Editing Equality (mathematics) (section)

=== Definitions ===
Equations are often used to introduce new terms or symbols for constants, [[Judgment (mathematical logic)|assert]] equalities, and introduce shorthand for complex expressions, which is called "equal by definition", and often denoted with (<math>:=</math>).<ref>{{Cite web |last1=Lankham |first1=Isaiah |last2=Nachtergaele |first2=Bruno |last3=Schilling |first3=Anne |date=January 21, 2007 |title=Some Common Mathematical Symbols and Abbreviations (with History) |url=https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Common_Math_Symbols.pdf |publisher=[[University of California, Davis]]}}</ref> It is similar to the concept of [[Assignment (computer science)|assignment]] of a [[Variable (computer science)|variable in computer science]]. For example, <math display = "inline">\mathbb{e} := \sum_{n=0}^\infty \frac{1}{n!}</math> defines [[Euler's number]],<ref>{{Cite encyclopedia |title=e |url=https://www.britannica.com/science/e-mathematics |access-date=2025-01-13 |encyclopedia=Encyclopædia Britannica}}</ref> and <math>i^2 = -1</math> is the defining property of the [[imaginary number]] <math>i.</math><ref>{{Cite book |last1=Marecek |first1=Lynn |last2=Mathis |first2=Andrea Honeycutt |date=2020-05-06 |title=Intermediate Algebra 2e |chapter=8.8 Use the Complex Number System |chapter-url=https://openstax.org/books/intermediate-algebra-2e/pages/8-8-use-the-complex-number-system |access-date=2025-03-04 |publisher=OpenStax |isbn=978-1-975076-49-8}}</ref>

In [[mathematical logic]], this is called an [[extension by definition]] (by equality) which is a [[conservative extension]] to a [[formal system]].{{Sfn|Mendelson|1964|pp=82–83}} This is done by taking the equation defining the new constant symbol as a new [[axiom]] of the [[Theory (mathematical logic)|theory]]. The first recorded symbolic use of "Equal by definition" appeared in ''Logica Matematica'' (1894) by [[Cesare Burali-Forti]], an Italian mathematician. Burali-Forti, in his book, used the notation (<math>=_\text{Def}   </math>).<ref>{{Cite book |last=Burali-Forti |first=Cesare |author-link=Cesare Burali-Forti |url=https://books.google.com/books?id=F5xJAAAAIAAJ |title=Logica matematica |date=1894 |publisher=[[Ulrico Hoepli]] |others=University of California |page=120 |language=it |trans-title=Mathematical logic |archive-url=https://archive.org/details/logicamatematic01buragoog/page/n131/mode/2up?q=def |archive-date=2009-08-01}}</ref><ref>{{Cite book |date=2013-11-07 |last1=Lankham |first1=Isaiah |last2=Nachtergaele |first2=Bruno |last3=Schilling |first3=Anne |title=Linear Algebra |chapter=13.3: Some Common Mathematical Symbols and Abbreviations |chapter-url=https://math.libretexts.org/Bookshelves/Linear_Algebra/Book:_Linear_Algebra_(Schilling_Nachtergaele_and_Lankham)/13:_Appendices/13.03:_Some_Common_Mathematical_Symbols_and_Abbreviations#:~:text=:=%20(the%20equal%20by%20definition,especially%20common%20in%20applied%20mathematics. |access-date=2025-03-04 |publisher=Mathematics LibreTexts, University of California, Davis}}</ref>