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| {{multiple image
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| | footer = On the left is a [[sphere]], whose volume is given by the mathematical formula {{math|1=''V'' {{=}} {{sfrac|4|3}} π ''r''<sup>3</sup>}}. On the right is the compound [[isobutane]], which has chemical formula (CH<sub>3</sub>)<sub>3</sub>CH.
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| | width = 175
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| | image1 = Sphere_wireframe_10deg_6r.svg
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| | alt1 = A sphere
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| | image2 = Isobutane-numbered-2D.png
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| | alt2 = Isobutane
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| }}
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| In [[science]], a '''formula''' is a concise way of expressing information symbolically as in a mathematical or [[chemical formula]]. The informal use of the 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 spelled either as ''formulas'' or ''formulae'' (from the original Latin).<ref name="oxford">[http://oxforddictionaries.com/view/entry/m_en_gb0311400 Oxford Dictionaries: formula].</ref>
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| In [[mathematics]], a formula is an entity constructed using the symbols and formation rules of a given [[formal language|logical language]].<ref>{{Citation
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| |last=Rautenberg
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| |first=Wolfgang
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| |authorlink=Wolfgang Rautenberg
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| |doi=10.1007/978-1-4419-1221-3
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| |title=A Concise Introduction to Mathematical Logic
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| |url=http://www.springerlink.com/content/978-1-4419-1220-6/
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| |publisher=[[Springer Science+Business Media]]
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| |location=[[New York City|New York, NY]]
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| |edition=3rd
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| |isbn=978-1-4419-1220-6
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| |year=2010
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| }}</ref> For example, determining the [[volume]] of a [[sphere]] requires a significant amount of [[integral calculus]] or its geometrical analogue, the [[method of exhaustion]];<ref>{{cite book
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| | last = Smith
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| | first = David E
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| | year = 1958
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| | title = History of Mathematics
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| | publisher = Dover Publications
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| | location = New York
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| | isbn = 0-486-20430-8
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| }}</ref> but, having done this once, mathematicians can produce a formula to describe the volume in terms of some other parameter (the [[radius]] for example). This particular formula is:
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| <center>{{math|big=1|''V'' {{=}} {{sfrac|4|3}} π ''r''<sup>3</sup>}}</center>
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| Having obtained this result, and knowing the radius of the sphere in question, we can quickly and easily determine its volume. Note 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 larger and more complex formulas.<ref>{{cite web
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| |url = http://math.stackexchange.com/questions/24241/why-do-mathematicians-use-single-letter-variables
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| |title = Why do mathematicians use single letter variables?
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| |date = 28 February 2011
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| |website = math.stackexchange.com
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| |accessdate = 31 December 2013
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| }}</ref> Mathematical formulas are often [[algebraic expression|algebraic]], [[closed-form expression|closed form]], and/or [[analytical expression|analytical]].
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| 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 and having a net negative charge.
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| In a general context, formulas are applied to provide a mathematical solution for real world problems. Some may be general: {{math|1='''F''' = ''m'''''a'''}}, which is one expression of [[Newton's laws of motion|Newton's second law]], is applicable to a wide range of physical situations. Other formulas may be specially created to solve a particular problem; for example, using the [[equation]] of a [[sine curve]] to model the movement of the tides in a bay. In all cases, however, formulas form the basis for all calculations.
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| [[Expression (mathematics)|Expression]]s are distinct from formulas in that they cannot contain an equals sign (=).<ref>{{Citation
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| |last1=Hamilton
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| |first1=A.G.
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| |title=Logic for Mathematicians
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| |publisher=Cambridge University Press
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| |location=Cambridge
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| |edition=2nd
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| |isbn=978-0-521-36865-0
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| |year=1988}}</ref> Whereas formulas are comparable to sentences, expressions are more like phrases.
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| ==Chemical formulas==
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| 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 formulas, 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 all be expressed as whole numbers. For example, the empirical formula of [[ethanol]] may be written C<sub>2</sub>H<sub>6</sub>O 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 with entirely whole-number empirical formulas. 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.
