Преглед на кодот на Степенување

{{Поврзано|полином|експоненцијална функција}}
'''Степенување''' е [[операција (математика)|математичката операција]], означувана со '''''b''<sup>''n''</sup>'''. Истата вклучува два броја, односно '''основа''' ''b'' и '''експонент''' ''n''. Друг збор за експонент е '''степен'''. Основата ''b'' се пишува на нивото и со истата големина како обичниот текст, а експонентот ''n'' се пишува непосредно десно од основата, за половина ниво погоре и со помала големина од обичниот текст, односно ''n'' се пишува како [[Горен и долен индекс|горен индекс]] на ''b''.<ref>{{Наведена мрежна страница | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, "Index (indices)" | first1=C.|last1=Clapham|first2=J.|last2=Nicholson | publisher =Addison-Wesley | year =2009|page=403|language=англиски|accessdate=1 септември 2013}}</ref>

*За ''b''<sup>''n''</sup> читаме:  &nbsp;  ''b'' на ''n''-ти  &nbsp; или  &nbsp; ''b'' на ''n''-ти степен.

==Експонентот е позитивен цел број ==
 '''Дефиниција''': <math>b^n= \underbrace{b \times \cdots \times b}_n \, , \,\,</math> &nbsp; каде што ''n'' е позитивен [[цел број]] 
*Значи, за ''n'' позитивен [[цел број]], степенување е повторно множење на ''b'' со себе ''n'' пати.
Ако ''n''=1, тогаш ''b''&sup1; = ''b'', т.е. ''b'' на ''први степен'' е ''b''.

Ако ''n''=2, тогаш ''b''&sup2; = ''b''&middot;''b'', т.е. ''b'' на ''втори степен'' или ''b'' на ''квадрат'' е ''b'' по ''b''.

Ако ''n''=3, тогаш ''b''&sup3; = ''b''&middot;''b''&middot;''b'', т.е. ''b'' на ''3-ти степен'' е ''b'' по ''b'' по ''b''.

...

*Клучниот збор е '''на'''. Изразот ''3 на 5-ти'' значи 3<sup>5</sup> = 3 &middot; 3 &middot; 3 &middot; 3 &middot; 3 = 243.
*Експонентот важи само за основата, т.е. само за бројот кој е непосредно на лево од него. 
*По договор и доколку нема загради, степенување се прави пред другите математички операции како множење (и делење) и собирање (и вадење).

[[Податотека: Exponents.png|thumb|right|''f''(''x'')=''x''&sup2; (сина) е полином; ''g''(''x'')=2<sup>''x''</sup> (црвена) е експоненцијална функција]]
'''Примери:'''
<div style="margin-left:15px">
{|  cellpadding=5 
|- align="center" 
|style="border: 1px solid grey"|(-17,4)<sup>2</sup>=(-17,4)&middot;(-17,4) = +302,76 = 302,76
|style="border: 1px solid grey"| -17,4<sup>2</sup> = - (17,4 &middot; 17,4) = -302,76
|- align="center" style="border: 5px solid grey"
|style="border: 1px solid grey"| 350 - 10<sup>2</sup> &middot; 4 = 350 - 100 &middot; 4 = 350 - 400 = -50
|style="border: 1px solid grey"| 0<sup>74</sup>=0
|}
</div>

*Функција каде што променливата e основата на експоненти кои се позитивни цели броеви се вика [[полином]]. Оваа класа на функции се многу важни.
'''Пример:''' ''f''(''x'')=''x''&sup2; е полином (од втор степен).

*Функција каде што променливата е експонент на основа која е позитивен реален број се вика [[експоненцијална функција]]. И оваа класа на функции се многу важни.
'''Пример:''' ''g''(''x'')=2<sup>''x''</sup> е експоненцијална функција (со основа 2).

==Експонентот е цел број==
 '''Дефиниција''': <math>b^{-n} = \frac{1}{b^n}</math>  &nbsp; каде што ''b''&ne;0 и ''n'' е [[цел број]] (позитивен, нула или негативен).
*Значи, при преместување на експоненцијален израз од едната страна на дробна црта на другата страна, знакот на експонентот се менува.
'''Примери:''' 
<div style="margin-left:15px">
{|  cellpadding=5
|- align="center"
|style="border: 1px solid grey"|<math>3x^{-2}=\tfrac{3}{x^2}</math>
|style="border: 1px solid grey"|<math>\tfrac{1}{(2x)^{-2}}=(2x)^2=2x \cdot 2x= 2 \cdot 2 \cdot x \cdot x = 4x^2</math>
|- align="center"
|style="border: 1px solid grey"|<math>\tfrac{1}{3x^{-7}}=\tfrac{x^7}{3}</math>
|style="border: 1px solid grey"|<math>\tfrac{6y}{(2x)^{-2}}=6y \cdot (2x)^2=6y \cdot 4x^2= 24x^2y</math>
|}
</div>

==Основни формули за експоненти==
 <math>b^m \cdot b^n=b^{m+n}\, , \,\, b \ne 0</math>
:Доказ: <math>b^m \cdot b^n = \underbrace{b \times \cdots \times b}_m \,\cdot \, \underbrace{b \times \cdots \times b}_n = \underbrace{b \times \cdots \times b}_{m+n}=b^{m+n}</math> 
 <math>(b^m)^n=b^{mn}\, , \,\, b \ne 0</math>
:Доказ: <math>(b^m)^n= \underbrace{\,\,\, \underbrace{b \times \cdots \times b}_m \,\cdot \,\underbrace{b \times \cdots \times b}_m  \cdot \, \cdots \, \cdot \,\underbrace{b \times \cdots \times b}_m \,\,\,}_n = \underbrace{b \times \cdots \times b}_{m \cdot n}=b^{mn}</math> 
 <math>b^0=1\, , \,\, b \ne 0</math>
:Доказ:<ref>{{Наведена мрежна страница|url=http://planetmath.org/introducing0thpower|title=Introducing 0th Power| publisher =Planet Math |language=англиски|accessdate=1 септември 2013}}</ref> <math>b^n \cdot 1=b^n=b^{n+0}=b^n \cdot b^0 </math>

<div style="margin-left:15px">
'''Примери'''
{|  cellpadding=5
|- align="center"
|style="border: 1px solid grey"|<math>\tfrac{3x^5}{x^2}=3x^5 \cdot x^{-2}= 3x^{3+(-2)}=3x^1=3x</math>
|style="border: 1px solid grey"|<math>{(-2x^2)}^3=(-2)^3 \cdot (x^2)^3=-8x^{2 \cdot 3}=-8x^6</math>
|- align="center"
|style="border: 1px solid grey"|<math>\tfrac{-2xy}{x^5y}=\tfrac{-2y\cdot y^{-1}}{x^{-1}x^5}=\tfrac{-2y^0}{x^4}=\tfrac{-2}{x^4}</math>
|style="border: 1px solid grey"|<math>\tfrac{-5}{3x^2+1}</math> не може да се упрости.
|- align="center"
|style="border: 1px solid grey"|<math>(\tfrac{2x^3}{y})^{-2}=(\tfrac{y}{2x^3})^{2}=\tfrac{y^2}{(2x^3)^2}=\tfrac{y^2}{4x^6}</math>
|style="border: 1px solid grey"|<math>0^{-3} </math> не постои.
|}
</div>

==Експонентот е рационален број (дропка)==
 Дефиниција: ''x'' = <sup>''n''</sup>√<span style="text-decoration:overline">''b''</span>, т.е. '''''n''-ти [[корен]]''' на реален број ''b'' е број ''x'' таков што ''x''<sup>''n''</sup> = ''b''.

Ако основата ''b'' е позитивен реален број и ''n'' е позитивен цел број, тогаш има точно едно реално решение, т.е. точно едно решение на равенката ''x<sup>n</sup>'' = ''b'' кое е реален број. Ова решение се вика '''главниот ''n''-ти корен''' на ''b''. Тоа се означува со <sup>''n''</sup>√<span style="text-decoration:overline">''b''</span>, каде што √<span style="text-decoration:overline">&ensp;</span> е симбол за [[коренување]]. Друго означување е со т.н. рационални експоненти, имено:

Нека ''b'' е позитивен реален број, ''m'' нека е цел број, а ''n'' нека е позитивен цел број:
  '''Означување''': <math>b^{^{1} / _{n}}=\sqrt[n]{b}</math>  &nbsp;и&nbsp;  <math>b^{^{m} / _{n}}=\sqrt[n]{b^m}</math>.
 
