Vedic Maths Sutra 05 : Śūnyam Sāmyasamuccaye

Sūtra 5 of 16
शून्यं साम्यसमुच्चये
Śūnyam Sāmyasamuccaye
“When the samuccaya is the same, that samuccaya is zero”

Meaning & Essence

This Sūtra means “When the samuccaya is the same, that samuccaya is zero” (or “When the sum is the same, that sum is zero”).

“Samuccaya” is a flexible word that can mean a common factor, the product of independent terms, the sum of numerators/denominators, etc. The Sūtra is mainly used for solving equations very quickly by recognising special patterns and setting the common total to zero.

Main Applications

The Sūtra has several useful interpretations:

\[ \text{If a common factor appears on both sides} \Rightarrow \text{set that factor = 0} \]
\[ (x + a)(x + b) = (x + c)(x + d) \quad \text{and} \quad a \times b = c \times d \quad \Rightarrow \quad x = 0 \]
\[ \frac{N_1}{D_1} = \frac{N_2}{D_2} \quad \text{and} \quad N_1 + N_2 = D_1 + D_2 \quad \Rightarrow \quad N_1 + N_2 = 0 \]

Worked Examples

Example 1 → Common factor on both sides
  • Equation: \( 7x + 3x = 4x + 5x \)
  • Every term has the common factor \( x \)
  • By the Sūtra → \( x = 0 \)
Answer: \( x = 0 \)
Example 2 → Common binomial factor
  • Equation: \( 5(x + 1) = 3(x + 1) \)
  • Common factor is \( (x + 1) \)
  • Set \( x + 1 = 0 \) → \( x = -1 \)
Answer: \( x = -1 \)
Example 3 → Product of independent terms is the same
  • Equation: \( (x + 7)(x + 9) = (x + 3)(x + 21) \)
  • Independent terms: \( 7 \times 9 = 63 \) and \( 3 \times 21 = 63 \)
  • Products are equal → by the Sūtra \( x = 0 \)
Answer: \( x = 0 \)
Example 4 → Sum of numerators = sum of denominators
  • Equation: \( \dfrac{2x + 9}{2x + 7} = \dfrac{2x + 7}{2x + 9} \)
  • Sum of numerators = \( (2x+9) + (2x+7) = 4x + 16 \)
  • Sum of denominators is the same → set \( 4x + 16 = 0 \)
  • \( x = -4 \)
Answer: \( x = -4 \)
Interactive Demo – Product of independent terms

Enter four numbers for the equation (x + a)(x + b) = (x + c)(x + d).
The demo checks whether a × b equals c × d. If yes, the solution is x = 0.

Please enter four valid numbers

    When to Use This Sūtra

    • Solving linear and simple quadratic equations that show special patterns
    • When a common factor appears on both sides of an equation
    • When the product of the constant terms is the same on both sides
    • When the sum of numerators equals the sum of denominators in a proportion
    • Excellent for rapid mental solution of many textbook-style equations

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