Algebraic identity

An algebraic identity is an equality that holds for every permitted value of its variables. It expresses a relation that remains true throughout its domain.

Algebraic expression: A mathematical phrase combining numbers, variables, and operations, without asserting an equality. An identity equates two expressions built from variables and operations.
Distributive property: The rule that multiplication distributes over addition: a(b+c)=ab+ac. Expanding products with this property often reveals an identity.
Difference of two squares: The factorization a²−b²=(a−b)(a+b). This compact polynomial identity turns a subtraction of squares into a product.
Equation: A mathematical statement asserting that two expressions are equal, often for particular unknown values. Every identity is an equation, but an equation need not hold for all permitted values.
Variable: A symbol that represents a value that may vary or is not yet specified. The variables in an identity range over its permitted domain.
Factorization: The rewriting of an expression as a product of simpler factors. Factoring can show that the difference between two sides is zero.
Pythagorean theorem: The Euclidean geometry theorem stating that a right triangle's hypotenuse satisfies c²=a²+b². Its equation holds for every right triangle, linking geometric relations to algebraic equality.
Conditional equation: An equation that is true only for particular values of its variables. Its solutions are a subset of the domain rather than the whole domain.
Equality: A relation stating that two mathematical objects have the same value. An identity is an equality that stays true across its domain.
Mathematical induction: A proof method establishing a statement for all natural numbers from a base case and an inductive step. Induction proves identities indexed by every natural number.

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