dedivision

Dedivision is a term used in algebra and symbolic computation to describe the process of removing division operations from expressions or equations by replacing divisions with multiplication by reciprocals, or by clearing denominators. It is used to simplify expressions and to transform equations into forms that are easier to solve computationally.

Common approaches include replacing a/b with a·b^{-1} (or 1/b, viewed as multiplication by a reciprocal), and clearing denominators by multiplying both sides of an equation by a common denominator. The latter yields an equivalent equation on the condition that the denominators are nonzero.

In inequalities, care must be taken because multiplying by expressions whose sign is not known can reverse the relation. In both equations and expressions, dedivision preserves equivalence only where denominators are nonzero, and domain constraints must be observed. Dedivision can also introduce extraneous solutions if performed without attention to these restrictions.

Examples: Solve x/(x-1) = 2. Dedivide by multiplying both sides by (x-1): x = 2(x-1) → x = 2x - 2 → x = 2, with the constraint x ≠ 1. Another example: 6/(2x) + 3 = 7. Multiply by 2x: 6 + 6x = 14x → 6 = 8x → x = 3/4, with x ≠ 0.

In practice, dedivision is common in teaching algebra and in computer algebra systems as a technique to simplify and to avoid division operations in intermediate steps.