This is a simplified and summarized version of – book written by the Indian Hindu Priest Bharati Krishna Tirthaji and first published in 1965. The Vedic maths contains 16 sutras (formulae) and 13 sub-sutras.

__The 16 Sutras of Vedic Maths__

**1. Ekadhikena Purvena - By one more than the previous one**

For any integer ending with 5, the square always ends with 25 and begins with the multiple of previous integer and one more than the integer.

**For example:**

Square of 25 is 2 x 3 .. 25 = 625.

Square of 85 is 8 x 9 .. 25 = 7225

**2. Nikhilam Navatascaravam Dasatah – All from 9 and the last from 10**

For multiplication of any numbers near to less than multiple of 10,

Step 1: Subtract the numbers with their closest multiple of 10 and multiply the results.

Step 2: Subtract the results with other numbers.

Step 3: Write result of Step 2 in the beginning and result of Step 1 at the end.

**For example:**

99 x 96 = ?

100 – 99 = 1 and 100 – 96 = 4

1 x 4 = 04

99 – 4 or 96 – 1 = 95

95 and 04

So, the answer is 9504.

**Example 2 : **

94 x 83 = ?

100 – 94 = 6 and 100 – 83 = 17

6 x 17 = 102

94 – 17 or 83 – 6 = 77

So, the answer is 7802.

**3. Urdva – Triyagbhyam – Vertically and crosswise**

For multiplication of any two two-digit numbers,

Step 1: Multiply the last digit

Step 2: Multiply numbers diagonally and add them

Step 3: Place Step 1 at the end and Step 2 at the beginning

Step 4: Multiply the first digit of both the number and put it at the most beginning

Step 5: For the result, more than 2 or more digits, add the beginning digits to the beginning numbers

**Example 1:**

45 x 87 = ?

5 x 7 = 35

(4 x 7) + (5 x 8) = 28 + 40 = 68

4 x 8 = 32

45 x 87 = 32 | 68 | 35 = 32 | 68 + 3 | 5 = 32 | 71 | 5 = 32 + 7 | 15 = 3915

**Example 2 :**

14 x 12 = ?

4 x 2 = 8

(1 x 2) + (4 x 1) = 6

1 x 1 = 1

14 x 12 = 168

**Example 3 : **

54 x 87 = ?

4 x 7 = 28

(5 x 7) + (4 x 8) = 35 + 32 = 67

5 x 8 = 40

54 x 87 = 40 | 67 | 28 = 4698

**4. Paraavartya Yojayet – Transpose and adjust**

For dividing large numbers by number greater than 10.

**For example:**

3784 divided by 12.

Step 1: Write the negative of the last number of the divisor under the dividend.

12

-2

Step 2: Separate the last digit of the dividend from the rest to calculate the remainder.

378 4

Step 3: Multiply the first digit with the above result i.e., -2.

3 x (-2) = -6

Step 4: Add the second digit with the result and continue till the last.

378 4 = 3 (7-6) 8 4 = 31 (8-2) 4 = 316 (4-12) = 315 (14 – 10)

Result: Quotient = 315 and Remainder = 4

**5. Sunyam Samya Samuccaye – When the sum is the same sum is zero**

This is related to equating with zero.

For example, as x is common factor in the equation “14x + 5x…… = 7x + 3x…..”, x will equal to 0.

i) 9(x+3) = 4(x+3)

According to the definition,

Since x+3 is a common factor,

x + 3 = 0

So, x = -3

Calculation with simple algebraic method,

9x + 27 = 4x + 12

5x = -15

x = -3

**6. Anurupye – Sunyamanyat – If one is in ratio the other one is 0**

We use this Sutra in solving a special type of simultaneous simple equations in which the coefficients of ‘one’ variable are in the same ratio to each other as the independent terms are to each other. In such a context the Sutra says the ‘other’ variable is zero from which we get two simple equations in the first variable (already considered) and of course give the same value for the variable.

**Example 1:**

3x + 7y = 2

4x + 21y = 6

Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is same as the ratio of independent terms i.e., 2 : 6 i.e., 1 : 3.

Hence the other variable x = 0 and 7y = 2 or 21y = 6 gives y = 2 / 7

**Example 2:**

323x + 147y = 1615 969x + 321y = 4845

**7. Sankalana – Vyavakalanabhyam – By addition and By subtraction**

In two general equation such as, ax + by = p and cx + dy = q, where x and y are unknown values,

x = (bq – pd) / (bc – ad)

y = (cp – aq) / (bc – ad)

**For example:**

i) 3x + 2y = 4 and 4x + 3y = 5

x = (10-12)/(8-9) = 2

y = (16-15)/(8-9) = -1

**8. Puranapuranabhyam – By the completion or non-completion**

This is a method of completion of polynomials to find its factors.

