SURYAMG

Complete 16 Vedic Mathematics Sutras (Multi-Context Masterclass)

1. Ekadhikena Purvena

Meaning: By one more than the previous one.
Formula: Left Part = N × (N + 1) | Right Part = Product of Units / 25

This trick is applied in two major contexts:

  1. Context A: Squaring numbers ending in 5.
  2. Context B: Multiplying numbers where tens digits are identical AND unit digits sum up to 10 (e.g., 87 × 83).
Context A: Squaring Numbers Ending in 5
Example 1: Find 35²
  • Step 1: Identify N = 3.
  • Step 2: Multiply N by one more than itself: 3 × (3 + 1) = 3 × 4 = 12.
  • Step 3: Right part is always 5 × 5 = 25.
  • Final Result: 1225
Example 2: Find 75²
  • Step 1: Identify N = 7.
  • Step 2: Calculate 7 × (7 + 1) = 7 × 8 = 56.
  • Step 3: Append 25 to the end.
  • Final Result: 5625
Context B: Tens Digits Identical & Units Sum to 10
Example 3: Calculate 87 × 83
  • Step 1: Verify condition: Tens digits are both 8. Units digits: 7 + 3 = 10.
  • Step 2 (Left Part): Multiply tens digit by one more: 8 × (8 + 1) = 8 × 9 = 72.
  • Step 3 (Right Part): Multiply unit digits directly: 7 × 3 = 21.
  • Final Result: 7221
Example 4: Calculate 64 × 66
  • Step 1: Tens digits are both 6. Units sum: 4 + 6 = 10.
  • Step 2 (Left Part): Multiply 6 × (6 + 1) = 6 × 7 = 42.
  • Step 3 (Right Part): Multiply unit digits: 4 × 6 = 24.
  • Final Result: 4224

Homework Problems

1. 95²
Answer: 9025
2. 43 × 47
Answer: 2021
3. 72 × 78
Answer: 5616
4. 113 × 117
Answer: 13221

2. Nikhilam Navatashcaramam Dashatah

Meaning: All from 9 and the last from 10.
Formula: Subtract digits from 9 and last from 10 OR (Number ± Deviation) | (Product of Deviations)

This Sutra is used across two main contexts:

  1. Context A: Subtracting large numbers from base powers of 10 (100, 1000, 10000).
  2. Context B: Multiplication of two numbers near a common base (e.g., base 100).
Context A: Subtraction from Base Numbers
Example 1: Calculate 1000 - 357
  • Step 1: Subtract 3 from 9 = 6.
  • Step 2: Subtract 5 from 9 = 4.
  • Step 3: Subtract last digit 7 from 10 = 3.
  • Final Result: 643
Example 2: Calculate 10000 - 4682
  • Step 1: 9 - 4 = 5
  • Step 2: 9 - 6 = 3
  • Step 3: 9 - 8 = 1
  • Step 4: 10 - 2 = 8
  • Final Result: 5318
Context B: Multiplication Near Base 100
Example 3: Calculate 97 × 94 (Below Base)
  • Step 1: Deviations from 100: -03 and -06.
  • Step 2 (Left Part): Cross-add: 97 + (-06) = 91.
  • Step 3 (Right Part): Multiply deviations: (-03) × (-06) = 18.
  • Final Result: 9118
Example 4: Calculate 105 × 108 (Above Base)
  • Step 1: Deviations from 100: +05 and +08.
  • Step 2 (Left Part): Cross-add: 105 + 08 = 113.
  • Step 3 (Right Part): Multiply deviations: 5 × 8 = 40.
  • Final Result: 11340

Homework Problems

1. 1000 - 482
Answer: 518
2. 98 × 95
Answer: 9310
3. 104 × 107
Answer: 11128
4. 10000 - 7234
Answer: 2766

3. Urdhva Tiryagbhyam

Meaning: Vertically and crosswise.
Formula: (a × c) | (a × d + b × c) | (b × d)

Universal technique used across two contexts:

