"India was the motherland of our race and Sanskrit the mother of Europe's languages. India was the mother of our philosophy, of much of our mathematics, of the ideals embodied in Christianity...of self-government and democracy. In many ways, Mother India is the mother of us all." - Will Durant, American Historian 1885-1981
What is an ancient system of mathematics that is being taught in some of the most prestigious institutions in England and Europe and not in India? It is Vedic mathematics-a long forgotten technique for mathematical calculations! It is amazing how with the help of 16 sutras and 16 upa-sutras you will be able to solve/calculate complex mathematical problems-mentally! The basic roots of Vedic mathematics lie in Vedas jus as basic roots of Hinduism. Vedic Maths form part of Jyothisha which is one of the six Vedangas. To many Indians Vedic and Sanskrit slokas/manthras are relevant only for religious purposes/occasions. But Vedas (written around 1500-900 BCE) in fact are a treasure house of knowledge and human experience-both secular and spiritual. Here you will will get an idea about the power of Vedic Mathematics.
Vedic Number Representation
Vedic knowledge is in the form of slokas or poems in Sanskrit verse. A number was encoded using consonant groups of the Sanskrit alphabet, and vowels were provided as additional latitude to the author in poetic composition. The coding key is given as Kaadi nav, taadi nav, paadi panchak, yaadashtak ta ksha shunyam Translated as below ·
Varnmala
ka kha ga gha gna
cha chha ja jha inja
ta tha da dha na
pa pha ba bha ma
ya ra la va sha
sha sa ha chjha tra gna
letter "ka" and the following eight letters
letter "ta" and the following eight letters
letter "pa" and the following four letters
letter "ya" and the following seven letters, and
letter "ksha" for zero.
In other words,
ka, ta, pa, ya = 1
kha, tha, pha, ra = 2
ga, da, ba, la = 3
gha, dha, bha, va = 4
gna, na, ma, scha = 5
cha, ta, sha = 6
chha, tha, sa = 7
ja, da, ha = 8
jha, dha = 9
ksha = 0
Thus pa pa is 11, ma ra is 52. Words kapa, tapa , papa, and yapa all mean the same that is 11.
from : http://www.sanalnair.org/articles/index-ved.htm
Sixteen Simple Mathematical Sutras (Phrases) From The Vedas
Sixteen Sutras and Their Corollaries
| | Sutras or Formulae | Sub-Sutras or Corollaries |
| | Ekadhikena Purvena (One more than the previous) P-2 Division eg. 1/19 | Anurupyena (Proportionately) P-20, 87 Multiplications |
| | Nikhilam Navatascaram Dashtah (All from 9 last from 10) P-13 Multiplication eg 9927 X 9999 | Sishyate Seshasanjnah |
| | Urdhva-tiryaghyam P- (Vertically and cross-wise) | Adhyam-adhyena antyam-antyena P-87 (First by the first and last by the last) |
| | Paravartya Yojayet P- (Transpose and apply) | Kevalaih Saptakam Gunyat |
| | Sunyam Samya samuccaye P-107 (when Samuccaya is the same, that Samuccaya is zero) | Veshtanam |
| | (Anurupye) Sunyamanyat | Yavadunam Tavadunam |
| | Sankalana-vyavakalana | Yavadunam Tavadunikrtya Vargamca Yojayet |
| | Purna purnabhyam | Antyayor Dasakepi (also for two numbers whose last digits together total 10) |
| | Calana-Kalanabhyam | Antyayoreva |
| | Yavadunam | Samuccayagunitah |
| | Vyashti samashti | Lopana Sthapanabhyam "by (alternate) Elimination and Retention" very useful in HCF, Solid Geometry, Coordinate Geometry, |
| | Seshanyankena caramena | Vikolanam |
| | Sopantya dvayamantyam | Gunita samuccayah Samucchaya gunitah P-89 *13 |
| | Ekanyunena purvena (one less than the previous) P-35 one of the multiplier-digits is all 9 | |
| | Gunita Samuccayah P- | |
16. | Gunaka Samuccayah | |
A few examples :
1st Example - 1 Divided by 19, 29, 39, …. 129 etc
- To Divide 1 by numbers ending in 9 like 1 divided by 19, 29, 39, ….. 119 etc.
Some of these numbers like 19, 29, 59 are prime numbers and so cannot be factorised and division becomes all the more difficult and runs into many pages in the present conventional method and the chances of making mistakes are many.
The Vedic Solution is obtained by applying the Sutra (theorem) Ekadhikena Purvena which when translated means "One more than the Previous"
Take for example 1 divided by 19. In the divisor 19, the previous is 1 and the factor is obtained by adding 1 to it which is 2. Similarly when we have to divide by 29, 39, … 119 the factors shall be 3,4,… 12 respectively. (Add 1 to the previous term in the divisor). After this divide 1 by the factor in a typical Vedic way and the answer is obtained in 1 step. Thus
1 divided by 19 = 0.0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
1 divided by 29 = 0.0 3 4 4 8 2 7 5 8 6 2 0 6 8 9 6 5 5 1 7 2 4 1 3 7 9 3 1
2nd Example - Square of Numbers ending in 5
Squares of 25, 35, 45, 85, 95, 105, 195 etc can be worked out mentally
Again the Sutra used here is Ekadhikena Purvena which means, "One more than the previous."