Внимавање треба кога основата ''b'' не е позитивен реален број при рационални (и реални) експоненти.
*За ''b''=0 и ''m''&ne;0, погорниот израз е 0.
*За ''b''&lt;0, погорниот израз е реален број само кога именителот на рационалниот експонент, т.е. кога ''n'' е непарен број (позитивен цел). Доколку именителот ''n'' е парен број, изразот е [[комплексен број]].
Меѓутоа, дигитрони и компјутерски апликации различно реагираат на експоненцијални изрази со негативна основа.

*Случајот кога и ''b''=0 и ''m''=0, т.е. 0<sup>0</sup> е многу сложено со различни можни вредности (и различни математичко толкување) и посебно се разгледуваат (види [[0на0]]).

==Експонентот е позитивен реален број==
Доколку основата ''b'' е позитивен реален број, а бидејќи секој ирационален број ''х'' може да се приближува со рационален број ''r'', степенување со експонент ''х'', т.е. ''b''<sup>''х''</sup> може да се дефинира преку [[непрекината функција|непрекинатост]] со <ref name=Denlinger>{{Наведена книга |title=Elements of Real Analysis |last=Denlinger |first=Charles G. |publisher=Jones and Bartlett |year=2011 |pages=278–283 |isbn=978-0-7637-7947-4}}</ref>
:<math> b^x = \lim_{r \to x} b^r\quad(r\in\mathbb Q,\,x\in\mathbb R)</math>
<!-- Тука треба нешто за кога b е негативен ирационален број...  -->

==Експоненти и функции==
За кратко пишување, позитивен целоброен експонент кај [[логаритамски функции|логаритамски]] и [[тригонометриски функции]] означуваат степенување. Имено и на пример:
<span style="line-height:30px">
 '''Означување''': <math>\sin^2(x) = \sin(x) \cdot \sin(x) =(\sin(x))^2 \,\, , \,\, \cos^2(x) = (\cos(x))^2 \,\, ...</math>
 '''Означување''': <math>\log^2(x) = \log(x) \cdot \log(x) =(\log(x))^2 \,\, , \,\, \ln^2(x) = (\ln(x))^2 \,\, ...</math>
</span>
Забележуваме дека при користење на повеќето компјутерски апликации и програми, овие кратенки '''не''' функционираат.

Од друга страна, кај општа функција ''f''(''x''), позитивен целоброен експонент обично означува повторна композиција на функцијата, т.е. ''f''&sup3;(''x'')=''f''(''f''(''f''(''x''))), а со експонентот (-1) се означува [[инверзна функција]] на ''f''.

Меѓутоа, нема конзистента дефиниција за експонент (-1) кај тригонометриски функции. На пример,
:sin<sup>-1</sup>(''x'')=arcsin(''x''), т.е. sin<sup>-1</sup>(''x'') e инверзната функција на sin(''x'') во [[САД]] и на повеќе дигитрони  или  sin<sup>-1</sup>(''x'')=<sup>1</sup>/<sub>sin(''x'')</sub> (во Р.М.).

[[Податотека: Win_calc.gif|right|frame|Степенување со Windows&reg; 7 дигитрон]]
[[Податотека: Calculator_big_chars_exponents.png|right|thumb|Копчиња за степенување со дигитрон]]
==Степенување со дигитрони==
Начинот на степенување со дигитрон зависи од типот на [[дигитрон]]от. Посебни дирки за степенување ги имаат т.н. [[дигитрон|научни дигитрони]] (види ги сликите).
<div style="line-height:1.2em">
*Секој научен дигитрон има едночекорен тастер: <span style="font-size:1.3em">{{тастер|''x''&sup2;}}</span>, т.е. по внесување на бројот ''b'' кој е основата, се притиска на овој тастер и пресметан е ''b''&sup2;. 
*Повеќето научни дигитрони имаат и едночекорен тастер: <span style="font-size:1.3em">{{тастер|''x''&sup3;}}</span> за трет степен.
*Секој научен дигитрон има едночекорен тастер: <span style="font-size:1.3em">{{тастер|<sup>1</sup>/<sub>''x''</sub>}}</span> или <span style="font-size:1.3em">{{тастер|''x''<sup>''-1''</sup>}}</span> за пресметување на [[реципрочен број]].
*Секој научен дигитрон има сложен тастер<span style="font-size:1.3em">{{тастер|''x''<sup>''y''</sup>}}</span> или <span style="font-size:1.3em">{{тастер|'''^'''}}</span> за пресметување со други експоненти освен 2 и 3. 
**Се внесува основата, па се притиска на овој тастер. Потоа се внесува експонентот, па се притиска на '''=''' или ''Enter''.
***За основа која е позитивен (реален) број нема некакви проблеми при користење на овој тастер.
***За основа која е негативен број, треба (а) рачно да се провери дека изразот е валиден (има решение) и (б) да се знае како работи дигитронот со негативни основи. 
</div>

'''Примери со обичен научен дигитрон:''' (види ја и анимацијата)
<div style="margin-left:15px">
{| border="1" cellspacing="0" cellpadding="3"
|-
|3&sup2; 
|Притисни: 3 &nbsp;<span style="font-size:1.3em">{{тастер|''x''&sup2;}}</span>  
|Резултатот е: 9
|-
|(-3)&sup2; 
|Притисни: 3 &nbsp;<span style="font-size:1.3em">{{тастер|&plusmn;}}</span> &nbsp;<span style="font-size:1.3em">{{тастер|''x''&sup2;}}</span>  
|Резултатот е: 9
|-
|2&sup3; 
|Притисни: 2 &nbsp;<span style="font-size:1.3em">{{тастер|''x''&sup3;}}</span>  
|Резултатот е: 8
|-
|4<sup>−1</sup>  
|Притисни: 4 &nbsp;<span style="font-size:1.3em">{{тастер|<sup>1</sup>/<sub>''x''</sub>}}</span>
|Резултатот е 0,25
|-
|<sup>3</sup>/<sub>4&sup2;</sub>  
|Притисни: 3 &nbsp; <span style="font-size:1.3em">{{тастер|/}}</span> &nbsp;<span style="font-size:1.3em">{{тастер|(}}</span> &nbsp;4&nbsp; <span style="font-size:1.4em">{{тастер|''x''&sup2;}}</span> &nbsp;<span style="font-size:1.3em">{{тастер|)}}</span>&nbsp; <span style="font-size:1.3em">{{тастер|&#61;}}</span>
|Резултатот е 0,1875
|-
|0,0625<sup>&frac34;</sup> 
|Притисни: 0&nbsp; .&nbsp; 0&nbsp; 6&nbsp; 2&nbsp; 5&nbsp; <span style="font-size:1.3em">{{тастер|''x''<sup>''y''</sup>}}</span>&nbsp; <span style="font-size:1.3em">{{тастер|(}}</span> &nbsp;3&nbsp; <span style="font-size:1.3em">{{тастер|/}}</span> &nbsp;4&nbsp; <span style="font-size:1.3em">{{тастер|)}}</span> &nbsp;<span style="font-size:1.3em">{{тастер|&#61;}}</span>. 
|Резултатот е: 0,125
|}
</div>

==Степенување во програмирање==
Означување на експонентот како [[Горен и долен индекс|горен индекс]] ''x''<sup>''y''</sup> е погодно за ракопис, но не е погодно за машинско пишување, особено во програмски јазици каде што сите карактери се на едно ниво (нема горен или долен индекс).
* <code>x ^ y</code>: BASIC, J, [[Геогебра]], [[Sage]], [[MATLAB]], [[R (програмски јазик)|R]], [[Microsoft Excel]], TeX и [[LaTeX]], TI-BASIC, bc (за цели бројни експоненти), Haskell (за ненегативни цели бројни експоненти), Lua, ASP и повеќето [[CAS|системи за алгебра со компјутери (CAS)]]. Ознаката '''^''' се вика [[карета]] и се добива на тастатурата со латинција со Shift+6.
* <code>x ** y</code>: Ada, Bash (Unix shell), COBOL, Fortran, FoxPro 2|FoxPro, Gnuplot, OCaml, F#, Perl, PL/I, Python, Rexx, Ruby, SAS, Seed7, Tcl, ABAP, Haskell (заекспоненти со подвижна запирка), Turing , VHDL
* <code>pow(x,y)</code>: C<sup>++</sup>, C<sup>#</sup>
* <code>x ↑ y</code>: Algol, Commodore BASIC
* <code>x ^^ y</code>: Haskell (за рационална основа, цели бројни експоненти), D 
* <code>pown x y</code>: F# (за цело бројна основа и експонент)
* <code>x⋆y</code>: APL

Многу програмски јазици немаат вградена синтакса за степенување, но имаат функции во нивните библиотеки.
<!--Слободно корегирај и цитирај наводи-->

Внимание: Во Bash, C, C++, C#, Java, JavaScript, Perl, PHP, Python и Ruby, Pascal, OCaml, каретата ^ означува друго, а не степенување. Треба да се води сметка за синтаксата на соодветниот јазик.

