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Ekādhikena Pūrvena,

Ekādhikena Pūrvena, ("By one more than the previous one") This  sūtra means that the prescribed arithmetical operation is either multiplication or division. Both are implied since we may proceed leftward and multiply (and carry-over the excess value to the next leftward column) or we may go rightward and divide (while prefixing the remainder). When dividing by a nine's family denominator, the digit to the left of the nine (previous) is increased by one to obtain the multiplier.

.Examples:


35×35 = ((3×3)+3),25 = 12,25 and 125×125 = ((12×12)+12),25 = 156,25
 
or by the sūtra, multiply "by one more than the previous one."
35×35 = ((3×4),25 = 12,25 and 125×125 = ((12×13),25 = 156,25
 
The latter portion is multiplied by itself (5 by 5) and the previous portion is square of first digit or first two digit (3×3) or (12×12) and adding the same digit in that figure (3or12) resulting in the answer 1225.
(Proof) This is a simple application of


 (a+b)^2=a^2+2ab+b^2 when a=10c and b=5, i.e.

(10c+5)^2=100c^2+100c+25=100c(c+1)+25.\,

It can also be applied in multiplications when the last digit is not 5 but the sum of the last digits is the base (10) and the previous parts are the same. Examples:
37 × 33 = (3 × 4),7 × 3 = 12,21
29 × 21 = (2 × 3),9 × 1 = 6,09 ?
 
This uses (a+b)(a-b)=a^2-b^2 twice combined with the previous result to produce:


(10c+5+d)(10c+5-d)=(10c+5)^2-d^2=100c(c+1)+25-d^2=100c(c+1)+(5+d)(5-d)
.




We illustrate this sūtra by its application to conversion of fractions into their equivalent decimal form. Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.




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