The last term is always 5 and the Previous terms are 2, 3, 4, 8, 9, 10, 19 etc and we have to add 1 to them. Square of the last term 5 is always 25.
Thus the Square of 25 is 2x3/25 = 625
the Square of 35 is 3x4/25 = 1225
the Square of 45 is 4x5/25 = 2025
the Square of 85 is 8x9/25 = 7225
the Square of 95, 105, 195 can be obtained in the same way.
Use the above formula to find the products of
23 multiplied by 27; 44 multiplied by 46; 192 multiplied by 198 and so on.
3rd Example - Multiplier-digits consist entirely of nines
The Sutra (theorem) used here is Ekanyunena Purvena, sound as if it were the converse of Ekadhik Sutra ie "one less"
777 multiplied by 999 = 776,223
(776 is one less than multiplicand 777 and 223 is the compliment of 776 from 9)
120 35 79 multiplied by 999 99 99 = 120 35 78, 879 64 21
1234 5678 09 multiplied 9999 9999 99 = 1234 5678 08 8765 4321 91
Such multiplications come up in advanced astronomy.
4th Example General Multiplication of any number by any number
The Sutra used is Urdhva-Tiryagbhyam which means, "Vertically and cross-wise" (P-39)
To multiply 12 by 13 mentally multiply
1 by 1, 3 by 1 and 2 by 1 and finally 2 by 3 and write the answer as 1 56
To multiply 37 by 42 mentally multiply
3 by 4, 3 by 2 and 7 by 4 and finally 7 by 2 and the answer is 12/34/14 = 1554
To multiply 1021 by 2103 mentally multiply as follows
1 by2 1by1+0by2 1by0+2by2+1by0 1by3+0by0+2by1+1by2 0by3+2by0+1by1 2by3+1by0 1by3 = 2147163
Multiplying 8 7 2 6 5 by 3 2 1 1 7 gives 2 8 0 2 6 9 0 0 0 5
5th Example Algebraic Divisions
Divide (12X2 –8X-32) by (X-2) using Urdhva-Tiryak, (Vertically and Cross-wise)
Just by observation we can say the quotient must be (12X+k)
We also know that –8x = kx - 24x Hence k = 16 cross-check –2k = -32 third term
Divide (X3 + 7X2 +6X +5) by (X-2)
X3 divided by X gives X2 which is therefore the 1st term of the equation.
X2 multiplied by –2 gives -2x2, But we have 7x2 in the Dividend. This means we have to get 9X2 more. This must result from the multiplication of X by 9X. Hence the 2nd term of the divisor must be 9X. Hence the quotient Q= X2+ 9x + k…
For the third term we already have 6X = kX – 18X Hence k = 24
To find the Remainder R: 2k+5 = 53
6th Example Division
Sutra: Paravartya Yojayet "Transpose and Apply" (P-63)
Divide (12x2 – 8x – 32) by (x - 2), factor is +2
(X – 2)/2 12x2 - 8x -32+24x + 32
--------------------------
12x + 16 R = 0
Divide 7x2 + 5x + 3 by x – 1 Q = 7x + 12 R = 15
Divide 7x2 + 5x + 3 by x + 1 Q = 7x –2 R = 5
Divide x3 + 7x2 + 6x + 5 by x – 2 Q = x2 + 9x + 24 R = 53
Divide x4 – x3 + x2 + 3x +5 by x2 – x – 1 Factors are x + 1
Q = x2 + 0x + 2 R = 5x + 7
Divide 6x4 + 13 x3 + 39 x2 + 37x + 45 by X2 – 2x – 9
Factors 2x + 9 Q = 6x2 + 25x + 143 R = 548x + 1332
Divide x4 + x2 + 1 by x2 – x +1 (add 0x3 & 0x)
X4 + 0x3 + x2 + 0x + 1 Factors are: x - 1
Q = x2 + x + 1 R = 0
Divide 1 2 3 4 by 1 1 2 Factors -1 –2 Q = 11 R = 2
Divide 1 3 4 5 6 by 1 1 2 3 Q = 12 R = - 20
The Reminder cannot be negative. Hence Q = 11 R = 1 1 0 3
Divide 1 3 9 0 5 by 1 1 3 Q = 1 24 R = -107
Q = 123 R = 6
7th Example Factorisations of Quadratics
Sutras: Anurupyena (P-84) "Proportionately"
Adhyam-Adhyena, Antyam-antyena (P-84) "first by the first and the last by the last"
8th Example - Verifying Correctness of answers
A Sub-sutra of immense utility for the purpose of verifying the correctness of our answers in multiplications, divisions and factorisations:
Gunita-Samuchhayah Samuchhaya-gunitah (P-86) means
"The product of the sum of the coefficients in the factors is equal to
the sum of the coefficients in the product"
Sc of the product = Product of the Sc in the factors
For example (x+7) (x+9) = (x2 + 16x + 63)
(1+7) (1+9) = (1 + 16 + 63) = 80
or (x+1) (x+2) (x+3) = (x3 + 6X2 + 11x + 6)
(1+1) (1+2) (1+3) = (1 + 6 + 11 + 6) = 24
9th Example Factorisations of Harder Quadratics
Lopana-stapana-bhyam (P-87) "by (alternate) Elimination and Retention"
It is very difficult to factorise the long quadratic (2x2 + 6y2 + 3z2 + 7xy + 11yz + 7zx)
But "Lopana-Sthapana" removes the difficulty. Eliminate z by putting z = 0.