==Ефикасно степенување==
Наједноставниот метод за пресметување на  ''b''<sup>''n''</sup> изискува {{nowrap|''n'' − 1}} множења, но може и поефикасно како на пример: За пресметување на 2<sup>100</sup>, користиме {{nowrap|1=100 = 64 + 32 + 4}}. Пресметувајќи редоследно:
# 2<sup>2</sup> = 4
# (2<sup>2</sup>)<sup>2</sup> = 2<sup>4</sup> = 16
# (2<sup>4</sup>)<sup>2</sup> = 2<sup>8</sup> = 256
# (2<sup>8</sup>)<sup>2</sup> = 2<sup>16</sup> = 65.536
# (2<sup>16</sup>)<sup>2</sup> = 2<sup>32</sup> = 4.294.967.296
# (2<sup>32</sup>)<sup>2</sup> = 2<sup>64</sup> = 18.446.744.073.709.551.616
# 2<sup>64</sup> 2<sup>32</sup> 2<sup>4</sup> = 2<sup>100</sup> = 1.267.650.600.228.229.401.496.703.205.376
Оваа низа на чекори бара само 8 множења наместо 100-1=99 (има две множења во последниот чекор).

Општо кажано, бројот на множења потребни за пресметување на ''b''<sup>''n''</sup> може да се редуцира на [[асимптопско означување|Θ]](log ''n'') користејќи ''квадратно степенување''.<!-- [[exponentiation by squaring]] or (more generally) [[addition-chain exponentiation]].  Finding the ''minimal'' sequence of multiplications (the minimal-length addition chain for the exponent) for ''b''<sup>''n''</sup> is a --> Денес нема ефикасен алгоритам за пресметување на минималната низа, но има повеќе ефикасни евристички алгоритми за степенување.<ref>Gordon, D. M. 1998. A survey of fast exponentiation methods. J. Algorithms 27, 1 (Apr. 1998), 129-146. doi:http://dx.doi.org/10.1006/jagm.1997.0913</ref>

<!--
==Историја на означувањето==
The term ''power'' was used by the [[Greek mathematics|Greek]] mathematician [[Euclid]] for the square of a line.<ref name=MacTutor/> [[Archimedes]] discovered and proved the law of exponents, 10<sup>a</sup>&nbsp;10<sup>b</sup>&nbsp;=&nbsp;10<sup>a+b</sup>, necessary to manipulate powers of 10.<ref>For further analysis see [[The Sand Reckoner]].</ref> In the 9th century, the Persian mathematician [[Muhammad ibn Mūsā al-Khwārizmī]] used the terms ''mal'' for a [[Square (algebra)|square]] and ''kab'' for a [[Cube (algebra)|cube]], which later [[Mathematics in medieval Islam|Islamic]] mathematicians represented in [[mathematical notation]] as ''m'' and ''k'', respectively, by the 15th century, as seen in the work of [[Abū al-Hasan ibn Alī al-Qalasādī]].<ref>{{MacTutor|id=Al-Qalasadi|title= Abu'l Hasan ibn Ali al Qalasadi}}</ref>

[[Nicolas Chuquet]] used a form of exponential notation in the 15th century, which was later used by [[Henricus Grammateus]] and [[Michael Stifel]] in the 16th century. [[Samuel Jeake]] introduced the term ''indices'' in 1696.<ref name=MacTutor>{{MacTutor|class=Miscellaneous|id=Mathematical_notation|title=Etymology of some common mathematical terms}}</ref> In the 16th century [[Robert Recorde]] used the terms square, cube, zenzizenzic (fourth power), sursolid (fifth), zenzicube (sixth), second sursolid (seventh), and [[zenzizenzizenzic]] (eighth).<ref>{{Наведена мрежна страница|url=http://www.worldwidewords.org/weirdwords/ww-zen1.htm|title=Zenzizenzizenzic - the eighth power of a number|publisher=World Wide Words|first=Michael|last=Quinion|accessdate=2010-03-19}}</ref> ''Biquadrate'' has been used to refer to the fourth power as well.

Some mathematicians (e.g., [[Isaac Newton]]) used exponents only for powers greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ''ax'' + ''bxx'' + ''cx''<sup>3</sup> + ''d''.

Another historical synonym, '''involution''',<ref>This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary [http://www.m-w.com/dictionary/involution].  The most recent usage in this sense cited by the OED is from 1806.</ref> is now rare and should not be confused with [[Involution (mathematics)|its more common meaning]].
-->

==Табела на целобројни експоненти ==
{|class="wikitable"
!''n'' !!''n''<sup>2</sup> !!''n''<sup>3</sup> !!''n''<sup>4</sup> !!''n''<sup>5</sup> !!''n''<sup>6</sup> !!''n''<sup>7</sup> !!''n''<sup>8</sup> !!''n''<sup>9</sup> !!''n''<sup>10</sup>
|-
|width="8%"|'''2''' || width="8%"|4|| width="7%"|8|| width="8%"|16|| width="9%"|32|| width="10%"|64|| width="11%"|128|| width="12%"|256|| width="12%"|512|| width="14%"|1,024
|-
|'''3''' ||9||27||81||243||729||2.187||6.561||19.683||59.049
|-
|'''4''' ||16||64||256||1.024||4.096||16.384||65.536||262.144||1.048.576
|-
|'''5''' ||25||125||625||3.125||15.625||78.125||390.625||1.953.125||9.765.625
|-
|'''6''' ||36||216||1.296||7.776||46.656||279.936||1.679.616||10.077.696||60.466.176
|-
|'''7''' ||49||343||2.401||16.807||117.649||823.543||5.764.801||40.353.607||282.475.249
|-
|'''8''' ||64||512||4.096||32.768||262.144||2.097.152||16.777.216||134.217.728||1.073.741.824
|-
|'''9''' ||81||729||6.561||59.049||531.441||4.782.969||43.046.721||387.420.489||3.486.784.401
|-
|'''10'''||100||1.000||10.000||100.000||1.000.000||10.000.000||100.000.000||1.000.000.000||10.000.000.000
|}

{|class="wikitable"
!''n'' !!''n''<sup>2</sup> !!''n''<sup>3</sup> !!''n''<sup>4</sup> !!''n''<sup>5</sup> !!''n''<sup>6</sup> 
|- align="center"
|width="100px"|<math>\tfrac{1}{2}</math>|| width="100px"|<math>\tfrac{1}{4}</math>|| width="100px"|<math>\tfrac{1}{8}</math>|| width="100px"|<math>\tfrac{1}{16}</math>|| width="100px"|<math>\tfrac{1}{32}</math>|| width="100px"|<math>\tfrac{1}{64}</math>
|-align="center"
|<math>\tfrac{1}{3}</math>'''||<math>\tfrac{1}{9}</math>||<math>\tfrac{1}{27}</math>||<math>\tfrac{1}{81}</math>||<math>\tfrac{1}{243}</math>||<math>\tfrac{1}{729}</math>
|-align="center"
|<math>\tfrac{1}{4}</math>||<math>\tfrac{1}{16}</math>||<math>\tfrac{1}{64}</math>||<math>\tfrac{1}{256}</math>||<math>\tfrac{1}{1024}</math>||<math>\tfrac{1}{4096}</math>
|-align="center"
|<math>\tfrac{1}{10}</math>'''&nbsp;=0,1'''||<math>\tfrac{1}{100}</math>&nbsp;=0,01||<math>\tfrac{1}{1000}</math>&nbsp;=0,001||0,0001||0,00001||0,000001
|}

