Vedic Mathematics

"By one more than the previous one"

Used to find squares of numbers ending in 5 and for division by numbers ending in 9. The sutra takes the previous digit, multiplies it by one more than itself, and appends 25 for squares.

"All from 9 and the last from 10"

The cornerstone sutra for multiplication of numbers close to a base (10, 100, 1000, etc.). Find the deficit/surplus from the base, cross-add/subtract, and multiply the deficits/surpluses.

"Vertically and crosswise"

The general formula for all multiplication. Multiply vertically and crosswise, then add. Works for any number of digits. This is the most important sutra — it covers multiplication, division, and even algebraic equations.

"Transpose and adjust"

Used for division when the divisor is slightly greater than a power of 10. Convert the divisor into its complement and use it as a multiplier. Also used for solving simultaneous equations.

"If the sum is the same, that sum is zero"

When the sum of the numerators or coefficients on both sides of an equation is the same, then that sum equals zero. Used for solving equations where a common factor can be identified quickly.

"If one is in ratio, the other is zero"

Used in simultaneous equations. If the coefficients of one variable are in the same ratio in both equations, that variable's value can be directly determined, and the other is found to be zero or by simple substitution.

"By addition and by subtraction"

Solve simultaneous equations by adding and subtracting the equations to eliminate variables. Simpler than traditional methods when coefficients are close in value.

"By the completion or non-completion"

Complete the expression to make it a perfect square or cube, solve, then adjust back. Used for solving quadratic equations and finding square roots.

"Differences and similarities"

Used in differential calculus and for finding successive differentials. Analyze the pattern of differences in a sequence to predict the next term or find the general formula.

"Whatever the extent of its deficiency"

Used for squaring numbers close to a base (10, 100, 1000). Find the deficit from the base, subtract it from the number to get the left part, then square the deficit for the right part.

"Part and whole"

The relationship between the individual (vyashti) and the collective (samansti). Used to find averages, totals from parts, and in factoring polynomials where individual factors relate to the whole.

"The remainders by the last digit"

Used for expressing fractions as recurring decimals. Particularly useful for finding repeating decimal expansions of fractions with denominators like 7, 13, 17, 19, etc.

"The ultimate and twice the penultimate"

Used in finding the sum of series and in solving certain types of equations involving rational expressions. The relationship between the last term and twice the second-to-last term gives a shortcut.

"By one less than the previous one"

Used for multiplication by numbers consisting entirely of 9s (like 9, 99, 999). Subtract 1 from the multiplicand for the left part, and subtract each digit from 9 for the right part.

"The product of the sum is equal to the sum of the product"

If the product of the sums of coefficients of factors equals the sum of the coefficients of the product, then the factorization is correct. Used as a verification tool for multiplication and factoring.

"The factors of the sum is equal to the sum of the factors"

The converse of the previous sutra. Used for verifying division and factoring. If the sum of the digits of the divisor times the sum of the digits of the quotient equals the sum of the digits of the dividend, the division is correct.