**For example:**

x³ + 9x² + 24x + 16 = 0 i.e. x³ + 9x² = -24x -16

We know that (x+3)³ = x³+9x²+27x+27 = 3x + 11 (Substituting above step).

i.e. (x+3)³ = 3(x+3) + 2 … (write 3x+11 in terms of LHS so that we substitute a term by a single variable).

Put y = x+3

So, y³ = 3y + 2

i.e. y³ – 3y – 2 = 0

Solving using the methods discussed (coeff of odd power = coeff of even power) before. We get

(y+1)² (y-2) = 0

So, y = -1 , 2

Hence, x = -4,-1

**9. Chalana – Kalanabhyam – Differences and Similarities**

i) In the first instance it is used to find the roots of a quadratic equation 7×2 – 11x – 7 = 0. Swamiji called the sutra as calculus formula. Its application at that point is as follows. Now by calculus formula we say: 14x–11 = ±√317

A Note follows saying every Quadratic can thus be broken down into two binomial factors. An explanation in terms of the first differential, discriminant with the sufficient number of examples are given in the chapter ‘Quadratic Equations’.

ii) At the Second instance under the chapter ‘Factorization and Differential Calculus’ for factorizing expressions of 3rd, 4th and 5th degree, the procedure is mentioned as ‘Vedic Sutras relating to Calana – Kalana – Differential Calculus’.

Further other Sutras 10 to 16 mentioned below are also used to get the required results. Hence the sutra and its various applications will be taken up at a later stage for discussion.

**10. Ekanyunena Purvena – By one less than the previous one**

For multiples of 9 as multiplier, the first digit is 1 less than the first digit of the multiplicand and the second digit is subtracting the lessened digit from multiple of 9.

**For example:**

i) 5 x 9 = (5-1), 9 = 4, 9-4 = 45

ii) 14 x 99 = (14-1), 99 – (14-1) = 1386

For multiplier less than the multiplicand, the first digit is additional 1 of first digit of the multiplicand less than the latter and the second digit is subtracting the second digit of multiplicand by 10.

**For example:**

i) 12 x 9 = (12-2), (10-2) = 108

ii) 28 x 9 = (28-3), (10-8) = 252

**11. Yaavadunam – Whatever the extent of its deficiency**

For square root of number close to multiple of 10,

Step 1: Subtract the number by multiple of 10

Step 2: First digit is the subtraction of the number from result of Step 1

Step 3: the Second digit is the square of result from Step 1

**For example:**

i) Square of 9 = (9-(10-9), (10-9) x (10 – 9) =81

ii) Square of 18 = 18 + 8, 8 x 8 = 26,64 = 2 (6+6) 4 = 324

**12. Vyashtisamanstih – Part and Whole**

This solves the equation of polynomials to find its factors by using Paravartya Sutra.

**For example:**

i) x³ + 7x² + 14x + 8 = 0 i.e. x³ + 7x² = – 14x – 8

We know that (x+3)³ = x³+9x²+27x+27 = 2x² + 13x + 19 (Substituting above step).

i.e. (x+3)³= 2x² + 13x + 19

Now we need factorize RHS in terms of (x+3). So apply Paravartya sutra.

Dividing 2x² + 13x + 19 by (x+3) gives 2x² + 13x + 19 = (x+3)(2x-7)-2

i.e. (x+3)³ = (x+3)(2x-7)-2

put y = x+3

So, y³ = y(2y+1) -2

Which gives y = 1,-1,2

Hence, x= -2, 4, 1, -1

**13. Shesanyankena Charamena – The remainders by the last digit**

Example: 1/7

• As seen earlier successive remainders are 1, 3, 2, 6, 4 and 5.

• We will write them as 3, 2, 6, 4, 5 and 1.

• Multiply them with the last digit of divisor (7): 21, 14, 42, 28, 35 and 7

• Now take their last digits and that’s the final answer: 0.142857. (another interesting concept).

Example: 1/13

• Successive remainders are 1, 10, 9, 12, 3 and 4.

• We will write them as 10, 9, 12, 3, 4 and 1.

• Multiply with the last digit of the divisor (3): 30, 27, 36, 9, 12 and 3.

• Taking just last digits gives final answer: 0.076923.

**14. Sopaantyadvayamantyam – The ultimate and twice the penultimate**

For the equation in the format 1/AB + 1/AC = 1/AD + 1/BC, the result is

2C(penultimate) + D(ultimate) = 0.