==Наводи==
{{наводи}}

==Поврзани теми==
*[[Коренување]]
*[[Рационален израз]]
*[[Полиноми]]
*[[Експоненцијална функција]]
*[[Логаритамска функција]]

== Надворешни врски ==
* {{Наведена мрежна страница|url=http://www.emathforall.com/wiki/RecnikT/NauchnoOznachuvanje|title=Научно означување|last1=Стојановска|first1=Л.|year=2010|language=македонски|accessdate=1 ноември 2013|archive-date=2016-03-05|archive-url=https://web.archive.org/web/20160305034600/http://www.emathforall.com/wiki/RecnikT/NauchnoOznachuvanje|url-status=dead}} 
* {{Наведена мрежна страница|url=http://www.purplemath.com/modules/exponent.htm|title=Exponents|publisher=Purple Math|last1=Staple|first1=E.|year=2013|language=англиски|accessdate=1 ноември 2013}} интерактивно (долу)
* {{Наведена мрежна страница|url=http://www.mathsisfun.com/algebra/exponent-laws.html|title=Laws of Exponents|last1=Pierce|first1=R.|year=2013|language=англиски|accessdate=1 ноември 2013}} интерактивно
* [http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0<sup>0</sup>?]
* [http://www.askamathematician.com/?p=4524 What does 0^0 (zero to the zeroth power) equal?] on AskAMathematician.com

{{Портал|Математика}}

{{Математички полиња}}

{{Нормативна контрола}}
<!--
==Complex exponents with positive real bases==
===Imaginary exponents with base e===
{{Main|Exponential function}}

[[Image:ExpIPi.gif|300px|thumb|right|The [[exponential function]] ''e''<sup>''z''</sup> can be defined as the [[limit of a sequence|limit]] of {{nowrap|(1 + ''z''/''N'')<sup>''N''</sup>}}, as ''N'' approaches infinity, and thus ''e''<sup>''iπ''</sup> is the limit of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}}. In this animation ''N'' takes various increasing values from 1 to 100. The computation of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}} is displayed as the combined effect of ''N'' repeated multiplications in the [[complex plane]], with the final point being the actual value of {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}}. It can be seen that as ''N'' gets larger {{nowrap|(1 + ''iπ''/''N'')<sup>''N''</sup>}} approaches a limit of −1. Therefore, {{nowrap|1=''e''<sup>''iπ''</sup> = −1,}} which is known as [[Euler's identity]].]]

The geometric interpretation of the operations on [[complex numbers]] and the definition of the exponential function is the clue to understanding ''e''<sup>''ix''</sup> for real ''x''. Consider the [[right triangle]] {{nowrap|(0, 1, 1 + ''ix''/''n'').}} For big values of ''n'' the triangle is almost a [[circular sector]] with a small central angle equal to ''x''/''n'' [[radian]]s. The triangles {{nowrap|(0, (1 + ''ix''/''n'')<sup>''k''</sup>, (1 + ''ix''/''n'')<sup>''k''+1</sup>)}} are mutually [[Similar triangles|similar]] for all values of ''k''. So for large values of ''n'' the limiting point of {{nowrap|(1 + ''ix''/''n'')<sup>''n''</sup>}} is the point on the [[unit circle]] whose angle from the positive real axis is ''x'' radians. The [[polar coordinates]] of this point are {{nowrap|1=(''r'', ''θ'') = (1, ''x''),}} and the [[cartesian coordinate]]s are (cos ''x'', sin ''x''). So {{nowrap|1=''e''<sup> ''ix''</sup> = cos ''x'' + ''i''sin ''x'',}} and this is [[Euler's formula]], connecting [[algebra]] to [[trigonometry]] by means of [[complex number]]s.

The solutions to the equation ''e''<sup>''z''</sup> = 1 are the integer multiples of 2π''i'':
:<math>\{ z : e^z = 1 \} = \{ 2k\pi i : k \in \mathbb{Z} \}</math>

More generally, if e<sup>''v''</sup> = ''w'', then every solution to ''e''<sup>''z''</sup> = ''w'' can be obtained by adding an integer multiple of 2π''i'' to ''v'':
:<math>\{ z : e^z = w \} = \{ v + 2k\pi i : k \in \mathbb{Z} \}</math>

Thus the complex exponential function is a [[periodic function]] with period 2π''i''.

More simply: ''e''<sup>''iπ''</sup> = −1; ''e''<sup>''x'' + ''iy''</sup> = ''e''<sup>''x''</sup>(cos ''y'' + ''i'' sin ''y'').

===Trigonometric functions===
{{Main|Euler's formula}}
It follows from Euler's formula stated above that the [[trigonometric functions]] cosine and sine are
:<math>\cos(z) = \frac{e^{iz} + e^{-iz}}{2}; \qquad \sin(z) = \frac{e^{iz} - e^{-iz}}{2i}</math>

Historically, cosine and sine were defined geometrically before the invention of complex numbers. The above formula reduces the complicated formulas for [[Trigonometric addition formulas|trigonometric functions of a sum]] into the simple exponentiation formula
:<math>e^{i(x+y)}=e^{ix}\cdot e^{iy}</math>

Using exponentiation with complex exponents may reduce problems in trigonometry to algebra.

===Complex exponents with base e===
The power {{nowrap|''z'' {{=}} ''e''<sup>''x'' + ''iy''</sup>}} can be computed as ''e''<sup>''x''</sup> · ''e''<sup>''iy''</sup>. The real factor ''e''<sup>''x''</sup> is the [[absolute value]] of ''z'' and the complex factor ''e''<sup>''iy''</sup> identifies the [[direction (geometry, geography)|direction]] of ''z''.

===Complex exponents with positive real bases===
If ''b'' is a positive real number, and ''z'' is any complex number, the power ''b''<sup>''z''</sup> is defined as ''e''<sup>''z''·ln(''b'')</sup>, where ''x'' = ln(''b'') is the unique real solution to the equation ''e''<sup>''x''</sup> = ''b''. So the same method working for real exponents also works for complex exponents.

For example:
:2<sup>''i''</sup> = ''e''<sup> ''i''·ln(2)</sup> = cos(ln(2)) + ''i''·sin(ln(2)) ≈ 0.76924 + 0.63896''i''
:''e''<sup>''i''</sup> ≈ 0.54030 + 0.84147''i''
:10<sup>''i''</sup> ≈ −0.66820 + 0.74398''i''
:(''e''<sup>2&pi;</sup>)<sup>''i''</sup> ≈ 535.49<sup>''i''</sup> ≈ 1

The identity <math>(b^z)^u=b^{zu}</math> is not generally valid for complex powers. A simple counterexample is given by:
:<math>(e^{2\pi i})^i=1^i=1\neq e^{-2\pi}=e^{2\pi i\cdot i}</math>

The identity is, however, valid when <math>z</math> is a real number, and also when <math>u</math> is an integer.

==Powers of complex numbers==
Integer powers of nonzero complex numbers are defined by repeated multiplication or division as above. If ''i'' is the [[imaginary unit]] and ''n'' is an integer, then ''i''<sup>''n''</sup> equals 1, ''i'', −1, or −''i'', according to whether the integer ''n'' is congruent to 0, 1, 2, or 3 modulo 4. Because of this, the powers of ''i'' are useful for expressing [[sequence]]s of [[Root of unity#Periodicity|period 4]].

Complex powers of positive reals are defined via ''e''<sup>''x''</sup> as in section [[#Complex powers of positive real numbers|Complex powers of positive real numbers]] above. These are continuous functions.

Trying to extend these functions to the general case of noninteger powers of complex numbers that are not positive reals leads to difficulties. Either we define discontinuous functions or [[multivalued function]]s. Neither of these options is entirely satisfactory.

The rational power of a complex number must be the solution to an algebraic equation. Therefore it always has a finite number of possible values. For example, ''w'' = ''z''<sup>1/2</sup> must be a solution to the equation ''w''<sup>2</sup> = ''z''. But if ''w'' is a solution, then so is −''w'', because (−1)<sup>2</sup> = 1. A unique but somewhat arbitrary solution called the [[principal value]] can be chosen using a general rule which also applies for nonrational powers.

