Editing Inequality (mathematics) (section)

==Notation==
There are several different notations used to represent different kinds of inequalities:
* The notation ''a'' < ''b'' means that ''a'' is '''less than''' ''b''.
* The notation ''a'' > ''b'' means that ''a'' is '''greater than''' ''b''.
In either case, ''a'' is not equal to ''b''. These relations are known as '''strict inequalities''',<ref name=":0" /> meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality is excluded.

In contrast to strict inequalities, there are two types of inequality relations that are not strict:
* The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' or ''a'' ≦ ''b'' means that ''a'' is '''less than or equal to''' ''b'' (or, equivalently, at most ''b'', or not greater than ''b'').
* The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' or  ''a'' ≧ ''b'' means that ''a'' is '''greater than or equal to''' ''b'' (or, equivalently, at least ''b'', or not less than ''b'').

In the 17th and 18th centuries, personal  notations  or  typewriting signs were used to signal inequalities.<ref>{{cite journal |first1=Elena | last1= Halmaghi |first2=Peter | last2= Liljedahl |title=Inequalities in the History of Mathematics: From Peculiarities to a Hard Discipline |journal=Proceedings of the 2012 Annual Meeting of the Canadian Mathematics Education Study Group}}</ref> For example, In 1670, [[John Wallis]] used a single horizontal bar ''above''  rather than below the < and >.
Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or  "less than (greater than) or equal to with double horizontal bars", first appeared in [[Pierre Bouguer]]'s work .<ref>{{cite web |title=Earliest Uses of Symbols of Relation |url=https://mathshistory.st-andrews.ac.uk/Miller/mathsym/relation/ |website=MacTutor |publisher=University of St Andrews, Scotland}}</ref> After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).

The relation '''not greater than''' can also be represented by <math>a \ngtr b,</math> the symbol for "greater than" bisected by a slash, "not". The same is true for '''not less than''', <math>a \nless b.</math>

The notation ''a'' ≠ ''b'' means that ''a'' is not equal to ''b''; this ''[[inequation]]'' sometimes is considered a form of strict inequality.<ref name=":1">{{Cite web|url=http://www.learnalberta.ca/content/memg/Division03/Inequality/index.html| title=Inequality| website=www.learnalberta.ca|access-date=2019-12-03}}</ref> It does not say that one is greater than the other; it does not even require ''a'' and ''b'' to be member of an [[ordered set]].

In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another,<ref name="Polyanin2006">{{cite book | last1=Polyanin | first1=A.D. | last2=Manzhirov | first2=A.V. | title=Handbook of Mathematics for Engineers and Scientists | publisher=CRC Press | year=2006 | isbn=978-1-4200-1051-0 | url=https://books.google.com/books?id=ge6nk9W0BCcC&pg=PR29 | access-date=2021-11-19 | page=29}}</ref> normally by several [[Order of magnitude|orders of magnitude]].  
* The notation ''a'' ≪ ''b'' means that ''a'' is '''much less than''' ''b''.<ref>{{Cite web|url=http://mathworld.wolfram.com/MuchLess.html|title=Much Less|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>
* The notation ''a'' ≫ ''b'' means that ''a'' is '''much greater than''' ''b''.<ref>{{Cite web|url=http://mathworld.wolfram.com/MuchGreater.html|title=Much Greater|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>
This implies that the lesser value can be neglected with little effect on the accuracy of an [[approximation]] (such as the case of [[ultrarelativistic limit]] in physics).

In all of the cases above, any two symbols mirroring each other are symmetrical; ''a'' < ''b'' and ''b'' > ''a'' are equivalent, etc.