus, increment it still further to that very extent; and also set up the square of that surplus. This sutra is very useful in calculating the sqaures of numbers nearer(greater) to powers of 10.

For instance: in computing the square of 103 we go through the following steps:
1. The nearest power of 10 to 103 is 100.
2. Therefore, let us take 100 as our base.
3. Since 103 is 3 more than 100(base), we call 3 as the surplus.
4. Increase the given number further by an amount equal to the surplus.
i.e., perform ( 103 + 3 ) = 106. This is the left side of our answer!!.
5. On the right hand side put the square of the surplus, that is square of 3 = 09.
6. Append the results from step 4 and 5 to get the result.

ANTYAYOR DAŚAKE'PI

The Sutra simply means - numbers of which the last digits added up give 10.
This sutra is helpful in multiplying numbers whose last digits add up to 10(or powers of 10). The remaining digits of the numbers should be identical.

EXAMPLE:  57 X 53
See the end digits sum 7 + 3 = 10 ; then by the sutras Antyayor dasakepi and Ekadhikena we have the answer.
Ekadhikena to the remaining digits means, increment the remaining digits by 1 and multiply it with the same.
57 x 53 = ( 5 + 1 )x5 / 7x3 ( the '/' is just a seperator and not a division mark)&nbsp = 30 / 21
= 3021.