Anurupye Shunyamanyat
आनुरूप्ये शून्यमन्यत्
Meaning: "If one is in ratio, the other is zero" or "When there is proportionality, the other (variable part) is zero". This sutra deals with situations involving ratios or proportions.
Primary Application: This sutra is typically applied to solve a specific system of two simultaneous linear equations. The conditions are:
- The ratio of the coefficients of 'x' in both equations is the same as the ratio of the coefficients of 'y'.
(i.e., for equations a1x + b1y = c1 and a2x + b2y = c2, if a1/a2 = b1/b2). - This common ratio (from condition 1) is different from the ratio of the constant terms.
(i.e., a1/a2 = b1/b2 ≠ c1/c2).
If these conditions are met, the sutra implies that for a unique solution to exist under such "proportionality" of variable coefficients but "disproportionality" of constants, the variables themselves (x and y) must be zero. Geometrically, this represents two parallel lines that are distinct (not overlapping), which can only "meet" (have a common solution) at the origin if forced by constants being zero, or have no solution if constants are non-zero.
Simplified Interpretation: If the variable parts of two equations are proportional (one is a multiple of the other), but the constant parts are not proportionally related in the same way, then the only way to satisfy both equations simultaneously (if a unique solution is sought) is if the variables are zero. This assumes a system that would otherwise be inconsistent.
Example: Consider the system:
2x + 3y = 5
4x + 6y = 12
Ratio of x-coefficients: 2/4 = 1/2.
Ratio of y-coefficients: 3/6 = 1/2.
Ratio of constants: 5/12.
Here, 1/2 ≠ 5/12. The coefficients of x and y are in ratio, but the constants are not. This system has no solution (parallel distinct lines). The sutra "Anurupye Shunyamanyat" would imply x=0, y=0 if we were trying to force a solution under specific constraints related to a theoretical point of commonality when the system should otherwise be inconsistent. In its direct application, if such a system (2x+3y=k1, 4x+6y=k2 with k1/k2 not 1/2) must have a solution, that solution is x=0,y=0, implying k1 and k2 must then also be 0. The sutra is subtle and applies to specific forms or interpretations where such proportionality leads to a zero value for the variables.
Let's analyze the system of equations: 3x + 4y = k1 and 6x + 8y = k2 using the "Anurupye Shunyamanyat" sutra.
Equation 1: 3x + 4y = k1
Equation 2: 6x + 8y = k2
Step 1: Check for Proportionality of Coefficients
- Ratio of x-coefficients (a1/a2): 3/6 = 1/2
- Ratio of y-coefficients (b1/b2): 4/8 = 1/2
We see that a1/a2 = b1/b2 = 1/2. The coefficients of x and y are proportional.
Step 2: Apply "Anurupye Shunyamanyat"
The sutra states: "If one is in ratio (coefficients of variables), the other is zero (the variables themselves under certain conditions)."
The condition for this application is that the ratio of coefficients of variables (1/2) is NOT equal to the ratio of the constant terms (k1/k2).
So, if 1/2 ≠ k1/k2
Then, for the system to have a unique, forced resolution under this "proportionality of variables but disproportionality of constants," the sutra implies that the variables themselves must be zero.
x = 0 and y = 0.
Step 3: Consequence if x=0 and y=0
If we substitute x=0 and y=0 back into the original equations:
- Equation 1: 3(0) + 4(0) = k1 => 0 = k1
- Equation 2: 6(0) + 8(0) = k2 => 0 = k2
This means that if the system with proportional variable coefficients but k1/k2 ≠ 1/2 were to have a solution, it must be x=0, y=0, which in turn forces k1=0 and k2=0. If k1 or k2 are non-zero and k1/k2 ≠ 1/2, the original system of equations actually has no solution (representing two distinct parallel lines).
Interpretation
The sutra highlights a specific condition. If coefficients of x and y are proportional (e.g., one equation's variable part is a multiple of the other's), but the constant terms don't follow the same proportion, then a "forced" solution point where the variables become zero (x=0, y=0) is suggested. This usually implies the constants (k1, k2) must also be zero for consistency. If they are not, the system is inconsistent.
Anurupye Shunyamanyat helps identify specific conditions leading to x=0, y=0. 🧐