View source for Zero to the power of zero

'''Zero to the power of zero''', denoted by '''0<sup>0</sup>''', is a [[Expression (mathematics)|mathematical expression]] with no agreed-upon [[Value (mathematics)|value]]. The most common possibilities are [[1#Mathematics|1]] or leaving the expression undefined, with justifications existing for each, depending on context.
In [[algebra]], [[combinatorics]], or [[set theory]], the generally agreed upon value is&nbsp;{{nowrap|1=0<sup>0</sup> = 1}}, whereas in [[mathematical analysis]], the expression is sometimes left undefined. [[#Treatment on computers|Computer program]]s also have differing ways of handling this expression.

==Discrete exponents==
There are many widely used formulas having terms involving [[natural number|natural-number]] exponents that require {{math|0<sup>0</sup>}} to be evaluated to {{math|1}}.  For example, regarding {{math|''b''<sup>0</sup>}} as an [[empty product]] assigns it the value {{math|1}}, even when {{math|1=''b'' = 0}}.  Alternatively, the [[Exponentiation#Combinatorial interpretation|combinatorial interpretation]] of {{math|''b''<sup>0</sup>}} is the number of [[empty tuple]]s of elements from a set with {{math|''b''}} elements; there is exactly one empty tuple, even if {{math|1=''b'' = 0}}.  Equivalently, the [[Exponentiation#Over sets|set-theoretic interpretation]] of {{math|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>

==Polynomials and power series==
Likewise, when working with [[polynomial]]s, it is often necessary to assign {{math|0<sup>0</sup>}} the value 1.  A polynomial is an expression of the form <math>a_0x^0+\cdots+a_nx^n</math> where {{math|''x''}} is an indeterminate, and the coefficients <math>a_n</math> are real numbers (or, more generally, elements of some [[ring (mathematics)|ring]]).  The set of all real polynomials in {{math|''x''}} is denoted by <math>\mathbb R[x]</math>.  Polynomials are added termwise, and multiplied by the applying the usual rules for exponents in the indeterminate {{math|''x''}} (see [[Cauchy product]]).  With these algebraic rules for manipulation, polynomials form a [[polynomial ring]].  The polynomial <math>x^0</math> is the [[identity element]] of the polynomial ring, meaning that it is the (unique) element such that the product of <math>x^0</math> with any polynomial <math>p(x)</math> is just <math>p(x)</math>.<ref>{{cite book|author=[[Nicolas Bourbaki]]|title=Algèbre|publisher=Springer|year=1970}}, §III.2 No. 9: "L'unique monôme de degré 0 est l'élément unité de <math>A[(X_i)_{i\in I}]</math>; on l'identifie souvent à l'élément unité 1 de {{math|''A''}}".</ref>  Polynomials can be evaluated by specializing the indeterminate {{math|''x''}} to be a real number.  More precisely, for any given real number <math>x_0</math> there is a unique unital [[ring homomorphism]] <math>\operatorname{ev}_{x_0}:\mathbb R[x]\to\mathbb R</math> such that <math>\operatorname{ev}_{x_0}(x^1)=x_0</math>.<ref>{{cite book|author=[[Nicolas Bourbaki]]|title=Algèbre|publisher=Springer|year=1970}}, §IV.1 No. 3.</ref>  This is called the ''evaluation homomorphism''.  Because it is a unital homomorphism, we have <math>\operatorname{ev}_{x_0}(x^0) = 1.</math>  That is, <math>x^0=1</math> for all specializations of {{math|''x''}} to a real number (including zero).

