Ekadhikena Purvena
"By one more than the one before"
Used for squaring numbers ending in 5 (e.g., 35²: 3×4 | 25) and for decimal conversions of fractions like 1/19.
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Nikhilam Navatashcaramam Dashatah
"All from 9 and the last from 10"
The universal formula for subtraction from powers of 10, and for multiplication of numbers near a base (e.g., 98 × 97).
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Urdhva-Tiryagbhyam
"Vertically and crosswise"
The general formula for multiplication of any two numbers. Also used for division, squaring, and square roots.
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Paravartya Yojayet
"Transpose and adjust"
Used for division when the divisor is slightly greater than a power of 10, and for solving simple equations (Algebra).
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Shunyam Saamyasamuccaye
"When the sum is the same, that sum is zero"
A powerful algebraic sutra. If a specific sum in an equation is common, it equates to zero (e.g., solving (x+1)/(x+2) = (x+3)/(x+4)).
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(Anurupye) Shunyam Anyat
"If one is in ratio, the other is zero"
Used in solving simultaneous linear equations. If x-coefficients and y-coefficients are in ratio, the constant term gives the solution.
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Sankalana-vyavakalanabhyam
"By addition and by subtraction"
Used for simultaneous equations where coefficients are interchanged (e.g., 45x + 23y = 113, 23x + 45y = 91).
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Puranapuranabhyam
"By the completion or non-completion"
Used for solving quadratic equations by completing the square or cubic equations.
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Chalana-Kalanabhyam
"Differences and Similarities"
Used in calculus for finding roots of equations and differential equations.
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Yavadunam
"Whatever the extent of its deficiency"
Used for squaring numbers near a base (e.g., 96²: deficiency is 4, so 96-4 | 4²).
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Vyashtisamanstih
"Part and Whole"
Used in factorization and relation between the individual parts and the whole set.
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Shesanyankena Charamena
"The remainders by the last digit"
Used for converting fractions to decimals (e.g., 1/7) by observing remainders.
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Sopantyadvayamantyam
"The ultimate and twice the penultimate"
Used for solving specific types of equations and fraction problems.
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Ekanyunena Purvena
"By one less than the one before"
The inverse of the first sutra. Used for multiplication by 9, 99, 999 (e.g., 46 × 99).
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Gunitasamuchyah
"The product of the sum is equal to the sum of the product"
Used for checking the correctness of multiplication and factorization results.
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Gunakasamuchyah
"The factors of the sum are equal to the sum of the factors"
Similar to the above, used for verifying algebraic operations.
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