Equality

Equality is a relation asserting that two mathematical expressions have the same value. It is written with the equals sign, “=”, and remains true when either side is replaced by an equal expression.

Mathematical expression: A combination of numbers, variables, and operations that represents a mathematical quantity. Each side of an equality is an expression whose value is being compared.
Equivalence relation: A relation that is reflexive, symmetric, and transitive. Equality has these three properties, which govern valid reasoning with equal expressions.
Approximate equality: A relation indicating that two quantities are close in value rather than exactly equal. The symbol ≈ signals a weaker numerical claim than the exact equality sign.
Linear equation: An equation in which each variable appears only to the first power. Its solutions are found by preserving equality while isolating a variable.
Equals sign: The symbol “=” used to assert that the expressions on its two sides have the same value. It is the standard notation that marks an equality between expressions.
Substitution property of equality: The rule that equal quantities can replace one another within mathematical statements without changing truth. It licenses replacing one side's value with the other's throughout a calculation.
Definitional equality: Equality established by the meaning or stipulated definition of an expression. It contrasts identities grounded in definition with equalities requiring proof or calculation.
Polynomial identity: An equality between polynomials that holds for every value of their variables. It exemplifies equality as a universal claim about expression values.
Variable: A symbol that represents a number or another mathematical object, often one whose value is unknown. An equality containing variables can hold for some values and fail for others.
Addition property of equality: The rule that adding the same quantity to both sides of an equality preserves it. It enables rearranging an equation while keeping its solutions unchanged.

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