This perspective is significant for many polynomial identities appearing in combinatorics.  For example, the [[binomial theorem]] <math>(1 + x)^n = \sum_{k = 0}^n \binom{n}{k} x^k</math> is not valid for {{math|1=''x'' = 0}} unless {{math|1=0<sup>0</sup> = 1}}.<ref>"Some textbooks leave the quantity {{math|0<sup>0</sup>}} undefined, because the functions {{math|''x''<sup>0</sup>}} and {{math|0<sup>''x''</sup>}} have different limiting values when {{math|''x''}} decreases to 0. But this is a mistake. We must define {{math|1=''x''<sup>0</sup> = 1}}, for all {{math|''x''}}, if the binomial theorem is to be valid when {{math|1=''x'' = 0}}, {{math|1=''y'' = 0}}, and/or {{math|1=''x'' = −''y''}}. The binomial theorem is too important to be arbitrarily restricted! By contrast, the function {{math|0<sup>''x''</sup>}} is quite unimportant".{{cite book|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>  Similarly, rings of [[power series]] require <math>x^0=1</math> to be true for all specializations of {{math|''x''}}.  Thus identities like <math>\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n</math> and <math>e^{x} = \sum_{n=0}^{\infty} \frac{x^n}{n!}</math> are only true as functional identities (including at {{math|1=''x'' = 0}}) if {{math|1=0<sup>0</sup> = 1}}.

In [[differential calculus]], the [[power rule]] <math>\frac{d}{dx} x^n = nx^{n-1}</math> is not valid for {{math|1=''n'' = 1}} at {{math|1=''x'' = 0}} unless {{math|1=0<sup>0</sup> = 1}}.

==Continuous exponents==
[[Image:X^y.png|right|thumb|300px|Plot of {{math|1=''z'' = ''x''{{i sup|''y''}}}}. The red curves (with {{math|''z''}} constant) yield different limits as {{math|(''x'', ''y'')}} approaches {{math|(0, 0)}}. The green curves (of finite constant slope, {{math|1=''y'' = ''ax''}}) all yield a limit of 1.]]

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>{{cite book|first=S. C.|last=Malik |author2=Savita Arora|year=1992|title=Mathematical Analysis|page=223|isbn=978-81-224-0323-7|quote=In general the limit of {{math|''φ''(''x'')/''ψ''(''x'')}} when {{math|1=''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 ({{math|0/0}}) then becomes meaningless. A case like this is known as an indeterminate form. Other such forms are {{math|∞/∞}}, {{math|0 × ∞}}, {{math|∞ − ∞}}, {{math|0<sup>0</sup>}}, {{math|1<sup>∞</sup>}} and {{math|∞<sup>0</sup>}}.|publisher=Wiley|location=New York}}</ref> In fact, when {{math|''f''(''t'')}} and {{math|''g''(''t'')}} are real-valued functions both approaching {{math|0}} (as {{math|''t''}} approaches a real number or {{math|±∞}}), with {{math|''f''(''t'') > 0}}, the function {{math|''f''(''t'')<sup>''g''(''t'')</sup>}} need not approach {{math|1}}; depending on {{math|''f''}} and {{math|''g''}}, the limit of {{math|''f''(''t'')<sup>''g''(''t'')</sup>}} can be any non-negative real number or {{math|+∞}}, or it can [[limit (mathematics)|diverge]]. For example, the functions below are of the form {{math|''f''(''t'')<sup>''g''(''t'')</sup>}} with {{math|''f''(''t''), ''g''(''t'') → 0}} as {{math|''t'' → 0<sup>+</sup>}} (a [[one-sided limit]]), 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>.

Thus, the two-variable function {{math|''x''<sup>''y''</sup>}}, though continuous on the set {{math|{(''x'', ''y'') : ''x'' > 0},}} [[Classification of discontinuities#Essential discontinuity|cannot be extended]] to a [[continuous function]] on <math>\{(x,y):x > 0\} \cup \{(0, 0)\}</math>, no matter how one chooses to define {{math|0<sup>0</sup>}}.<ref>{{cite journal|author=L. J. Paige|title=A note on indeterminate forms|jstor=2307224|journal=American Mathematical Monthly|volume=61|issue=3|date=March 1954|pages=189–190|doi=10.2307/2307224}}</ref> However, under certain conditions, such as when {{math|''f''}} and {{math|''g''}} are both [[analytic functions]] and {{math|''f''}} is positive on the open interval {{math|(0, ''b'')}} for some positive {{math|''b''}}, the limit approaching from the right is always {{math|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 {{math|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 {{math|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>