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| When the chemical compound of the formula consists of simple [[molecule]]s, chemical formulas often employ ways to suggest the structure of the molecule. These types of formulas are variously known as '''molecular formulas''' and '''[[structural formula|condensed formulas]]'''. 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. However, except for very simple substances, molecular chemical formulas lack needed structural information, and are ambiguous.
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| == In computing ==
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| 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 [[computer]] [[Instruction (computer science)|instruction]] such as.
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| : ''Degrees Celsius'' = (5/9)*(''Degrees Fahrenheit'' -32)
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| In computer [[spreadsheet]] software, a formula indicating how to compute the value of a [[cell reference|cell]], say ''A3'', is written such as
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| : ''=A1+A2''
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| 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 and stating its name would be redundant.
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| ==Formulas with prescribed units ==
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| A [[physical quantity]] can be expressed as the product of a number and a [[Units of measurement|physical unit]]. A formula expresses a relationship between physical quantities. A necessary condition for a formula to be valid is that all terms [[Dimensional analysis#Commensurability|have the same dimension]], meaning every term in the formula could be potentially converted to contain the identical unit (or product of identical units).<ref>{{cite isbn|159126099X}}</ref>
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| In the example above, for the volume of a sphere, we may wish to compute with ''r'' = 2.0 cm, which yields
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| :<math>V = \frac{4}{3}\pi(2.0 \mbox{ cm})^3 = 33.47 \mbox{ cm}^{3}.</math> | |
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| There is vast educational training about retaining units in computations, and converting units to a desirable form, such as in [[units conversion by factor-label]].
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| However, the vast majority of computations with measurements are done in computer programs with no facility for retaining a symbolic computation of the units. Only the numerical quantity is used in the computation. This requires that the universal formula be converted to a formula that is intended to be used only with prescribed units, meaning the numerical quantity is implicitly assumed to be multiplying a particular unit. The requirements about the prescribed units must be given to users of the input and the output of the formula.
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| For example suppose the formula is to require that <math> V \equiv \mathrm{VOL}~\bold{tbsp}</math>, where '''tbsp''' is the U.S. tablespoon (as seen in [[conversion of units]]) and VOL is the name for the number used by the computer. Similarly, the formula is to require
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| <math> r \equiv \mathrm{RAD}~\bold{cm}</math>. The derivation of the formula proceeds as:
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| : <math> \mathrm{VOL}~\bold{tbsp} = \frac{4}{3} \pi \mathrm{RAD}^3~ \bold{cm}^3.</math> | |
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| Given that <math>1~\bold{tbsp} = 14.787~\bold{cm}^3 </math>,
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| the formula with prescribed units is
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| : <math> \mathrm{VOL} = 0.2933~\mathrm{RAD}^3.</math>
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| The formula is not complete without words such as:
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| "VOL is volume in '''tbsp''' and RAD is radius in '''cm'''".
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| Other possible words are "VOL is the ratio of <math>V</math> to '''tbsp''' and RAD is the ratio of <math>r</math> to '''cm'''."
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| The formula with prescribed units could also appear with simple symbols,
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| perhaps even the identical symbols as in the original dimensional formula:
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| : <math> V = 0.2833~r^3.</math>
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| and the accompanying words could be: "where V is volume ('''tbsp''') and r is radius ('''cm''')".
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| If the physical formula is not dimensionally homogeneous, and therefore erroneous,
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| the falsehood becomes apparent in the impossibility
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| to derive a formula with prescribed units. It would not be possible to
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| derive a formula consisting only of numbers and dimensionless ratios.
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| ===In Science===
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| Formulas used in science almost always require a choice of units.<ref>{{cite isbn|1466571144}}</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.
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| 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]]:
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| :<math>S = k \cdot \log W \! </math> (1) S= k ln W
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| where ''k'' is [[Boltzmann constant|Boltzmann's constant]] equal to 1.38062 x 10<sup>−23</sup> joule/kelvin and ''W'' is the number of [[Microstate (statistical mechanics)|microstate]]s consistent with the given [[macrostate]].
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| == See also ==
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| {{wiktionary|formula}}
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| * [[Formula (mathematical logic)]]
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| * [[Formula unit]]
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| * [[Mathematical notation]]
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| * [[Symbol (chemical element)]]
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| ==References==
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| <references />
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| [[Category:Mathematical notation]]
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