In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. Equality between A and B is denoted with an equals sign as A = B, and read "A equals B". A written expression of equality is called an equation or identity depending on the context. Two objects that are not equal are said to be distinct.

Equality is often considered a primitive notion, meaning it is not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"), reflecting a general conceptual difficulty in fully characterizing the concept. Basic properties about equality like reflexivity, symmetry, and transitivity have been understood intuitively since at least the ancient Greeks, but were not symbolically stated as general properties of relations until the late 19th century by Giuseppe Peano. Other properties like substitution and function application weren't formally stated until the development of symbolic logic.

There are generally two ways that equality is formalized in mathematics: through logic or through set theory. In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the law of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality. After the foundational crisis in mathematics at the turn of the 20th century, set theory (specifically Zermelo–Fraenkel set theory) became the most common foundation of mathematics. In set theory, any two sets are defined to be equal if they have all the same members. This is called the axiom of extensionality.

In English, the word equal is derived from the Latin aequālis ('like', 'comparable', 'similar'), which itself stems from aequus ('level', 'just'). The word entered Middle English around the 14th century, borrowed from Old French equalité (modern égalité). More generally, the interlingual synonyms of equal have been used more broadly throughout history (see § Geometry).

Before the 16th century, there was no common symbol for equality, and equality was usually expressed with a word, such as aequales, aequantur, esgale, faciunt, ghelijck, or gleich, and sometimes by the abbreviated form aeq, or simply ⟨æ⟩ and ⟨œ⟩. Diophantus's use of ⟨ἴσ⟩, short for ἴσος (ísos 'equals'), in Arithmetica (c. 250 AD) is considered one of the first uses of an equals sign.

The sign =, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte (1557), just one year before his death. The original form of the symbol was much wider than the present form. In his book, Recorde explains his symbol as "Gemowe lines", from the Latin gemellus ('twin'), using two parallel lines to represent equality because he believed that "no two things could be more equal."

Recorde's symbol was not immediately popular. After its introduction, it wasn't used again in print until 1618 (61 years later), in an anonymous Appendix in Edward Wright's English translation of Descriptio, by John Napier. It wasn't until 1631 that it received more than general recognition in England, being adopted as the symbol for equality in a few influential works. Later used by several influential mathematicians, most notably, both Isaac Newton and Gottfried Leibniz, and due to the prevalence of calculus at the time, it quickly spread throughout the rest of Europe.

The first three properties are generally attributed to Giuseppe Peano for being the first to explicitly state these as fundamental properties of equality in his Arithmetices principia (1889). However, the basic notions have always existed; for example, in Euclid's Elements (c. 300 BC), he includes 'common notions': "Things that are equal to the same thing are also equal to one another" (transitivity), "Things that coincide with one another are equal to one another" (reflexivity), along with some function-application properties for addition and subtraction. The function-application property was also stated in Peano's Arithmetices principia, however, it had been common practice in algebra since at least Diophantus (c. 250 AD). The substitution property is generally attributed to Gottfried Leibniz (c. 1686), and often called Leibniz's Law.