View source for Formula

{{Short description|Concise way of expressing information symbolically}}
{{other uses}}
{{multiple image
| footer    = On the left is a [[sphere]], whose volume {{math|''V''{{spaces|thin}}}} is given by the mathematical formula {{math|1=''V'' {{=}} {{sfrac|4|3}} &pi; ''r''<sup>3</sup>}}. On the right is the compound [[isobutane]], which has chemical formula (CH<sub>3</sub>)<sub>3</sub>CH.
| width     = 130

| image1    = Sphere_wireframe_10deg_6r.svg
| alt1      = A sphere

| image2    = Isobutane_numbered_2D.svg
| alt2      = Isobutane
}}
[[File:Edsger Dijkstra 1994.jpg|thumb|upright|One of the most influential figures of [[computing science]]'s [[list of pioneers in computer science|founding generation]], [[Edsger Dijkstra]] at the blackboard during a conference at [[ETH Zurich]] in 1994. In Dijkstra's own words, "[[A picture is worth a thousand words|A picture may be worth a thousand words]], a formula is worth a thousand pictures."<ref>[[Edsger Dijkstra|Dijkstra, E.W.]] (July 1996), ''[https://www.cs.utexas.edu/users/EWD/transcriptions/EWD12xx/EWD1239.html A first exploration of effective reasoning]'' [EWD896]. (E.W. Dijkstra Archive, Center for American History, [[University of Texas at Austin]])</ref>]]

In [[science]], a '''formula''' is a concise way of expressing information symbolically, as in a '''mathematical formula''' or a ''[[chemical formula]]''. The informal use of the [[terminology|term]] ''formula'' in science refers to the [[Commensurability (philosophy of science)|general construct of a relationship between given quantities]].  

The plural of ''formula'' can be either ''formulas'' (from the most common [[English plurals#Regular plurals|English plural noun form]]) or, under the influence of [[scientific Latin]], ''formulae'' (from the [[Latin influence in English|original Latin]]).<ref name="oxford">{{OED|formula}}</ref>

==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|orig-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.

==Chemical formulas==
{{main|Chemical formula}}

In [[Chemistry#Principles of modern chemistry|modern chemistry]], a [[chemical formula]] is a way of expressing information about the proportions of [[atom]]s that constitute a particular [[chemical compound]], using a single line of chemical [[chemical symbols|element symbols]], [[numbers]], and sometimes other symbols, such as parentheses, brackets, and plus (+) and minus (−) signs.<ref>Atkins, P.W., Overton, T., Rourke, J., Weller, M. and Armstrong, F. ''Shriver and Atkins inorganic chemistry'' (4th edition) 2006 ([[Oxford University Press]]) {{isbn|0-19-926463-5}}</ref> For example, H<sub>2</sub>O is the chemical formula for [[water]], specifying that each [[molecule]] consists of two [[hydrogen]] (H) atoms and one [[oxygen]] (O) atom. Similarly, O{{sup sub|−|3}} denotes an [[ozone]] molecule consisting of three oxygen atoms<ref>{{Cite web|url=http://www.chm.bris.ac.uk/motm/ozone/CHEM.htm|title=Ozone Chemistry|website=www.chm.bris.ac.uk|access-date=2019-11-26}}</ref> and a net [[negative charge]].

{{Image frame|width=300|content=<chem>H-\overset{\displaystyle H \atop |}{\underset{| \atop \displaystyle H}{C}}-\overset{\displaystyle H \atop |}{\underset{| \atop \displaystyle H}{C}}-\overset{\displaystyle H \atop |}{\underset{| \atop \displaystyle H}{C}}-\overset{\displaystyle H \atop |}{\underset{| \atop \displaystyle H}{C}}-H</chem>|caption=The '''[[structural formula]]''' for [[butane]]. There are three common non-pictorial types of chemical formulas for this molecule:{{unordered list
 | the '''empirical formula''' C<sub>2</sub>H<sub>5</sub>
 | the '''molecular formula''' C<sub>4</sub>H<sub>10</sub> and
 | the '''condensed formula''' (or ''semi-structural formula'') CH<sub>3</sub>CH<sub>2</sub>CH<sub>2</sub>CH<sub>3</sub>.}}
|alt=A diagram showing one line of four connected Cs (carbon atoms) branching out to ten Hs (hydrogen atoms)}}

A [[chemical formula]] identifies each constituent [[chemical element|element]] by its [[chemical symbol]], and indicates the proportionate number of atoms of each element.