==Complex exponents==
In the [[complex domain]], the function {{math|''z''{{i sup|''w''}}}} may be defined for nonzero {{math|''z''}} by choosing a [[complex logarithm#Branches of the complex logarithm|branch]] of {{math|log ''z''}} and defining {{math|''z''{{i sup|''w''}}}} as {{math|''e''{{i sup|''w'' log ''z''}}}}. This does not define {{math|0<sup>''w''</sup>}} since there is no branch of {{math|log ''z''}} defined at {{math|1=''z'' = 0}}, let alone in a neighborhood of {{math|0}}.<ref>"Since {{math|log(0)}} does not exist, {{math|0<sup>''z''</sup>}} is undefined. For {{math|Re(''z'') > 0}}, we define it arbitrarily as {{math|0}}." George F. Carrier, Max Krook and Carl E. Pearson, ''Functions of a Complex Variable: Theory and Technique'', 2005, p.&nbsp;15 {{ISBN|0-89871-595-4}}</ref><ref>"For {{math|1=''z'' = 0}}, {{math|''w'' ≠ 0}}, we define {{math|1=0<sup>''w''</sup> = 0}}, while {{math|0<sup>0</sup>}} is not defined." Mario Gonzalez, ''Classical Complex Analysis'', Chapman & Hall, 1991, p.&nbsp;56. {{ISBN|0-8247-8415-4}}</ref><ref>"... Let's start at {{math|1=''x'' = 0}}. Here {{math|''x''<sup>''x''</sup>}} is undefined."  Mark D. Meyerson, The {{math|''x''<sup>''x''</sup>}} Spindle, ''Mathematics Magazine'' '''69''', no. 3 (June 1996), 198-206. {{doi|10.1080/0025570X.1996.11996428}}</ref>

==History of differing points of view==
The debate over the definition of  <math>0^0</math> has been going on at least since the early 19th century. At that time, most mathematicians agreed that <math>0^0 = 1</math>, 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 <math>0^0</math> along with expressions like <math>\textstyle \frac{0}{0}</math> in a [[Indeterminate form#List of indeterminate forms|table of indeterminate 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 <math>0^0 = 1</math>, and [[August Ferdinand Möbius|Möbius]]<ref name="Möbius1834">{{cite journal |author =  A. F. Möbius |title = Beweis der Gleichung 0<sup>0</sup> = 1, nach J. F. Pfaff |trans-title = Proof of the equation 0<sup>0</sup> = 1, according to J. F. Pfaff|url = http://babel.hathitrust.org/cgi/pt?id=mdp.39015036988163;view=1up;seq=142 |journal = Journal für die reine und angewandte Mathematik |volume = 12 |year = 1834 |pages = 134–136}}</ref> sided with him, erroneously claiming that <math>\textstyle \lim_{t \to 0^+} f(t)^{g(t)} \;=\; 1</math> whenever <math>\textstyle \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 <math>\textstyle (e^{-1/t})^t</math>, and this quieted the debate for some time. More historical details can be found in Knuth (1992).<ref name="Knuth1992" />