  1. Context A: General 2-digit by 2-digit number multiplication.
  2. Context B: Algebraic binomial expansion (ax + b)(cx + d).
Context A: Arithmetic Multiplication
Example 1: Calculate 21 × 32
  • Step 1 (Right): Vertically multiply unit digits: 1 × 2 = 2.
  • Step 2 (Middle): Crosswise sum: (2×2) + (1×3) = 4 + 3 = 7.
  • Step 3 (Left): Vertically multiply tens digits: 2 × 3 = 6.
  • Final Result: 672
Example 2: Calculate 34 × 23
  • Step 1 (Right): 4 × 3 = 12 (Write 2, carry 1).
  • Step 2 (Middle): (3×3) + (4×2) + 1 = 9 + 8 + 1 = 18 (Write 8, carry 1).
  • Step 3 (Left): (3×2) + 1 = 7.
  • Final Result: 782
Context B: Algebraic Expansion
Example 3: Expand (2x + 3)(4x + 1)
  • Step 1 (Right): Multiply constants: 3 × 1 = 3.
  • Step 2 (Middle): Cross-multiply x terms: (2x×1) + (3×4x) = 2x + 12x = 14x.
  • Step 3 (Left): Multiply x terms vertically: 2x × 4x = 8x².
  • Final Result: 8x² + 14x + 3
Example 4: Expand (3x + 2)(x + 5)
  • Step 1 (Right): 2 × 5 = 10.
  • Step 2 (Middle): (3x×5) + (2×x) = 15x + 2x = 17x.
  • Step 3 (Left): 3x × x = 3x².
  • Final Result: 3x² + 17x + 10

Homework Problems

1. 12 × 14
Answer: 168
2. 42 × 21
Answer: 882
3. Expand (x + 4)(3x + 2)
Answer: 3x² + 14x + 8
4. Expand (2x + 5)(2x + 3)
Answer: 4x² + 16x + 15

4. Paraavartya Yojayet

Meaning: Transpose and adjust.
Formula: Transpose signs of divisor terms OR transpose constants in equations.

Applied in two distinct contexts:

  1. Context A: Division when divisor is slightly greater than a base of 10.
  2. Context B: Solving simple linear algebraic equations.
Context A: Fast Division
Example 1: Divide 134 by 12
  • Step 1: Divisor = 12 (Base 10). Deviation = +2. Transpose sign: -2.
  • Step 2: Set up: 1 3 | 4. Bring down 1.
  • Step 3: 1 × (-2) = -2. Add to 3: 3 - 2 = 1 (Quotient = 11).
  • Step 4: 1 × (-2) = -2. Add to 4: 4 - 2 = 2 (Remainder = 2).
  • Final Result: Quotient = 11, Remainder = 2
Example 2: Divide 143 by 11
  • Step 1: Divisor = 11. Transposed deviation = -1.
  • Step 2: Bring down 1. 1 × (-1) = -1. Add to 4: 4 - 1 = 3 (Quotient = 13).
  • Step 3: 3 × (-1) = -3. Add to 3: 3 - 3 = 0 (Remainder = 0).
  • Final Result: Quotient = 13, Remainder = 0
Context B: Solving Linear Equations
Example 3: Solve 2x + 5 = 13
  • Step 1: Transpose +5 to RHS: 13 - 5 = 8.
  • Step 2: Transpose coefficient 2: x = 8 / 2.
  • Final Result: x = 4
Example 4: Solve 5x - 7 = 18
  • Step 1: Transpose -7 to RHS: 18 + 7 = 25.
  • Step 2: Transpose coefficient 5: x = 25 / 5.
  • Final Result: x = 5

Homework Problems

1. Divide 124 by 11
Answer: Q = 11, R = 3
2. Divide 154 by 12
Answer: Q = 12, R = 10
3. Solve 3x + 4 = 19
Answer: x = 5
4. Solve 4x - 9 = 27
Answer: x = 9

5. Shunyam Saamyasamuccaye

Meaning: When the sum is the same, that sum is zero.
Formula: If LHS sum of independent terms = RHS sum of independent terms, set variable = 0.

Context A: Solves symmetrical algebraic equations by inspection.