**For example:**

i) 1/(x+2)(x+3) + 1/(x+2)(x+4) = 1/(x+2)(x+5) + 1/(x+3)(x+4)

Applying the formula, 2(x+4) + (x+5) = 0

or, x = -13/3

**15. Gunitasamuchyah – The product of the sum is equal to the sum of the product**

For quadratic equation, in order to verify the result, the product of the sum of the coefficients of ‘x’ in the factors is equal to the sum of the coefficients of ‘x’ in the product.

**For example:**

i) (x + 3) (x + 2) = x² + 5x + 6

or, (1+3) (1+2) = 1 + 5 + 6

or, 12 = 12 ; thus verified.

ii) (x – 4) (2x + 5) = 2x² – 3x – 20

or, (1 – 4) (2 + 5) = 2 – 3 – 20

or -21 = – 21 ; thus verified.

**16. Gunakasamuchyah – The factors of the sum is equal to the sum of the factors**

For quadratic equation, the factor of the sum of the coefficients of ‘x’ in the product is equal to the sum of the coefficients of ‘x’ in the factors.

__The 13 sub-sutras of Vedic Maths__

**1. Anurupyena – Proportionality**

This Sutra is highly useful to find products of two numbers when both of them are near the Common bases i.e powers of base 10.

**For example:**

i) 46 x 43

We take 50 as a base point because it is the nearest powers of base 10.

46 = 50 – 4

43 = 50 – 7

Multiplying the difference from 50 to the numbers,

4 x 7 = 28 …… (a)

From cross subtraction,

43 – 4 = 39 or 46 – 7 = 39…….(b)

Adding (a) with the base i.e., 28+50 = 78 (goes on right hand side)

Dividing 39 by 2 give 19 with remainder 1. (goes on left hand side)

Thus, 46 x 43 = 1978

ii) 56 x 53

56 = 50 + 6

53 = 50 + 3

56 x 53 = (56 + 3) or (53 + 6), (6 x 3) + 50

=59,18 = 59/2, 18 = 2968

**2. Adyamadyenantya – mantyena – The first by the first and the last by the last**

**For example: **

i) 2x² + 5x -3

Divide the first term’s coeff (2) of equation by 1st term of factor (1) and divide last

term of equation (-3) by 2nd term of factor (3).

So, 2nd factor: 2x-1

**3. Yavadunam Tavadunikrtya Varganca Yojayet – Whatever the deficiency subtract that deficit and write alongside the square of**

For find squares of numbers close to base 10, we subtract the number from base 10 and take a square of the result. Then we subtract the result from the number and cross the results.

**For example:**

i) Square of 8

10 – 8 = 2, square of 2 is 4

8 – 2 = 6

Thus, Square of 8 = 64

ii) Square of 6

10 – 6 = 4, square of 4 is 16

6 – 4 = 2

Thus, Square of 6 = 2(1)6=36

**4. Antyayor Desakepi**

This is the multiplication of two numbers in the same structure of numbers to make their sum being the multiple of 10. For example: 43 x 47, 116 x 114, 1125 x 1125, etc.

For this you apply ekadhikena for the first digits of left hand side leaving the last digit and multiply with the first number of right hand side. The you multiply the last digit of the left hand side with the last digit of right hand side. Practically explained as below.

i) 43 x 47 = (4+1) x 4 | 3 x 7

=5 x 4 | 21

=20 | 21

= 2021

ii) 116 x 114 = (11 + 1) x 11 | 6 x 4

=132 | 24

= 13224

iii) 1125 x 1125 = 113 x 112 | 5 x 5

= 12656 | 25

= 1265625

**5. Antyayoreva – Only the last terms**

For the equation in the format (AC + D) / (BC + E) = A/B, the result is A/B = D/E.

**For example:**

i) (x²+x+1) / (x²+3x+3) = (x+1) / (x+3)

or, {x(x+1)+1} / {x(x+3)+1} = (x+1) / (x+3)

or, (x+1) / (x+3) = 1 / 3

or, x = 0

**6. Lopana Sthapanabhyam – By alternate elimination and retention**

This is to factorize complex equation by eliminating any of the variable.

**For example:**

i) Factorize 2x² + 6y² + 3z² + 7xy + 11yz + 7zx

We have 3 variables x,y,z.

Remove any of the variable like z by putting z=0.

Hence the given expression

E = 2×2 + 6y2 + 7xy = (x+2y) (2x+3y) … (Combination of Anurupyena & Adyamadyenantyamantya).

Similarly, if y=0, then E = 2×2 + 3z2 + 7zx = (x+3z) (2x+z)

As x and 2x are present separately and uniquely.

Hence we may map to get factors.

