Editing Tilde (section)

====As a relational operator====

In [[mathematics]], the tilde operator (which can be represented by a tilde or the dedicated character {{unichar|223C|TILDE OPERATOR}}), sometimes called "twiddle", is often used to denote an [[equivalence relation]] between two objects. Thus "{{math|{{mvar|x}} ~ {{mvar|y}}}}" means "{{mvar|x}} is [[equivalence relation|equivalent]] to {{mvar|y}}". It is a weaker statement than stating that {{mvar|x}} [[equality (mathematics)|equals]] {{mvar|y}}. The expression "{{math|{{mvar|x}} ~ {{mvar |y}}}}" is sometimes read aloud as "{{mvar|x}} twiddles {{mvar|y}}", perhaps as an analogue to the verbal expression of "{{math |1={{mvar |x}} = {{mvar|y}}}}".<ref>{{Citation | last = Derbyshire | first = J | title = Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics | place = New York | publisher = Penguin | year = 2004 | url = http://mathworld.wolfram.com/Tilde.html}}.</ref>

The tilde can indicate approximate equality in a variety of ways. It can be used to denote the [[asymptotic analysis|asymptotic equality]] of two functions. For example, {{math|{{mvar|f}} ({{mvar|x}}) ~ {{mvar |g}}({{mvar |x}})}} means that <math>\lim_{x \to \infty} \frac{f(x)}{g(x)} = 1</math>.<ref name=wolfram_tilde />

A tilde is also used to indicate "[[Approximation|approximately]] equal to" (e.g. 1.902 ~= 2). This usage probably developed as a typed alternative to the [[:Image:Libra.svg|libra symbol]] used for the same purpose in written mathematics, which is an equal sign with the upper bar replaced by a bar with an upward hump, bump, or loop in the middle (︍︍♎&#xFE0E;) or, sometimes, a tilde (≃).{{Citation needed|date=October 2024}} The symbol "≈" is also used for this purpose.

In [[physics]] and [[astronomy]], a tilde can be used between two expressions (e.g. {{math|{{mvar|h}} ~ 10<sup>−34</sup> J s}}) to state that the two are of the same [[order of magnitude]].<ref name=wolfram_tilde />

In [[statistics]] and [[probability theory]], the tilde means "is distributed as";<ref name = wolfram_tilde /> see [[random variable]] (e.g. {{math|''X'' ~ ''B''(''n'', ''p'')}} for a [[binomial distribution]]).

A tilde can also be used to represent geometric [[Similarity (geometry)|similarity]] (e.g. {{math |∆{{mvar |ABC}} ~ ∆{{mvar|DEF}}}}, meaning [[triangle]] {{mvar|ABC}} is similar to {{mvar|DEF}}). A triple tilde ('''≋''') is often used to show [[congruence (geometry)|congruence]], an equivalence relation in geometry.{{Citation needed|date=October 2024}}

In [[graph theory]], the tilde can be used to represent adjacency between vertices. The edge <math>(x,y)</math> connects vertices <math>x</math> and <math>y</math> which can be said to be adjacent, and this adjacency can be denoted <math>x \sim y</math>.