Context A: Symmetrical Equations
Example 1: Solve (x + 2) + (x + 3) = (x + 1) + (x + 4)
  • Step 1: Check LHS constant sum: 2 + 3 = 5.
  • Step 2: Check RHS constant sum: 1 + 4 = 5.
  • Step 3: Since sums are equal, set variable = 0.
  • Final Result: x = 0
Example 2: Solve 12x + 5x = 8x + 9x
  • Step 1: LHS coefficient sum: 12 + 5 = 17.
  • Step 2: RHS coefficient sum: 8 + 9 = 17.
  • Final Result: x = 0

Homework Problems

1. (x + 4) + (x + 5) = (x + 2) + (x + 7)
Answer: x = 0
2. 14x + 3x = 10x + 7x
Answer: x = 0
3. (x + 1) + (x + 8) = (x + 3) + (x + 6)
Answer: x = 0
4. 9x + 11x = 15x + 5x
Answer: x = 0

6. Anurupye Shunyamanyat

Meaning: If one is in ratio, the other is zero.
Formula: If coefficient ratio of a variable = ratio of constant terms, the OTHER variable = 0.

Context A: Fast solution for linear simultaneous equations without substitution.

Context A: Simultaneous Equations
Example 1: Solve 3x + 2y = 6 and 6x + 5y = 12
  • Step 1: Ratio of x coefficients: 3 / 6 = 1 / 2.
  • Step 2: Ratio of constants: 6 / 12 = 1 / 2.
  • Step 3: Since x ratio = constant ratio, set y = 0.
  • Step 4: Substitute y = 0 in 3x = 6 -> x = 2.
  • Final Result: x = 2, y = 0
Example 2: Solve 4x + 3y = 8 and 12x + 7y = 24
  • Step 1: Ratio of x: 4 / 12 = 1 / 3.
  • Step 2: Ratio of constants: 8 / 24 = 1 / 3.
  • Step 3: Therefore, y = 0 and 4x = 8 -> x = 2.
  • Final Result: x = 2, y = 0

Homework Problems

1. 2x + 5y = 4 & 4x + 9y = 8
Answer: x = 2, y = 0
2. 5x + 4y = 10 & 15x + 7y = 30
Answer: x = 2, y = 0
3. 3x + 8y = 9 & 6x + 11y = 18
Answer: x = 3, y = 0
4. 7x + 2y = 14 & 21x + 5y = 42
Answer: x = 2, y = 0

7. Sankalana Vyavakalanabhyam

Meaning: By addition and by subtraction.
Formula: Step 1: Add equations to get (x + y). Step 2: Subtract equations to get (x - y).

Context A: Symmetric simultaneous linear equations where x and y coefficients are swapped.

Context A: Swapped Coefficient Equations
Example 1: Solve 45x - 23y = 113 and 23x - 45y = 23
  • Step 1 (Add): 68x - 68y = 136 -> x - y = 2.
  • Step 2 (Subtract): 22x + 22y = 90 -> x + y = 4.
  • Step 3: Solve x - y = 2 and x + y = 4 -> 2x = 6 -> x = 3, y = 1.
  • Final Result: x = 3, y = 1
Example 2: Solve 21x + 17y = 55 and 17x + 21y = 59
  • Step 1 (Add): 38x + 38y = 114 -> x + y = 3.
  • Step 2 (Subtract): 4x - 4y = -4 -> x - y = -1.
  • Step 3: Add both simplified equations: 2x = 2 -> x = 1, y = 2.
  • Final Result: x = 1, y = 2

Homework Problems

1. 12x + 13y = 25 & 13x + 12y = 25
Answer: x = 1, y = 1
2. 15x + 11y = 41 & 11x + 15y = 37
Answer: x = 2, y = 1
3. 10x + 5y = 35 & 5x + 10y = 25
Answer: x = 3, y = 1
4. 7x + 3y = 23 & 3x + 7y = 17
Answer: x = 2.9, y = 0.9

8. Puranapuranabhyam

Meaning: By the completion or non-completion.
Formula: Complete numbers to nearest multiples of 10 by borrowing from other terms.

Context A: Mental addition and simplification by creating complete round numbers.

Context A: Mental Addition
Example 1: Calculate 49 + 27
  • Step 1: 49 needs 1 to complete 50.
  • Step 2: Borrow 1 from 27 (27 - 1 = 26).
  • Step 3: 50 + 26 = 76.
  • Final Result: 76
Example 2: Calculate 98 + 85
  • Step 1: 98 needs 2 to complete 100.
  • Step 2: Take 2 from 85 (85 - 2 = 83).
  • Step 3: 100 + 83 = 183.
  • Final Result: 183

Homework Problems

1. 39 + 45
Answer: 84
2. 97 + 68
Answer: 165
3. 298 + 145
Answer: 443
4. 59 + 84
Answer: 143

9. Chalana Kalanabhyam

Meaning: Differences and Similarities.
Formula: Derivative = ± sqrt(Discriminant) for quadratic roots.