Hence the given expression E = 2x2 + 6y2 + 7xy = (x+2y) (2x+3y)
Similarly, if y=0, then E = 2x2 + 3z2 + 7zx = (x+3z) (2x+z)
Hence E = (x+2y+3z) (2x+3y+z)
Factorise 2x2 + 2y2 + 5xy + 2x- 5y –12 = (x+3) (2x-4) and (2y+3) (y-4)
Hence, E = (x+2y+3) (2x+y-4)
* This "Lopana-sthapana" method (of alternate elimination and retention) will be found highly useful in HCF, in Solid Geometry and in Co-ordinate Geometry of the straight line, the Hyperbola, the conjugate Hyperbola, the Asymptotes etc.
10th Example - Factorisations of Harder Quadratics – Special Cases
Sunyam Samya samuccaye P-107 (when Samuccaya is the same, that Samuccaya is zero) Samuccaya is a technical term which has several meanings.
First Meaning: It is a term which occurs as a common factor in all the terms concerned
Thus 12x + 3x = 4x + 5x x is common, hence x = 0
Or 9 (x+1) = 7 (x+1) here (x+1) is common; hence x +1= 0
Second Meaning: Here Samuccaya means "the product of the independent terms"
Thus, (x +7) (x +9) = (x +3) (x +21)
Here 7 x9 = 3 x 21. Therefore x = 0
Third Meaning Samuccaya thirdly means the sum of the Denominators of two fractions
having the same numerical numerator
Thus, 1/(2x –1) + 1/(3x –1) = 0 Hence 5x – 2 =0 or x = 2/5
Fourth Meaning: Here Samuccaya means combination (or TOTAL).
In this context it is used in different contexts. These are
If the sum of the Numerators and the sum of the Denominators be the same, then that sum = 0
(2x +9)/ (2x +7) = (2x +7)/ (2x +9)
N1 + N2 = D1 + D2 = 2x + 9 + 2x + 7 = 0
Hence 4x + 16 = 0 hence x = -4
Note: If there is a numerical factor in the algebraic sum, that factor should be removed.
(3x +4)/ (6x +7) = (x +1)/ (2x +3)
Here N1 +N2 = 4x +5; D1 +D2 = 8x + 10; 4x +5 =0 x= -5/4
Fifth Meaning: Here Samuccaya means TOTAL ie Addition & subtraction
Thus, (3x +4)/ (6x +7) = (5x +6)/ (2x +3)
Here N1+N2 = D1 + D2 = 8x + 10 =0 hence x = - 5/4
D1 – D2 = N2 – N1 = 2x + 2 = 0 x = -1
Sixth Meaning: Here Samuccaya means TOTAL; used in Harder equations
Thus, 1/ (x-7) + 1/(x-9) = 1/(x-6) + 1/(x-10)
Vedic Sutra says, (other elements being equal), the sum-total of the denominators on LHS and the total on the RHS are the same, then the total is zero.
Here, D1 + D2 = D3 + D4 = 2x-16 =0 hence x = 8
Examples 1/(x+7) + 1/(x+9) = 1/(x+6) + 1/(x+10) x = - 8
1/(x-7) + 1(x+9) = 1/(x+11) + 1/(x-9) x = - 1
1/(x-8) + 1/(x-9) = 1/(x-5) + 1/(x-12) x = 8-1/2
1/(x-b) - 1/(x-b-d) = 1/(x-c+d) - 1/(x-c) x = 1/2(b+c)
Special Types of seeming Cubics (x- 3)3 + (x –9)3 = 2(x –6)3
current method is very lengthy, but Vedic method says, (x-3) + (x-9) = 2x – 12 Hence x = 6
(x-149)3 + (x-51)3 = 2(x-100)3 Hence 2x-200 =0 & x = 100
(x+a+b-c)3 + (x+b+c-a)3 = 2(x+b)3 x = -b
source: http://www.ourkarnataka.com/vedicm/vedicms.htm#_Toc450379902
Interactive Vedic Maths site : http://www.mathresource.iitb.ac.in/VedicMaths/VedicMathsSutras.html
