Identity (mathematics)

Definition and notation

A typical way to express an identity is to write an equality together with an implicit or explicit universal quantifier, for example "for all x, f(x)=g(x)". In informal writing the ordinary equals sign "=" is often used; some authors prefer the symbol "≡" to stress that the equality is an identity rather than a conditional equation. Note that "≡" is also commonly used for congruence in number theory, so context matters. For a discussion of equality in logic and algebra, consult the entry on equality.

Common types and examples

• Algebraic identities: polynomial equalities such as (a+b)^2 = a^2 + 2ab + b^2, valid for all numbers a and b in the chosen domain.
• Trigonometric identities: relations like sin^2 x + cos^2 x = 1, true for every real x.
• Functional identities: equalities between functions treated as formal objects, e.g. the logarithm rule log(ab)=log a + log b for positive a,b.
• Formal or ring identities: two polynomials or expressions that are identical as formal expressions because their coefficients coincide.

History and development

The idea of identities appears throughout the history of algebra and analysis as mathematicians recognized certain equalities that hold universally, which led to systematic study of symbolic manipulation. Classical trigonometric and algebraic identities were catalogued and used for simplification and computation long before modern formalizations in abstract algebra clarified when two expressions are equal as formal objects.

Uses and significance

Identities are tools for simplification, proof, and computation. They let one transform expressions, solve classes of problems, derive series and limits, and verify structural properties in algebraic systems. In computer algebra systems the distinction between an identity and a conditional equation guides algorithms for simplification and pattern matching.

Distinctions and related concepts

Important distinctions include identity versus conditional equation: an identity holds for all admissible variable values while a conditional equation holds only for some. Related but different notions are the identity function (the map that leaves every element unchanged) and the identity element of an algebraic structure (an element e with ae = ea = a); these are distinct senses of the word "identity" in mathematics.

For further reading on foundational terms and how mathematicians mark universal equalities, see the general references in identity discussions and introductory mathematics texts.