Definition:Equals

Definition
The symbol $=$ means equals.
 * $x = y$ means $x$ is the same object as $y$, and is read $x$ equals $y$, or $x$ is equal to $y$.


 * $x \ne y$ means $x$ is not the same object as $y$, and is read $x$ is not equal to $y$.

The expression:
 * $a = b$

means:
 * $a$ and $b$ are names for the same object.

Equality
The word equality is the noun derived from the verb equals.

Note on Terminology
Two objects being equal is not necessarily the same as two objects being congruent. This distinction is often not made. When such a difference is important the symbol $=$ may be used for equals and $\cong$ for congruent.

Also see

 * Equals is Equivalence Relation
 * Axiom:Leibniz's Law
 * Definition:Set Equality

Historical Note
The equals sign was introduced by in his 1557 work:
 * The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers.

Placing two hyphens together, one above the other, he wrote:
 * "To avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe: $= \!\!\! = \!\!\! = \!\!\! = \!\!\! =$, bicause noe .2. thynges, can be moare equalle."

The word gemowe comes from the Latin geminus meaning twin.

used the symbol $\sim$, while used $\propto$. Both were in due course supplanted by $=$.