Complex powers and logarithms are more naturally handled as single valued functions on a [[Riemann surface]]. Single valued versions are defined by choosing a sheet. The value has a discontinuity along a [[branch cut]]. Choosing one out of many solutions as the principal value leaves us with functions that are not continuous, and the usual rules for manipulating powers can lead us astray.

Any nonrational power of a complex number has an infinite number of possible values because of the multi-valued nature of the [[complex logarithm]] (see [[#Complex logarithms|below]]). The principal value is a single value chosen from these by a rule which, amongst its other properties, ensures powers of complex numbers with a positive real part and zero imaginary part give the same value as for the corresponding real numbers.

Exponentiating a real number to a complex power is formally a different operation from that for the corresponding complex number. However in the common case of a positive real number the principal value is the same.

The powers of negative real numbers are not always defined and are discontinuous even where defined. In fact, they are only defined when the exponent is a rational number with the denominator being an odd integer. When dealing with complex numbers the complex number operation is normally used instead.

===Complex exponents with complex bases===
For complex numbers ''w'' and ''z'' with ''w''&nbsp;≠&nbsp;0, the notation ''w''<sup>''z''</sup> is ambiguous in the same sense that [[complex logarithm|log&nbsp;''w'']] is.

To obtain a value of ''w''<sup>''z''</sup>, first choose a logarithm of ''w''; call it log&nbsp;''w''.  Such a choice may be the [[complex logarithm#Definition of principal value|principal value]] Log&nbsp;''w'' (the default, if no other specification is given), or perhaps a value given by some other [[complex logarithm#Branches of the complex logarithm|branch of log&nbsp;''w'']] fixed in advance.  Then, using the complex exponential function one defines

:<math>w^z = e^{z \log w}</math>

because this agrees with the [[#Real exponents|earlier definition]] in the case where ''w'' is a positive real number and the (real) principal value of log&nbsp;''w'' is used.

If ''z'' is an [[integer]], then the value of ''w''<sup>''z''</sup> is independent of the choice of log&nbsp;''w'', and it agrees with the [[#Positive integer exponents|earlier definition of exponentation with an integer exponent]].

If ''z'' is a [[rational number]] ''m''/''n'' in lowest terms with ''z''&nbsp;>&nbsp;0, then the infinitely many choices of log&nbsp;''w'' yield only ''n'' different values for ''w''<sup>''z''</sup>; these values are the ''n'' complex solutions ''s'' to the equation ''s''<sup>''n''</sup> = ''w''<sup>''m''</sup>.

If ''z'' is an [[irrational number]], then the infinitely many choices of log&nbsp;''w'' lead to infinitely many distinct values for ''w''<sup>''z''</sup>.

The computation of complex powers is facilitated by converting the base ''w'' to [[polar form]], as described in detail [[#Computing complex powers|below]].

A similar construction is employed in [[Quaternion#Exponential, logarithm, and power|quaternions]].

===Complex roots of unity===
{{Main|Root of unity}}

[[File:One3Root.svg|thumb|right|The three 3rd roots of 1]]
A complex number ''w'' such that ''w''<sup>''n''</sup> = 1 for a positive integer ''n'' is an ''' ''n''th root of unity'''.  Geometrically, the ''n''th roots of unity lie on the unit circle of the complex plane at the vertices of a regular ''n''-gon with one vertex on the real number 1.

If ''w''<sup>''n''</sup> = 1 but ''w''<sup>''k''</sup> ≠ 1 for all natural numbers ''k'' such that 0 < ''k'' < ''n'', then ''w'' is called a '''primitive ''n''th root of unity.'''   The negative unit −1 is the only primitive square root of unity. The [[imaginary unit]] ''i'' is one of the two primitive 4-th roots of unity; the other one is −''i''.

The number ''e''<sup>2''πi'' ({{frac|1|''n''}})</sup> is the primitive ''n''th root of unity with the smallest positive [[complex argument]].  (It is sometimes called the '''principal ''n''th root of unity''', although this terminology is not universal and should not be confused with the [[principal value]] of <sup>''n''</sup>√<span style="text-decoration:overline">1</span>, which is 1.<ref>This definition of a principal root of unity can be found in:
*{{Наведена книга | title = Introduction to Algorithms | edition = second | author = Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein | publisher = MIT Press | year = 2001 | isbn = 0-262-03293-7}} [http://highered.mcgraw-hill.com/sites/0070131511/student_view0/chapter30/glossary.html Online resource]
*{{Наведена книга | title = Difference Equations: From Rabbits to Chaos | edition = Undergraduate Texts in Mathematics | author = Paul Cull, Mary Flahive, and Robby Robson | year = 2005 | publisher = Springer | isbn = 0-387-23234-6 }} Defined on page 351, available on Google books.
* "[http://mathworld.wolfram.com/PrincipalRootofUnity.html Principal root of unity]", MathWorld.</ref>)

The other ''n''th roots of unity are given by
:<math>\left( e^{ \frac{2}{n} \pi i } \right) ^k = e^{ \frac{2}{n} \pi i k }</math>

for 2 ≤ ''k'' ≤ ''n''.

===Roots of arbitrary complex numbers===
Although there are infinitely many possible values for a general complex logarithm, there are only a finite number of values for the power ''w<sup>q</sup>'' in the important special case where ''q'' = 1/''n'' and ''n'' is a positive integer. These are the '''''n''th roots''' of ''w''; they are solutions of the equation ''z<sup>n</sup>'' = ''w''.  As with real roots, a second root is also called a square root and a third root is also called a cube root.

It is conventional in mathematics to define ''w''<sup>1/''n''</sup> as the principal value of the root. If ''w'' is a positive real number, it is also conventional to select a positive real number as the principal value of the root ''w''<sup>1/''n''</sup>.  For general complex numbers, the ''n''th root with the smallest argument is often selected as the principal value of the ''n''th root operation, as with principal values of roots of unity.

The set of ''n''th roots of a complex number ''w'' is obtained by multiplying the principal value ''w''<sup>1/''n''</sup> by each of the ''n''th roots of unity. For example, the fourth roots of 16 are 2, −2, 2''i'', and −2''i'', because the principal value of the fourth root of 16 is 2 and the fourth roots of unity are 1, −1, ''i'', and −''i''.

===Computing complex powers===

It is often easier to compute complex powers by writing the number to be exponentiated in [[Principal argument|polar form]].  Every complex number ''z'' can be written in the polar form
:<math>z = re^{i\theta} = e^{\ln(r) + i\theta}</math>

where ''r'' is a nonnegative real number and θ is the (real) [[complex argument|argument]] of ''z''.  The polar form has a simple geometric interpretation: if a complex number ''u'' + ''iv'' is thought of as representing a point (''u'', ''v'') in the [[complex plane]] using [[Cartesian coordinate system|Cartesian coordinates]], then (''r'', θ) is the same point in [[polar coordinates]].  That is, ''r'' is the "radius" ''r''<sup>2</sup> = ''u''<sup>2</sup> + ''v''<sup>2</sup> and θ is the "angle" θ = [[atan2]](''v'', ''u'').  The polar angle θ is ambiguous since any integer multiple of 2π could be added to θ without changing the location of the point.  Each choice of θ gives in general a different possible value of the power.  A [[branch cut]] can be used to choose a specific value.  The principal value (the most common branch cut), corresponds to θ chosen in the interval (−π, π]. For complex numbers with a positive real part and zero imaginary part using the principal value gives the same result as using the corresponding real number.

In order to compute the complex power ''w''<sup>''z''</sup>, write ''w'' in polar form:
:<math>w = r e^{i\theta}</math>

Then
:<math>\log w = \log r + i \theta</math>

and thus
:<math>w^z = e^{z \log w} = e^{z(\log r + i\theta)}</math>

If ''z'' is decomposed as ''c'' + ''di'', then the formula for ''w''<sup>''z''</sup> can be written more explicitly as
:<math>\left( r^c e^{-d\theta} \right) e^{i (d \log r + c\theta)} = \left( r^c e^{-d\theta} \right) \left[ \cos(d \log r + c\theta) + i \sin(d \log r + c\theta) \right]</math><!-- e^{c \log r - d\theta + i (d \log r + c\theta)}

This final formula allows complex powers to be computed easily from decompositions of the base into polar form and the exponent into Cartesian form.  It is shown here both in polar form and in Cartesian form (via Euler's identity).

