Sūtra 8 of 16

Pūraṇāpūraṇābhyām

By the completion or non-completion

Completing a number to the next round figure (like making 97 into 100).

When to use

Algebraic factorisation, completing the square, mental arithmetic where rounding helps.

How it works

Add what is needed to complete the number to a nicer base, perform the operation, then subtract the addition.

Memory formula

Complete to a convenient form, solve there, then compensate.

Worked examples

Problem 1

Solve x² + 4x − 21 = 0

0/3

Where this ancient technique still lives in today's world.

Coding

Used in optimisation (gradient descent on quadratic loss), Kalman filters, and Gaussian-distribution math.

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