Reexpression

Reexpression is the act of expressing something in a form different from the original while preserving its meaning or value. The term is used in mathematics, logic, linguistics, and computer science to describe transforming an object into an equivalent representation.

In mathematics, reexpression refers to rewriting an expression or function in an alternative but equivalent form. Techniques include applying algebraic identities, substitutions, or changes of variables to simplify, reveal structure, or improve numerical properties. Common targets are polynomials, rational functions, and trigonometric or exponential functions, which can often be represented in multiple equivalent forms.

Examples include reexpressing sin^2 x as (1 − cos 2x)/2, rewriting a rational function by partial fractions, converting e^x to its power series, or expressing a polynomial in factored form rather than expanded form. Such reexpressions preserve the value of the expression for all valid inputs.

In linear algebra, a vector or linear operator can be reexpressed in a different basis or coordinate system, yielding a new set of coordinates that describe the same object. This change of basis is a common tool for simplifying problems or aligning with specific geometric or computational contexts.

In linguistics and rhetoric, reexpression (often called paraphrase) involves conveying the same content with different wording or structure. This can aid clarity, emphasis, or accessibility, while maintaining the original meaning.

In computer science, program transformation and optimization may involve reexpression of code to alternative forms that preserve behavior but enhance performance or readability.

Note that the exact usage of the term varies by field, but the core idea is the generation of equivalent representations.