The following examples use the principal value, the branch cut which causes θ to be in the interval (−π, π]. To compute ''i''<sup>''i''</sup>, write ''i'' in polar and Cartesian forms:
:<math>\begin{align}
  i &= 1 \cdot e^{\frac{1}{2} i \pi} \\
  i &= 0 + 1i
\end{align}</math>

Then the formula above, with ''r'' = 1, θ = {{frac|π|2}}, ''c'' = 0, and ''d'' = 1, yields:
:<math>i^i = \left( 1^0 e^{-\frac{1}{2}\pi} \right) e^{i \left[1 \cdot \log 1 + 0 \cdot \frac{1}{2}\pi \right]} = e^{-\frac{1}{2}\pi} \approx 0.2079</math>

Similarly, to find (−2)<sup>3 + 4''i''</sup>, compute the polar form of −2,
:<math>-2 = 2e^{i \pi}</math>

and use the formula above to compute
:<math>(-2)^{3 + 4i} = \left( 2^3 e^{-4\pi} \right) e^{i[4\log(2) + 3\pi]} \approx (2.602 - 1.006 i) \cdot 10^{-5}</math>

The value of a complex power depends on the branch used.  For example, if the polar form ''i'' = 1''e''<sup>''i''({{frac|5π|2}})</sup> is used to compute ''i'' <sup>''i''</sup>, the power is found to be ''e''<sup>−{{frac|5π|2}}</sup>; the principal value of ''i'' <sup>''i''</sup>, computed above, is ''e''<sup>−{{frac|π|2}}</sup>. The set of all possible values for ''i'' <sup>''i''</sup> is given by:<ref>[http://www.cut-the-knot.org/do_you_know/complex.shtml Complex number to a complex power may be real] at Cut The Knot gives some references to ''i''<sup>''i''</sup></ref>
:<math>\begin{align}
  i   &= 1 \cdot e^{\frac{1}{2} i\pi + i 2 \pi k} \big| k \isin \mathbb{Z} \\
  i^i &= e^{i \left(\frac{1}{2} i\pi + i 2 \pi k\right)} \\
      &= e^{-\left(\frac{1}{2} \pi + 2 \pi k\right)}
\end{align}</math>

So there is an infinity of values which are possible candidates for the value of ''i''<sup>''i''</sup>, one for each integer ''k''. All of them have a zero imaginary part so one can say ''i''<sup>''i''</sup> has an infinity of valid real values.

===Failure of power and logarithm identities===
Some identities for powers and logarithms for positive real numbers will fail for complex numbers, no matter how complex powers and complex logarithms are defined ''as single-valued functions''. For example:

* The identity log(''b''<sup>''x''</sup>) = ''x'' · log&thinsp;''b'' holds whenever ''b'' is a positive real number and ''x'' is a real number.  But for the [[principal branch]] of the complex logarithm one has
*:: <math> i\pi = \log(-1) = \log\left[(-i)^2\right] \neq 2\log(-i) = 2\left(-\frac{i\pi}{2}\right) = -i\pi</math>
*: Regardless of which branch of the logarithm is used, a similar failure of the identity will exist. The best that can be said (if only using this result) is that:
*:: <math>\log(w^z) \equiv z \cdot \log(w) \pmod{2 \pi i}</math>
*: This identity does not hold even when considering log as a multivalued function. The possible values of log(''w''<sup>''z''</sup>) contain those of ''z'' · log&thinsp;''w'' as a subset. Using Log(''w'') for the principal value of log(''w'') and ''m'', ''n'' as any integers the possible values of both sides are:
*:: <math>\begin{align}
            \left\{\log(w^z)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n + 2 \pi i m \right\} \\
      \left\{z \cdot \log(w)\right\} &= \left\{ z \cdot \operatorname{Log}(w) + z \cdot 2 \pi i n \right\}
    \end{align}</math>
* The identities (''bc'')<sup>''x''</sup> = ''b''<sup>''x''</sup>''c''<sup>''x''</sup> and (''b''/''c'')<sup>''x''</sup> = ''b''<sup>''x''</sup>/''c''<sup>''x''</sup> are valid when ''b'' and ''c'' are positive real numbers and ''x'' is a real number.  But a calculation using principal branches shows that
*:: <math>1 = (-1\times -1)^\frac{1}{2}   \not = (-1)^\frac{1}{2}(-1)^\frac{1}{2} = -1</math>
*: and
*::<math>i = (-1)^\frac{1}{2} = \left (\frac{1}{-1}\right )^\frac{1}{2} \not = \frac{1^\frac{1}{2}}{(-1)^\frac{1}{2}} = \frac{1}{i} = -i</math>
*: On the other hand, when ''x'' is an integer, the identities are valid for all nonzero complex numbers.
*: If exponentiation is considered as a multivalued function then the possible values of (−1×−1)<sup>1/2</sup> are {1,&nbsp;−1}. The identity holds but saying {1}&nbsp;=&nbsp;{(−1×−1)<sup>1/2</sup>} is wrong.
* The identity (e<sup>''x''</sup>)<sup>''y''</sup> = e<sup>''xy''</sup> holds for real numbers ''x'' and ''y'', but assuming its truth for complex numbers leads to the following [[Mathematical fallacy|paradox]], discovered in 1827 by [[Thomas Clausen (mathematician)|Clausen]]:<ref name="Clausen1827">{{cite journal |author=Steiner J, Clausen T, Abel NH |title=Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen |url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=270662 |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=2 |year=1827 |pages=286–287}}</ref>
*: For any integer ''n'', we have:
*:#<math>e^{1 + 2 \pi i n} = e^{1} e^{2 \pi i n} = e \cdot 1 = e</math>
*:#<math>\left( e^{1+2\pi i n} \right)^{1 + 2 \pi i n} = e</math>
*:#<math>e^{1 + 4 \pi i n - 4 \pi^{2} n^{2}} = e</math>
*:#<math>e^1 e^{4 \pi i n} e^{-4 \pi^2 n^2} = e</math>
*:#<math>e^{-4 \pi^2 n^2} = 1</math>
*: but this is false when the integer ''n'' is nonzero.
*: There are a number of problems in the reasoning:
*: The major error is that changing the order of exponentiation in going from line two to three changes what the principal value chosen will be.
*: From the multi-valued point of view, the first error occurs even sooner. Implicit in the first line is that ''e'' is a real number, whereas the result of  ''e''<sup>1+2π''in''</sup> is a complex number better represented as ''e''+0''i''. Substituting the complex number for the real on the second line makes the power have multiple possible values. Changing the order of exponentiation from lines two to three also affects how many possible values the result can have. <math>\scriptstyle (e^z)^w \;\ne\; e^{z w}</math>, but rather <math>\scriptstyle (e^z)^w \;=\; e^{(z \,+\, 2\pi i n) w}</math> multivalued over integers ''n''.

==Zero to the power of zero==
[[Image:X^y.png|right|thumb|300px|Plot of {{nowrap|1=''z'' = ''x''<sup>''y''</sup>}}. The red curves (with ''z'' constant) yield different limits as (''x'',''y'') approaches (0,0). The green curves (of finite constant slope, {{nowrap|1=''y'' = ''ax''}}) all yield a limit of 1.]]

