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{{short description|Function or value which does not change during a process}}
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'''Константа''' је [[број]] или именована бројевна вредност која се у току рачуна не мења и представља супротност [[Променљива (математика)|променљивој]] чија вредност се у сваком тренутку може променити. Константа може имати одређену или неодређену вредност.
Генерално у [[математика|математици]], реч ''константа'' може да има вишеструка значања. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other [[Value (mathematics)|value]]); as a noun, it has two different meanings:
* A fixed and well-defined [[number]] or other non-varying [[mathematical object]].<ref name=":0">{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-08|website=Math Vault|language=en-US}}</ref> The terms ''[[mathematical constant]]'' or ''[[physical constant]]'' are sometimes used to distinguish this meaning.
* A [[Function (mathematics)|function]] whose value remains unchanged (i.e., a [[constant function]]).<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Constant|url=https://mathworld.wolfram.com/Constant.html|access-date=2020-08-08|website=mathworld.wolfram.com|language=en}}</ref> Such a constant is commonly represented by a [[variable (mathematics)|variable]] which does not depend on the main variable(s) in question. This is the case, for example, for a [[constant of integration]], which is an arbitrary constant function (i.e., one that does not depend on the variable of integration) added to a particular [[antiderivative]] to get all the antiderivatives of the given function.
For example, a general [[quadratic function]] is commonly written as:
:<math>a x^2 + b x + c\, ,</math>
where {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are constants (or parameters), and {{mvar|x}} a [[Variable (mathematics)|variable]]—a placeholder for the [[Argument of a function|argument]] of the function being studied. A more explicit way to denote this function is
:<math>x\mapsto a x^2 + b x + c \, ,</math>
which makes the function-argument status of {{mvar|x}} (and by extension the constancy of {{mvar|a}}, {{mvar|b}} and {{mvar|c}}) clear. In this example {{mvar|a}}, {{mvar|b}} and {{mvar|c}} are [[coefficient]]s of the [[polynomial]]. Since {{mvar|c}} occurs in a term that does not involve {{mvar|x}}, it is called the [[Constant term|constant term of the polynomial]] and can be thought of as the coefficient of {{mvar|x<sup>0</sup>}}. More generally, any polynomial term or expression of [[Degree of a polynomial|degree]] zero is a constant.<ref>{{cite book | last = Foerster | first = Paul A. | title = Algebra and Trigonometry: Functions and Applications, Teacher's Edition | edition = Classics | year = 2006 | isbn = 0-13-165711-9 | publisher = [[Prentice Hall]] | location = Upper Saddle River, NJ | url-access = registration | url = https://archive.org/details/algebratrigonome00paul_0 }}</ref>{{rp|18}}
== Константна функција ==
A constant may be used to define a [[constant function]] that ignores its arguments and always gives the same value.<ref>{{cite book|title=Encyclopedia of Mathematics|last1=Tanton|first1=James|year=2005|publisher=Facts on File, New York|isbn=0-8160-5124-0|page=94}}</ref><ref>{{cite web | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, Constant Function | author=C.Clapham, J.Nicholson | publisher =Addison-Wesley | year =2009|page=175|access-date=January 12, 2014}}</ref><ref>{{cite book|title=CRC Concise Encyclopedia of Mathematics|last1=Weisstein|first1=Eric|publisher=CRC Press, London|isbn=0-8493-9640-9|year=1999|page=313}}</ref> A constant function of a single variable, such as <math>f(x)=5</math>, has a [[graph of a function|graph]] of a horizontal straight line parallel to the ''x''-axis. Such a function always takes the same value (in this case, 5), because its argument does not appear in the expression defining the function.
[[Арност]] је the number of [[argument of a function|arguments]] or [[operand]]s taken by a [[function (mathematics)|function]] or [[operation (mathematics)|operation]] in [[logic]], [[mathematics]], and [[computer science]]. In mathematics, arity may also be named ''rank'',<ref name="Hazewinkel2001">{{cite book|author-link=Michiel Hazewinkel|first=Michiel|last=Hazewinkel|title=Encyclopaedia of Mathematics, Supplement III|url=https://books.google.com/books?id=47YC2h295JUC&pg=PA3|year=2001|publisher=Springer|isbn=978-1-4020-0198-7|page=3}}</ref><ref name="Schechter1997">{{cite book|first=Eric|last=Schechter|title=Handbook of Analysis and Its Foundations|url=https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA356|year=1997|publisher=Academic Press|isbn=978-0-12-622760-4|page=356}}</ref> but this word can have many other meanings in mathematics. In logic and philosophy, it is also called '''adicity''' and '''degree'''.<ref name="DetlefsenBacon1999">{{cite book|first1=Michael |last1=Detlefsen|first2=David Charles|last2=McCarty|first3=John B.|last3=Bacon|title=Logic from A to Z|url=https://archive.org/details/logicfromtoz0000detl|url-access=registration |year=1999|publisher=Routledge|isbn=978-0-415-21375-2|page=[https://archive.org/details/logicfromtoz0000detl/page/7 7]}}</ref><ref name="CocchiarellaFreund2008">{{cite book|first1=Nino B.|last1=Cocchiarella|first2=Max A.|last2=Freund|title=Modal Logic: An Introduction to its Syntax and Semantics|url=https://books.google.com/books?id=zLmxqytfLhgC&pg=PA121|year=2008|publisher=Oxford University Press|isbn=978-0-19-536658-7|page=121}}</ref> In [[linguistics]], it is usually named '''[[valency (linguistics)|valency]]'''.<ref name="Crystal2008">{{cite book|first=David|last=Crystal|title=Dictionary of Linguistics and Phonetics|year=2008|publisher=John Wiley & Sons|isbn=978-1-405-15296-9|page=507|edition=6th}}</ref>
== Зависност од контекста ==
The context-dependent nature of the concept of "constant" can be seen in this example from elementary calculus:
:<math>\begin{align}
\frac{d}{dx} 2^x & = \lim_{h\to 0} \frac{2^{x+h} - 2^x} h = \lim_{h\to 0} 2^x\frac{2^h - 1} h \\[8pt]
& = 2^x \lim_{h\to 0} \frac{2^h - 1} h & & \text{since } x \text{ is constant (i.e. does not depend on } h\text{)} \\[8pt]
& = 2^x \cdot\mathbf{constant,} & & \text{where }\mathbf{constant}\text{ means not depending on } x.