In [[empirical formula]]s, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound—as ratios to the key element. For molecular compounds, these ratio numbers can always be expressed as whole numbers. For example, the empirical formula of [[ethanol]] may be written C<sub>2</sub>H<sub>6</sub>O,<ref>{{Cite web|url=https://pubchem.ncbi.nlm.nih.gov/compound/702|title=Ethanol|last=PubChem|website=pubchem.ncbi.nlm.nih.gov|language=en|access-date=2019-11-26}}</ref> because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom. Some types of ionic compounds, however, cannot be written as empirical formulas which contains only the whole numbers. An example is [[boron carbide]], whose formula of CB<sub>n</sub> is a variable non-whole number ratio, with n ranging from over 4 to more than 6.5.

When the chemical compound of the formula consists of simple [[molecule]]s, chemical formulas often employ ways to suggest the structure of the molecule. There are several types of these formulas, including [[molecular formula]]s and [[condensed formula]]s. A molecular formula enumerates the number of atoms to reflect those in the molecule, so that the molecular formula for [[glucose]] is C<sub>6</sub>H<sub>12</sub>O<sub>6</sub> rather than the glucose empirical formula, which is CH<sub>2</sub>O. Except for the very simple substances, molecular chemical formulas generally lack needed structural information, and might even be ambiguous in occasions.

A [[structural formula]] is a drawing that shows the location of each atom, and which atoms it binds to.

==In computing==
In [[computing]], a formula typically describes a [[calculation]], such as addition, to be performed on one or more variables. A formula is often implicitly provided in the form of a [[Instruction (computer science)|computer instruction]] such as.

: ''Degrees Celsius'' = (5/9)*(''Degrees Fahrenheit''&nbsp; - 32)

In computer [[spreadsheet]] software, a formula indicating how to compute the value of a [[cell reference|cell]], say ''A3'', could be written as

: ''=A1+A2''

where ''A1'' and ''A2'' refer to other cells (column A, row 1 or 2) within the spreadsheet. This is a shortcut for the "paper" form ''A3 = A1+A2'', where ''A3'' is, by convention, omitted because the result is always stored in the cell itself, making the stating of the name redundant.

==Units==
Formulas used in science almost always require a choice of units.<ref>{{Cite book | isbn = 978-1466571143 | title = CRC Handbook of Chemistry and Physics, 94th Edition | editor1-last = Haynes | editor1-first= William M. | year = 2013 | orig-year = 1914 | publisher = CRC Press | location = Boca Raton  }}</ref> Formulas are used to express relationships between various quantities, such as temperature, mass, or charge in physics; supply, profit, or demand in economics; or a wide range of other quantities in other disciplines.

An example of a formula used in science is [[Boltzmann's entropy formula]]. In [[statistical thermodynamics]], it is a probability equation relating the [[entropy]] ''S'' of an ideal gas to the quantity ''W'', which is the number of [[Microstate (statistical mechanics)|microstates]] corresponding to a given [[macrostate]]:
: <math>S = k \cdot \ln W </math>
where ''k'' is the [[Boltzmann constant]], equal to {{physconst|k|ref=no}}, and ''W'' is the number of [[Microstate (statistical mechanics)|microstate]]s consistent with the given [[macrostate]].

== See also ==
* [[Formula editor]]
* [[Formula unit]]
* [[Law (mathematics)]]
* [[Mathematical notation]]
* [[Scientific law]]
* [[Symbol (chemical element)]]
* [[Theorem]]
* [[Well-formed formula]]

== References ==
{{reflist}}

[[Category:Mathematical notation]]
[[Category:Elementary algebra]]