Editing Formula (section)

==In mathematics==
In [[mathematics]], a formula generally refers to an [[equation]] or [[Inequality (mathematics)|inequality]] relating one [[mathematical expression]] to another, with the most important ones being [[Mathematical theorem|mathematical theorems]]. For example, determining the [[volume]] of a [[sphere]] requires a significant amount of [[integral calculus]] or its [[Geometry|geometrical]] analogue, the [[method of exhaustion]].<ref>{{cite book
 | last = Smith
 | first = David E.
 | author-link = David Eugene Smith
 | year = 1958
 | title = History of Mathematics
 | publisher = [[Dover Publications]]
 | location = [[New York City|New York]]
 | isbn = 0-486-20430-8
}}</ref> However, having done this once in terms of some [[parameter]] (the [[radius]] for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius:

: <math>V = \frac{4}{3} \pi r^3.</math>

Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume ''V'' and the radius ''r'' are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex.<ref>{{cite web
 |url        = https://math.stackexchange.com/q/24241
 |title      = Why do mathematicians use single letter variables?
 |date       = 28 February 2011
 |website    = [[Stackexchange.com|math.stackexchange.com]]
 |access-date = 31 December 2013
}}</ref><!-- StackExchange is a forum, not a proper citation source. If proper source cannot be found, then the source should be removed.  --> Mathematical formulas are often [[algebraic equation|algebraic]], [[analytical expression|analytical]] or in [[closed-form expression|closed form]].<ref>{{cite web
 |url        = https://www.andlearning.org/math-formula/
 |title      = List of Mathematical formulas
 |date       = 24 August 2018
 |website    = andlearning.org
}}</ref>

In a general context, formulas often represent mathematical models of real world phenomena, and as such can be used to provide solutions (or approximate solutions) to real world problems, with some being more general than others. For example, the formula

: <math>F = ma</math>

is an expression of [[Newton's laws of motion|Newton's second law]], and is applicable to a wide range of physical situations. Other formulas, such as the use of the [[equation]] of a [[sine curve]] to model the [[Tidal movement|movement of the tides]] in a [[bay]], may be created to solve a particular problem. In all cases, however, formulas form the basis for calculations.

[[Expression (mathematics)|Expression]]s are distinct from formulas in the sense that they don't usually contain [[Relation (mathematics)|relations]] like [[Equality (mathematics)|equality]] (=) or [[Inequality (mathematics)|inequality]] (<).  Expressions denote a [[mathematical object]], where as formulas denote a statement about mathematical objects.<ref>{{cite book|first=Robert R.|last=Stoll|year=1963|title=Set Theory and Logic|publisher=Dover Publications|location=San Francisco, CA|isbn=978-0-486-63829-4}}</ref><ref>{{Citation
  |last1=Hamilton
  |first1=A. G.
  |title=Logic for Mathematicians
  |publisher=[[Cambridge University Press]]
  |location=[[Cambridge]]
  |edition=2nd
  |isbn=978-0-521-36865-0
  |year=1988}}</ref>{{dubious|Expressions containing equals signs|date=May 2023}} This is analogous to natural language, where a [[noun phrase]] refers to an object, and a whole [[Sentence (linguistics)|sentence]] refers to a fact. For example, <math>8x-5</math> is an expression, while <math>8x-5 \geq 3 </math> is a formula.

However, in some areas mathematics, and in particular in [[computer algebra]], formulas are viewed as expressions that can be evaluated to ''[[Logical truth|true]]'' or ''[[False (logic)|false]]'', depending on the values that are given to the variables occurring in the expressions. For example <math>8x-5 \geq 3</math> takes the value ''false'' if {{mvar|x}} is given a value less than 1, and the value ''true'' otherwise. (See [[Boolean expression]])

===In mathematical logic===
In [[mathematical logic]], a formula (often referred to as a ''[[well-formed formula]]'') is an entity constructed using the symbols and formation rules of a given [[formal language|logical language]].<ref>{{Citation
  |last=Rautenberg
  |first=Wolfgang
  |author-link=Wolfgang Rautenberg
  |doi=10.1007/978-1-4419-1221-3
  |title=A Concise Introduction to Mathematical Logic
  |publisher=[[Springer Science+Business Media]]
  |location=[[New York City|New York, NY]]
  |edition=3rd
  |isbn=978-1-4419-1220-6
  |year=2010
}}</ref> For example, in [[first-order logic]],
:<math>\forall x \forall y (P(f(x)) \rightarrow\neg (P(x) \rightarrow Q(f(y),x,z)))</math>
is a formula, provided that <math>f</math> is a unary function symbol, <math>P</math> a unary predicate symbol, and <math>Q</math> a ternary predicate symbol.