===For discrete exponents===
In most settings not involving [[Continuous function|continuity]] in the exponent, interpreting 0<sup>0</sup> as 1 simplifies formulas and eliminates the need for special cases in theorems. (See the next paragraph for some settings that ''do'' involve continuity.)
For example:
*Regarding ''b''<sup>0</sup> as an [[empty product]] assigns it the value 1, even when ''b''&nbsp;=&nbsp;0.
*The [[#Combinatorial interpretation|combinatorial interpretation]] of 0<sup>0</sup> is the number of [[empty tuple]]s of elements from the empty set. There is exactly one empty tuple.
*Equivalently, the [[#Exponentiation over sets|set-theoretic interpretation]] of 0<sup>0</sup> is the number of functions from the empty set to the empty set.  There is exactly one such function, the [[empty function]].<ref name="Bourbaki">N. Bourbaki, Elements of Mathematics, Theory of Sets, Springer-Verlag, 2004, III.§3.5.</ref>
*The notation <math>\scriptstyle \sum a_nx^n</math> for  [[polynomial]]s and [[power series]] rely on defining 0<sup>0</sup> = 1. Identities like <math>\scriptstyle \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n</math> and <math>\scriptstyle e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math> and the [[binomial theorem]] <math>\scriptstyle (1 + x)^n = \sum_{k = 0}^n \binom{n}{k} x^k</math> are not valid for {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.<ref>"Some textbooks leave the quantity 0<sup>0</sup> undefined, because the functions ''x''<sup>0</sup> and 0<sup>''x''</sup> have different limiting values when ''x'' decreases to 0. But this is a mistake. We must define {{nowrap|1=''x''<sup>0</sup> = 1}}, for all ''x'', if the binomial theorem is to be valid when {{nowrap|1=''x'' = 0}}, {{nowrap|1=''y'' = 0}}, and/or {{nowrap|1=''x'' = −''y''}}. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function 0<sup>''x''</sup> is quite unimportant".{{Наведена книга|title=[[Concrete Mathematics]]|edition=1st|publisher=Addison Wesley Longman Publishing Co|date=1989-01-05|isbn=0-201-14236-8|author=[[Ronald Graham]], [[Donald Knuth]], and [[Oren Patashnik]]|page=162|chapter=Binomial coefficients}}</ref>
*In [[differential calculus]], the [[power rule]] <math>\scriptstyle \frac{d}{dx} x^n = nx^{n-1}</math> is not valid for {{nowrap|1=''n'' = 1}} at {{nowrap|1=''x'' = 0}} unless {{nowrap|1=0<sup>0</sup> = 1}}.

===In analysis===
On the other hand, when 0<sup>0</sup> arises when trying to determine a [[limit of a function|limit]] of the form <math>\scriptstyle \lim_{x\rarr 0} f(x)^{g(x)}</math>, it must be handled as an [[indeterminate form]].
*Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.<ref>{{Наведена книга|first=S. C.|last=Malik |coauthors=Savita Arora|year=1992|title=Mathematical Analysis|page=223|isbn=978-81-224-0323-7|quote=In general the limit of φ(''x'')/ψ(''x'') when ''x''=''a'' in case the limits of both the functions exist is equal to the limit of the numerator divided by the denominator. But what happens when both limits are zero? The division (0/0) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are ∞/∞ 0&nbsp;&times;&nbsp;∞,  ∞&nbsp;−&nbsp;∞, 0<sup>0</sup>, 1<sup>∞</sup> and ∞<sup>0</sup>.|publisher=Wiley|location=New York}}</ref> In fact, when ''f''(''t'') and ''g''(''t'') are real-valued functions both approaching 0 (as ''t'' approaches a real number or ±∞), with ''f''(''t'') > 0, the function ''f''(''t'')<sup>''g''(''t'')</sup> need not approach 1; depending on ''f'' and ''g'', the limit of ''f''(''t'')<sup>''g''(''t'')</sup> can be any nonnegative real number or +∞, or it can be [[undefined (mathematics)|undefined]]. For example, the functions below are of the form ''f''(''t'')<sup>''g''(''t'')</sup> with ''f''(''t''),''g''(''t'')&nbsp;→&nbsp;0 as [[one-sided limit|''t''&nbsp;→&nbsp;0<sup>+</sup>]], but the limits are different:
::<math> \lim_{t \to 0^+} {t}^{t} = 1, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^t = 0, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t^2}}\right)^{-t} = +\infty, \quad \lim_{t \to 0^+} \left(e^{-\frac{1}{t}}\right)^{at} = e^{-a}</math>.

:So 0<sup>0</sup> is an indeterminate form.  This behavior shows that the two-variable function ''x''<sup>''y''</sup>, though continuous on the set {(''x'',''y''): ''x'' > 0}, cannot be extended to a [[continuous function]] on any set containing (0,0), no matter how 0<sup>0</sup> is defined.<ref>{{cite journal|author=L. J. Paige|title=A note on indeterminate forms|jstor=2307224|journal=American Mathematical Monthly|volume=61|issue=3|month=March | year=1954|pages=189–190|doi=10.2307/2307224}}</ref> However, under certain conditions, such as when ''f'' and ''g'' are both [[analytic functions]] and ''f'' is positive on the open interval (0,''b'') for some positive ''b'', the limit approaching from the right is always 1.<ref>[http://www.faqs.org/faqs/sci-math-faq/specialnumbers/0to0/ sci.math FAQ: What is 0^0?]</ref><ref>{{Cite journal | doi = 10.2307/2689754| jstor = 2689754  | pages = 41–42  | title = The Indeterminate Form 0<sup>0</sup>  | journal = [[Mathematics Magazine]] | volume = 50  | issue = 1  | publisher = [[Mathematical Association of America]] | year = 1977 | author1 = Rotando | first1 = Louis M. | author2 = Korn | first2 = Henry }}</ref><ref>{{Cite journal | doi = 10.2307/3595845 | jstor = 3595845  | pages = 55–56  | title = On the Indeterminate Form 0<sup>0</sup> | journal = [[The College Mathematics Journal]] | volume = 34  | issue = 1  | publisher = [[Mathematical Association of America]]  | year = 2003 | author1 = Lipkin | first1 = Leonard J.}}</ref>
*In the [[complex domain]], the function ''z''<sup>''w''</sup> is defined for nonzero ''z'' by choosing a [[complex logarithm#Branches of the complex logarithm|branch]] of log ''z'' and setting ''z''<sup>''w''</sup> := ''e''<sup>''w''&nbsp;log&nbsp;''z''</sup>, but there is no branch of log ''z'' defined at ''z'' = 0, let alone in a neighborhood of 0.<ref>"... Let's start at ''x'' = 0. Here ''x''<sup>''x''</sup> is undefined."  Mark D. Meyerson, The ''x''<sup>''x''</sup> Spindle, ''Mathematics Magazine'' '''69''', no. 3 (June 1996), 198-206.</ref>

===History of differing points of view===
Different authors interpret the situation above in different ways:
* Some argue that the best value for 0<sup>0</sup> depends on context, and hence that [[defined and undefined|defining]] it once and for all is problematic.<ref>Examples include Edwards and Penny (1994). ''Calculus'', 4th ed,, Prentice-Hall, p. 466, and Keedy, Bittinger, and Smith (1982). ''Algebra Two.'' Addison-Wesley, p. 32.</ref>  According to Benson (1999), "The choice whether to define 0<sup>0</sup> is based on convenience, not on correctness."<ref>Donald C. Benson, ''The Moment of Proof : Mathematical Epiphanies.'' New York Oxford University Press (UK), 1999. ISBN 978-0-19-511721-9</ref>
* Others argue that 0<sup>0</sup> should be defined as 1.  According to p.&nbsp;408 of [[Donald Knuth|Knuth]] (1992), it "''has'' to be 1", although he may have been referring to the specific context of mappings between sets, and goes on to say that "Cauchy had good reason to consider 0<sup>0</sup> as an undefined ''limiting form''" and that "in this much stronger sense, the value of 0<sup>0</sup> is less defined than, say, the value of 0&nbsp;+&nbsp;0" (emphases in original).<ref name="Knuth1992">Donald E. Knuth, Two notes on notation, ''[[American Mathematical Monthly|Amer. Math. Monthly]]'' '''99''' no. 5 (May 1992), 403–422 ([http://arxiv.org/abs/math/9205211 arXiv:math/9205211 [math.HO]]).</ref>