Context A: Solving quadratic equations using calculus derivatives.

Context A: Quadratic Roots
Example 1: Solve x² - 7x + 12 = 0
  • Step 1: Derivative: 2x - 7.
  • Step 2: Discriminant square root: sqrt(49 - 48) = ±1.
  • Step 3: 2x - 7 = ±1 -> 2x = 8 or 2x = 6.
  • Final Result: x = 4, x = 3
Example 2: Solve x² - 5x + 6 = 0
  • Step 1: Derivative: 2x - 5.
  • Step 2: Discriminant square root: sqrt(25 - 24) = ±1.
  • Step 3: 2x - 5 = ±1 -> 2x = 6 or 2x = 4.
  • Final Result: x = 3, x = 2

Homework Problems

1. x² - 9x + 20 = 0
Answer: x = 5, x = 4
2. x² - 3x + 2 = 0
Answer: x = 2, x = 1
3. x² - 8x + 15 = 0
Answer: x = 5, x = 3
4. x² - 10x + 21 = 0
Answer: x = 7, x = 3

10. Yaavadunam

Meaning: Whatever the extent of its deficiency.
Formula: Square = (Number ± Deficiency) | (Deficiency²)

Applied in two contexts:

  1. Context A: Squaring numbers below a base (deficiency).
  2. Context B: Squaring numbers above a base (surplus).
Context A: Numbers Below Base
Example 1: Find 96²
  • Step 1: Base = 100. Deficiency d = 4.
  • Step 2 (Left): Number - Deficiency = 96 - 4 = 92.
  • Step 3 (Right): Deficiency² = 4² = 16.
  • Final Result: 9216
Example 2: Find 94²
  • Step 1: Base = 100. Deficiency d = 6.
  • Step 2: 94 - 6 = 88.
  • Step 3: 6² = 36.
  • Final Result: 8836
Context B: Numbers Above Base
Example 3: Find 108²
  • Step 1: Base = 100. Surplus s = 8.
  • Step 2 (Left): Number + Surplus = 108 + 8 = 116.
  • Step 3 (Right): Surplus² = 8² = 64.
  • Final Result: 11664
Example 4: Find 104²
  • Step 1: Base = 100. Surplus s = 4.
  • Step 2: 104 + 4 = 108.
  • Step 3: 4² = 16.
  • Final Result: 10816

Homework Problems

1. 98²
Answer: 9604
2. 92²
Answer: 8464
3. 107²
Answer: 11449
4. 112²
Answer: 12544

11. Vyashtisamanstih

Meaning: Part and Whole.
Formula: Solve place values individually then integrate into whole sum.

Context A: Positional component breakdown for fast multi-digit additions.

Context A: Positional Breakdown
Example 1: Add 234 + 561
  • Step 1: Hundreds: 200 + 500 = 700.
  • Step 2: Tens: 30 + 60 = 90.
  • Step 3: Units: 4 + 1 = 5.
  • Final Result: 795
Example 2: Add 342 + 435
  • Step 1: Hundreds: 300 + 400 = 700.
  • Step 2: Tens: 40 + 30 = 70.
  • Step 3: Units: 2 + 5 = 7.
  • Final Result: 777

Homework Problems

1. 512 + 264
Answer: 776
2. 123 + 654
Answer: 777
3. 415 + 372
Answer: 787
4. 621 + 158
Answer: 779

12. Shesanyankena Charamena

Meaning: The remainder by the last digit.
Formula: Sum of digits gives the remainder modulo 9.

Context A: Finding remainders instantly when dividing by 9.

Context A: Modulo 9 Remainders
Example 1: Find remainder of 4325 ÷ 9
  • Step 1: Sum digits: 4 + 3 + 2 + 5 = 14.
  • Step 2: Sum again: 1 + 4 = 5.
  • Final Result: Remainder = 5
Example 2: Find remainder of 7812 ÷ 9
  • Step 1: Sum digits: 7 + 8 + 1 + 2 = 18.
  • Step 2: 1 + 8 = 9 (Divisible with 0 remainder).
  • Final Result: Remainder = 0

Homework Problems

1. Remainder of 5241 ÷ 9
Answer: 3
2. Remainder of 8132 ÷ 9
Answer: 5
3. Remainder of 6004 ÷ 9
Answer: 1
4. Remainder of 9871 ÷ 9
Answer: 7

13. Sopantyadvayamantyam

Meaning: The ultimate and twice the penultimate.
Formula: Factorize polynomials x² + (a+b)x + ab into (x + a)(x + b).

