Equality (mathematics)

In mathematics, equality is a fundamental relationship between two quantities or expressions. It states that they possess the same value or represent the same mathematical object.

When we write this relationship using a symbol, we call the result an equation. If two objects do not share this relationship, they are described as distinct. While it seems simple, equality is often treated as a primitive notion. This means it is not formally defined by other ideas, but is instead seen as a basic relation that every thing bears to itself.

To understand how equality functions, we look at its core logical properties. The first is reflexivity, which states that every object is equal to itself. The second is symmetry, meaning if one value equals a second, then the second must equal the first. The third is transitivity, which dictates that if A equals B and B equals C, then A must equal C.

Beyond these, there is the substitution property. This allows a mathematician to replace one part of an expression with an equal value without changing the overall meaning. These rules ensure that mathematical logic remains consistent and predictable.

Mathematicians formalize equality in two primary ways: through logic or through set theory. In the field of logic, equality is treated as a primitive predicate. This includes the reflexive property, known as the law of identity, and the substitution property. From these starting points, all other necessary properties can be derived. In contrast, set theory provides a different foundation. In this system, specifically Zermelo–Fraenkel set theory, two sets are defined as equal if they contain exactly the same members. This specific rule is known as the axiom of extensionality.

The history of equality spans thousands of years of human thought. Around 350 BC, the Greek philosopher Aristotle discussed equality in his work, *Categories*. He argued that equality is a mark of quantity, such as number, length, or volume. He believed that qualities, like the whiteness of an object, could be similar but not truly equal. Much later, in the late 19th century, Giuseppe Peano began to explicitly state the fundamental properties of equality as formal mathematical rules.

These developments moved equality from an intuitive concept to a rigorous logical tool.

Our modern symbol for equality has a specific origin. Before the 16th century, mathematicians used words like *aequales* or *gleich* to show equality. The equals sign (=) was first recorded by the Welsh mathematician Robert Recorde in his 1557 book, *The Whetstone of Witte*.

Recorde chose two parallel lines because he believed no two things could be more equal than two lines of the same length. His original symbol was much wider than the one we use today. It was not immediately popular, appearing again in print only 61 years later in 1618. Eventually, influential mathematicians like Isaac Newton and Gottfried Leibniz helped it spread across Europe.

Equations serve many different purposes in mathematical practice. An equation can be a simple statement that is either true or false, such as 2 + 2 = 4. In algebra, equations often contain variables, which are unknown values. Solving an equation means finding the specific values that make the equality true.

This is often visualized using a balance scale, where both sides must remain level. There is also a distinction between an equation and an identity. An identity is a special kind of equality that remains true for every possible value of its variables. For example, certain mathematical rules work for every real number, making them identities rather than conditional equations.

Equality connects deeply to many advanced branches of mathematics. In calculus, the study of change, equations often involve functions and their derivatives. A functional equation is one where the unknown is a function rather than a single number. When these equations involve derivatives, they are called differential equations. Furthermore, equality is essential to the concept of functions. The extensionality of functions allows mathematicians to view an identity as an equality between two different functional mappings. This interconnectedness makes equality the bedrock of mathematical reasoning and discovery.