E = (x+2y+3x) (2x+3y+z)

**7. Vilokanam – By mere observation**

Generally we come across problems which can be solved by mere observation. But we follow the same conventional procedure and obtain the solution. But the hint behind the Sutra enables us to observe the problem completely and find the pattern and finally solve the problem by just observation.

**For example:**

i) x + 1/x = 5/2

x + 1/x = 2 + 1/2

Hence, x = 2 and 1/x = 1/2

**8. Gunitahsamuccayah Samuccayagunitah**

For quadratic equation, in order to verify the result, the product of the sum of the coefficients of ‘x’ in the factors is equal to the sum of the coefficients of ‘x’ in the product.

**For example:**

i) (x + 3) (x + 2) = x² + 5x + 6

or, (1+3) (1+2) = 1 + 5 + 6

or, 12 = 12 ; thus verified.

ii) (x – 4) (2x + 5) = 2x² – 3x – 20

or, (1 – 4) (2 + 5) = 2 – 3 – 20

or -21 = – 21 ; thus verified.

**9. Sisyate Sesasamjnah – The remainder remains constant**

For multiplication of any numbers near to more than multiple of 10,

• Step 1: Subtract the the closest multiple of 10 from the number and multiply the results.

• Step 2: Add the results with other numbers.

• Step 3: Write result of Step 2 in the beginning and result of Step 1 at the end.

**For example: **

104 x 101 = ?

104 – 100 = 4 and 101 – 100 = 1

4 x 1 = 04

101 + 4 or 104 + 1 = 105

105 and 04

So, the answer is 10504.

**10. Gunakasamuccayah**

For quadratic equation, the factor of the sum of the coefficients of ‘x’ in the product is equal to the sum of the coefficients of ‘x’ in the factors.

**11. Vestanam – By Osculation**

Let us check whether 21 is divisible by 7.

For 7, Ekadhika (positive osculator) is 5

So as per the mentioned process, multiply 5 with 1 and add 2 to the product.

• 21; 1×5+2 = 7 (Divisible by 7)

• 91; 1×5+9 = 14 (Divisible by 7). Can be continued further as

14; 4×5 + 1 = 21; and 21;1×5+2 = 7

• 112; 2×5+11= 21. (seen earlier)

• 2107; 7×5 + 210 = 245

245; 5×5+24= 49 (Divisible by 7 or continue further).

**12. Yavadunam Tavadunam**

Consider following 2 general equations

ax + by = p

cx + dy = q

Solving,

x = (bq – pd) / (bc – ad)

y = (cp – aq) / (bc – ad)

Notice that for calculation of numerators (x any y) cyclic method is used and Denominators remains same for both x and y.

**Examples: **

2x + 3y =6

3x + 4y = 3

Applying above formula:

x = (9 – 24)/ (9 – 8) = -15

y = (18 – 6) (9 – 8) = 12

-3x + 5y = 2

4x + 3y = -5

Applying above formula:

x = (-25 -6) / (20+9) = -31/29

y = (8-15) / (20+9) = -7/29

**13. Kevalaih Saptakam Gunyat**

For 7 the Multiplicand is 143 (Kevala: 143, Sapta: 7).

On the basis of 1/7, without any multiplication, we can calculate 2/7, 3/7, 4/7, 5/7 and 6/7. For that 1/7=0.142857 is to be remembered. But since remembering 0. 142857 is difficult we remember Kevala(143). This is the only use of this sutra (for remembrance).

1/7 = 0.142857 and By Ekanyuna, 143 x 999 = 142857

We need to remember 142857 for following reasons:

• 2/7 = 0.285714

• 3/7 = 0.428571

• 4/7 = 0.571428

• 5/7 = 0.714285

• 6/7 = 0.857142

All the values are in cyclic order.

So if we can remember 1/7 then we can obtain 2/7, 3/7 and etc as

• 2/7: the last digit of answer must be 4 (2*7=14)

• 3/7: the last digit of answer must be 1 (3*7=21)

• 4/7: the last digit of answer must be 8 (4*7=28)

• 5/7: the last digit of answer must be 5 (5*7=35)

• 6/7: the last digit of answer must be 2 (6*7=42)

We are aware that this attempt is only to make you familiar with a few special methods of Vedic Mathematics. The methods discussed, and organization of the content here are intended for any reader with some basic mathematical background. That is why the serious mathematical issues, higher level mathematical problems are not taken up in this article, even though many aspects like four fundamental operations, squaring, cubing, linear equations, simultaneous equations. factorization, H.C.F, recurring decimals, etc are dealt with. Many more concepts and aspects are omitted unavoidably, keeping in view the scope and limitations of the present volume.

Source: detechter.com