# Definition:Equals

## Definition

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

The expression:

- $a = b$

means:

- $a$ and $b$ are names for the same object.

## 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 **equal** and $\cong$ for **congruent**.

## Also see

- Equality is Equivalence Relation
- Axiom:Leibniz's Law
- Definition:Diagonal Relation
- Definition:Set Equality

## Historical Note

The **equals sign** was introduced by Robert Recorde in his $1557$ work *The Whetstone of Witte*.

Placing two long 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**.

François Viète used the symbol $\sim$, while René Descartes used $\propto$.

Both were in due course supplanted by $=$, a shortened and hence more efficient version of Recorde's invention.

It is suggested by some sources that this was mainly through the influence of Leibniz.

## Linguistic Note

The word **equality** is the noun derived from the adjective **equal**.

The word **equals** has the verb form **equate**, which means to state that one expression is **equal to** another expression.

Hence **to equate** means **to form an equation**.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) $\S 16: \ 19$ - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 2.1$. Relations on a set - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$. Sets - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Equality - 1972: Patrick Suppes:
*Axiomatic Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.2$ Logic and Notation - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**equals sign** - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**equals sign** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 6$ Significance of the results