\end{align}</math>
"Constant" means not depending on some variable; not changing as that variable changes. In the first case above, it means not depending on ''h''; in the second, it means not depending on ''x''. A constant in a narrower context could be regarded as a variable in a broader context.<ref name=":0" />
== Значајне математичке константе ==
Some values occur frequently in mathematics and are conventionally denoted by a specific symbol.<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols: Constants|url=https://mathvault.ca/hub/higher-math/math-symbols/#Constants|access-date=2020-08-08|website=Math Vault|language=en-US}}</ref><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Constant|url=https://mathworld.wolfram.com/Constant.html|access-date=2020-08-08|website=mathworld.wolfram.com|language=en}}</ref> These standard symbols and their values are called mathematical constants. Examples include:
* 0 ([[zero]]).
* 1 ([[one]]), the [[natural number]] after zero.
* {{pi}} ([[pi]]), the constant representing the [[ratio]] of a circle's circumference to its diameter, approximately equal to 3.141592653589793238462643.<ref>{{cite book | last = Arndt | first = Jörg | last2 = Haenel | first2 = Christoph | title = Pi – Unleashed | url = https://archive.org/details/piunleashed00jarn | url-access = limited | page = [https://archive.org/details/piunleashed00jarn/page/n253 240] | year = 2001 | publisher = Springer | isbn = 978-3540665724}}</ref>
* [[e (mathematical constant)|{{mvar|e}}]], approximately equal to 2.718281828459045235360287.
* {{mvar|i}}, the [[imaginary unit]] such that {{math|''i''<sup>2</sup> {{=}} −1}}.
* <math alt="Square root of 2">\sqrt{2}</math> ([[square root of 2]]), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.
* {{mvar|φ}} ([[golden ratio]]), approximately equal to 1.618033988749894848204586, or algebraically, <math>1+ \sqrt{5} \over 2</math>.<ref name=":0" />
== Константе у рачуну ==
In [[calculus]], constants are treated in several different ways depending on the operation. For example, the [[derivative]] of a constant function is zero. This is because the derivative measures the rate of change of a function with respect to a variable, and since constants, by definition, do not change, their derivative is hence zero.
Conversely, when [[Antiderivative|integrating]] a constant function, the constant is multiplied by the variable of integration. During the evaluation of a [[limit (mathematics)|limit]], the constant remains the same as it was before and after evaluation.
Integration of a function of one variable often involves a [[constant of integration]].<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-14|website=Math Vault|language=en-US}}</ref><ref>{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals | publisher=[[Brooks/Cole]] | edition=6th | year=2008 | isbn=0-495-01166-5 | url-access=registration | url=https://archive.org/details/calculusearlytra00stew_1 }}</ref><ref>{{cite book | last1=Larson | first1=Ron | authorlink=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=[[Brooks/Cole]] | edition=9th | year=2009 | isbn=0-547-16702-4}}</ref><ref>{{Cite web|title=Definition of constant of integration {{!}} Dictionary.com|url=https://www.dictionary.com/browse/constant-of-integration|access-date=2020-08-14|website=www.dictionary.com|language=en}}</ref> This arises due to the fact that the [[integral operator]] is the [[Inverse function|inverse]] of the [[derivative|differential operator]], meaning that the aim of integration is to recover the original function before differentiation. The differential of a constant function is zero, as noted above, and the differential operator is a linear operator, so functions that only differ by a constant term have the same derivative. To acknowledge this, a constant of integration is added to an [[indefinite integral]]; this ensures that all possible solutions are included. The constant of integration is generally written as 'c', and represents a constant with a fixed but undefined value.
=== Примери ===
If {{math|''f''}} is the constant function such that <math>f(x) = 72</math> for every {{math|''x''}} then
:<math>\begin{align}
f'(x) &= 0 \\
\int f(x) \,dx &= 72x + c
\end{align}</math>
== Види још ==
Линија 7 ⟶ 66:
* [[Константа (програмирање)|Константа у програмским језицима]]
== Реферефенце ==
{{reflist}}
== Литература ==
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* Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]'' Springer-Verlag. {{ISBN|3-540-90578-2}}.
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{{refend}}
== Спољашње везе ==
{{Commonscat|Constants}}
* {{cite web | url = http://dictionary.reference.com/browse/equation | title = Equation | work = Dictionary.com | publisher = Dictionary.com, LLC | access-date = 2009-11-24 }}
* {{cite web|title=College Algebra|last1=Dawkins|first1=Paul|year=2007|publisher= Lamar University|url=http://tutorial.math.lamar.edu/Classes/Alg/Alg.aspx|page=224|access-date=January 12, 2014}}
* {{Cite web|last=Weisstein|first=Eric W.|title=Constant of Integration|url=https://mathworld.wolfram.com/ConstantofIntegration.html|access-date=2020-08-14|website=mathworld.wolfram.com|language=en}}
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