Context A: Fast mental factorization of simple quadratic trinomials.

Context A: Quadratic Factorization
Example 1: Factorize x² + 7x + 10
  • Step 1: Constant = 10, Middle coefficient = 7.
  • Step 2: Find factors of 10 that sum to 7: 5 and 2.
  • Final Result: (x + 5)(x + 2)
Example 2: Factorize x² + 9x + 20
  • Step 1: Constant = 20, Middle coefficient = 9.
  • Step 2: Factors of 20 summing to 9: 5 and 4.
  • Final Result: (x + 5)(x + 4)

Homework Problems

1. Factorize x² + 5x + 6
Answer: (x + 3)(x + 2)
2. Factorize x² + 8x + 12
Answer: (x + 6)(x + 2)
3. Factorize x² + 10x + 24
Answer: (x + 6)(x + 4)
4. Factorize x² + 11x + 30
Answer: (x + 6)(x + 5)

14. Ekanyunena Purvena

Meaning: By one less than the previous one.
Formula: Left = Number - 1 | Right = 999... - Left Part

Context A: Multiplying any number by a series consisting entirely of nines (9, 99, 999).

Context A: Multiplication by Nines
Example 1: Calculate 48 × 99
  • Step 1 (Left Part): Subtract 1: 48 - 1 = 47.
  • Step 2 (Right Part): 99 - 47 = 52.
  • Final Result: 4752
Example 2: Calculate 354 × 999
  • Step 1: Left Part = 354 - 1 = 353.
  • Step 2: Right Part = 999 - 353 = 646.
  • Final Result: 353646

Homework Problems

1. 63 × 99
Answer: 6237
2. 87 × 99
Answer: 8613
3. 241 × 999
Answer: 240759
4. 752 × 999
Answer: 751248

15. Gunitasamuccayah

Meaning: The product of the sum is equal to the sum of the product.
Formula: Product of Digit Sums of Factors = Digit Sum of Product

Context A: Verifying algebraic factorization correctness.

Context A: Factorization Verification
Example 1: Verify (x + 2)(x + 3) = x² + 5x + 6
  • Step 1: LHS factor sums: (1 + 2) × (1 + 3) = 3 × 4 = 12.
  • Step 2: RHS coefficient sum: 1 + 5 + 6 = 12.
  • Final Result: 12 = 12 (Verified correct)
Example 2: Verify (2x + 1)(x + 4) = 2x² + 9x + 4
  • Step 1: LHS sums: (2 + 1) × (1 + 4) = 3 × 5 = 15.
  • Step 2: RHS sum: 2 + 9 + 4 = 15.
  • Final Result: 15 = 15 (Verified)

Homework Problems

1. Verify (x+1)(x+4) = x²+5x+4
Answer: LHS = 10, RHS = 10
2. Verify (x+3)(x+5) = x²+8x+15
Answer: LHS = 32, RHS = 32
3. Verify (2x+3)(x+1) = 2x²+5x+3
Answer: LHS = 10, RHS = 10
4. Verify (x+2)(x+4) = x²+6x+8
Answer: LHS = 18, RHS = 18

16. Gunakasamuccayah

Meaning: The factor of the sum is equal to the sum of the factors.
Formula: Extract HCF/common variables to group algebraic expressions.

Context A: Simplifying algebraic terms by factoring out common terms.

Context A: Algebraic Grouping
Example 1: Factorize 6x + 9
  • Step 1: Find HCF of 6 and 9: 3.
  • Step 2: Factor out 3: 3 × (2x + 3).
  • Final Result: 3(2x + 3)
Example 2: Factorize x²y + xy²
  • Step 1: Common factor: xy.
  • Step 2: Factor out xy: xy(x + y).
  • Final Result: xy(x + y)

Homework Problems

1. Factorize 8x + 12
Answer: 4(2x + 3)
2. Factorize 15x - 20
Answer: 5(3x - 4)
3. Factorize a²b + ab²
Answer: ab(a + b)
4. Factorize 14x² + 21x
Answer: 7x(2x + 3)