The debate has been going on at least since the early 19th century. At that time, most mathematicians agreed that 0<sup>0</sup> = 1, until in 1821 [[Cauchy]]<ref>Augustin-Louis Cauchy, ''Cours d'Analyse de l'École Royale Polytechnique'' (1821).  In his ''Oeuvres Complètes'', series 2, volume 3.</ref> listed 0<sup>0</sup> along with expressions like {{Frac|0|0}} in a table of undefined forms.
In the 1830s Libri<ref>Guillaume Libri, Note sur les valeurs de la fonction 0<sup>0<sup>x</sup></sup>, ''[[Crelle's Journal|Journal für die reine und angewandte Mathematik]]'' '''6''' (1830), 67–72.</ref><ref>Guillaume Libri, Mémoire sur les fonctions discontinues, ''Journal für die reine und angewandte Mathematik'' '''10''' (1833), 303–316.</ref> published an unconvincing argument for 0<sup>0</sup> = 1, and [[August Ferdinand Möbius|Möbius]]<ref>A. F. Möbius, Beweis der Gleichung 0<sup>0</sup> = 1, nach [[Johann Friedrich Pfaff|J. F. Pfaff]], ''Journal für die reine und angewandte Mathematik'' '''12''' (1834), 134–136.</ref> sided with him, erroneously claiming that
<math>\scriptstyle \lim_{t \to 0^+} f(t)^{g(t)} \;=\; 1</math> whenever <math>\scriptstyle \lim_{t \to 0^+} f(t) \;=\; \lim_{t \to 0^+} g(t) \;=\; 0</math>. A commentator who signed his name simply as "S" provided the counterexample of (''e''<sup>−1/''t''</sup>)<sup>''t''</sup>, and this quieted the debate for some time. More details can be found in Knuth (1992).<ref name="Knuth1992" />

===Treatment on computers===

====IEEE floating point standard====
The [[IEEE 754-2008]] floating point standard is used in the design of most floating point libraries. It recommends a number of different functions for computing a power:<ref>{{Наведена книга|title=Handbook of Floating-Point Arithmetic|publisher=Birkhäuser Boston|year=2009|isbn=978-0-8176-4704-9|page=216|author9=Jean-Michel Muller et al}}</ref>
*<tt>pow</tt> treats 0<sup>0</sup> as 1. This is the oldest defined version. If the power is an exact integer the result is the same as for <tt>pown</tt>, otherwise the result is as for <tt>powr</tt> (except for some exceptional cases).
*<tt>pown</tt> treats 0<sup>0</sup> as 1. The power must be an exact integer. The value is defined for negative bases; e.g., <tt>pown(−3,5)</tt> is −243.
*<tt>powr</tt> treats 0<sup>0</sup> as [[NaN]] (Not-a-Number – undefined). The value is also NaN for cases like <tt>powr(−3,2)</tt> where the base is less than zero. The value is defined by ''e''<sup>power'×log(base)</sup>.

====Programming languages====
Most programming language with a power function are implemented using the IEEE <tt>pow</tt> function and therefore evaluate 0<sup>0</sup> as 1. The later C<ref>{{cite journal|title=Rationale for International Standard—Programming Languages—C|version=Revision 5.10|date=April 2003|author=John Benito|url=http://www.open-std.org/jtc1/sc22/wg14/www/C99RationaleV5.10.pdf|page=182|format=PDF}}</ref> and C++ standards describe this as the [[normative]] behaviour. The [[Java (programming language)|Java]] standard<ref>{{Наведена мрежна страница|url=http://download.oracle.com/javase/1.4.2/docs/api/java/lang/Math.html#pow%28double,%20double%29 |title=Math (Java 2 Platform SE 1.4.2) pow |publisher=Oracle}}</ref> mandates this behavior. The [[.NET Framework]] [[method (computer science)|method]] <code>System.Math.Pow</code> also treats 0<sup>0</sup> as 1.<ref>{{Наведена мрежна страница |url=http://msdn.microsoft.com/en-us/library/system.math.pow.aspx |title=.NET Framework Class Library Math.Pow Method |publisher=Microsoft}}</ref>

====Mathematics software====
*[[Sage (mathematics software)|Sage]] simplifies ''b''<sup>0</sup> to 1,  even if no constraints are placed on ''b''.<ref>{{Наведена мрежна страница |url=http://sagenb.org/home/pub/2433/ | title=Sage worksheet calculating x^0 |publisher=Jason Grout}}</ref> It does not simplify 0<sup>''x''</sup>, and it takes 0<sup>0</sup> to be 1.
*[[Maple (software)|Maple]] simplifies ''b''<sup>0</sup> to 1 and 0<sup>''x''</sup> to 0, even if no constraints are placed on ''b'' (the latter simplification is only valid for ''x'' > 0), and evaluates 0<sup>0</sup> to 1.{{Citation needed|date=July 2010}}
*[[Macsyma]] also simplifies ''b''<sup>0</sup> to 1 and 0<sup>''x''</sup> to 0, even if no constraints are placed on ''b'' and ''x'', but issues an error for 0<sup>0</sup>.{{Citation needed|date=July 2010}}
*[[Mathematica]] and [[Wolfram Alpha]] simplify ''b''<sup>0</sup> into 1, even if no constraints are placed on ''b''.<ref>{{Наведена мрежна страница |url=http://www.wolframalpha.com/input/?i=a^0 | title=Wolfram Alpha calculates b^0 |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref> While Mathematica does not simplify 0<sup>''x''</sup>, Wolfram Alpha returns two results, 0 and indeterminate.<ref>{{Наведена мрежна страница |url=http://www.wolframalpha.com/input/?i=0^a | title=Wolfram Alpha calculates 0^a |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref> Both Mathematica and Wolfram Alpha take 0<sup>0</sup> to be an [[indeterminate form]].<ref>{{Наведена мрежна страница |url=http://www.wolframalpha.com/input/?i=0^0 | title=Wolfram Alpha calculates 0^0 |publisher=Wolfram Alpha LLC, accessed July 24, 2011}}</ref>

==Limits of powers==
The section [[#Zero to the power of zero|zero to the power of zero]] gives a number of examples of limits which are of the [[indeterminate form]] 0<sup>0</sup>. The limits in these examples exist, but have different values, showing that the two-variable function ''x''<sup>''y''</sup> has no limit at the point (0,0). One may ask at what points this function does have a limit.

More precisely, consider the function ''f''(''x'',''y'')&nbsp;=&nbsp;''x''<sup>''y''</sup> defined on ''D''&nbsp;=&nbsp;{(''x'',''y'')&nbsp;∈&nbsp;'''R'''<sup>2</sup>&nbsp;:&nbsp;''x''&nbsp;>&nbsp;0}. Then ''D'' can be viewed as a subset of {{overline|'''R'''}}<sup>2</sup> (that is, the set of all pairs (''x'',''y'') with ''x'',''y'' belonging to the [[extended real number line]] {{overline|'''R'''}}&nbsp;=&nbsp;[−∞,&nbsp;+∞], endowed with the [[product topology]]), which will contain the points at which the function ''f'' has a limit.

In fact, ''f'' has a limit at all [[accumulation point]]s of ''D'', except for (0,0), (+∞,0), (1,+∞) and (1,−∞).<ref>N. Bourbaki, ''Topologie générale'', V.4.2.</ref> Accordingly, this allows one to define the powers ''x''<sup>''y''</sup> by continuity whenever 0&nbsp;≤&nbsp;''x''&nbsp;≤&nbsp;+∞, −∞ ≤&nbsp;y&nbsp;≤&nbsp;+∞, except for 0<sup>0</sup>, (+∞)<sup>0</sup>, 1<sup>+∞</sup> and 1<sup>−∞</sup>, which remain indeterminate forms.

Under this definition by continuity, we obtain:

* ''x''<sup>+∞</sup> = +∞ and ''x''<sup>−∞</sup> = 0, when 1 < ''x'' ≤ +∞.
* ''x''<sup>+∞</sup> = 0 and ''x''<sup>−∞</sup> = +∞, when 0 ≤ ''x'' < 1.
* 0<sup>''y''</sup> = 0 and (+∞)<sup>''y''</sup> = +∞, when 0 < ''y'' ≤ +∞.
* 0<sup>''y''</sup> = +∞ and (+∞)<sup>''y''</sup> = 0, when −∞ ≤ ''y'' < 0.

These powers are obtained by taking limits of ''x''<sup>''y''</sup> for ''positive'' values of ''x''. This method does not permit a definition of ''x''<sup>''y''</sup> when ''x''&nbsp;<&nbsp;0, since pairs (''x'',''y'') with ''x''&nbsp;<&nbsp;0 are not accumulation points of ''D''.

On the other hand, when ''n'' is an integer, the power ''x''<sup>''n''</sup> is already meaningful for all values of ''x'', including negative ones. This may make the definition 0<sup>''n''</sup>&nbsp;=&nbsp;+∞ obtained above for negative ''n'' problematic when ''n'' is odd, since in this case ''x''<sup>''n''</sup>&nbsp;→&nbsp;+∞ as ''x'' tends to 0 through positive values, but not negative ones.
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