More recent authors interpret the situation above in different ways:
* Some argue that the best value for <math>0^0</math> 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 <math>0^0</math> is based on convenience, not on correctness. If we refrain from defining <math>0^0</math>, then certain assertions become unnecessarily awkward. [...] The consensus is to use the definition <math>0^0=1</math>, although there are textbooks that refrain from defining {{nowrap begin}}<math>0^0</math>."<ref>Donald C. Benson, ''The Moment of Proof : Mathematical Epiphanies.'' New York Oxford University Press (UK), 1999. {{ISBN|978-0-19-511721-9}}</ref>{{nowrap end}}
* Others argue that <math>0^0</math> should be defined as 1. [[Donald Knuth|Knuth]] (1992) contends strongly that <math>0^0</math> "''has'' to be 1", drawing a distinction between the ''value'' <math>0^0</math>, which should equal 1 as advocated by Libri, and the ''limiting form'' <math>0^0</math> (an abbreviation for a limit of <math>\textstyle f(x)^{g(x)}</math> where <math>\textstyle f(x), g(x) \to 0</math>), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."<ref name="Knuth1992">Donald E. Knuth, Two notes on notation, ''[[American Mathematical Monthly|Amer. Math. Monthly]]'' '''99''' no. 5 (May 1992), 403–422 ({{arXiv|math/9205211}}).</ref>  Vaughn gives several other examples of theorems whose (simplest) statements require 0<sup>0</sup> = 1 as a convention.<ref>{{Cite web|url=https://www.maa.org/book/export/html/116806|title=What is 0^0?|website=www.maa.org|access-date=2019-07-26}}</ref>

==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 operations for computing a power:<ref name="Muller_2010">{{cite book |author-last1=Muller |author-first1=Jean-Michel |author-last2=Brisebarre |author-first2=Nicolas |author-last3=de Dinechin |author-first3=Florent |author-last4=Jeannerod |author-first4=Claude-Pierre |author-last5=Lefèvre |author-first5=Vincent |author-last6=Melquiond |author-first6=Guillaume |author-last7=Revol |author-first7=Nathalie |author-last8=Stehlé |author-first8=Damien |author-last9=Torres |author-first9=Serge |title=Handbook of Floating-Point Arithmetic |year=2010 |publisher=[[Birkhäuser]] |edition=1 |doi=10.1007/978-0-8176-4705-6 |lccn=2009939668|page=216}}  {{isbn|978-0-8176-4705-6}} (online), {{isbn|0-8176-4704-X}} (print)</ref> 
*<tt>pow</tt> treats 0<sup>0</sup> as 1. 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>.

The <tt>pow</tt> variant is inspired by the <tt>pow</tt> function from [[C99]], mainly for compatibility.<ref>{{cite web |url=http://grouper.ieee.org/groups/754/email/msg03270.html |title=More transcendental questions |website=grouper.ieee.org |archive-url=https://web.archive.org/web/20171114040441/http://grouper.ieee.org/groups/754/email/msg03270.html |archive-date=2017-11-14 |access-date=2019-05-27}} (Beginning of the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)</ref> It is useful mostly for languages with a single power function. The <tt>pown</tt> and <tt>powr</tt> variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above).<ref>{{cite web |url=http://grouper.ieee.org/groups/754/email/msg03292.html |title=Re: A vague specification |website=grouper.ieee.org |archive-url=https://web.archive.org/web/20171114040300/http://grouper.ieee.org/groups/754/email/msg03292.html |archive-date=2017-11-14 |access-date=2019-05-27}} (Suggestion of variants in the discussion about the power functions for the revision of the IEEE 754 standard, May 2007.)</ref>

===Programming languages===
The C and C++ standards do not specify the result of 0<sup>0</sup> (a domain error may occur), but as of [[C99]], if the [[normative]] annex&nbsp;F is supported, the result is required to be 1 because there are significant applications for which this value is more useful than [[NaN]]<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> (for instance, with [[#Discrete_exponents|discrete exponents]]). The [[Java (programming language)|Java]] standard<ref>{{cite web|url=http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html#pow-double-double- |title=Math (Java Platform SE 8) pow |publisher=Oracle}}</ref> and the [[.NET Framework]] [[method (computer science)|method]] <code>System.Math.Pow</code><ref>{{cite web |url=http://msdn.microsoft.com/en-us/library/system.math.pow.aspx |title=.NET Framework Class Library Math.Pow Method |publisher=Microsoft}}</ref> also treat 0<sup>0</sup> as 1. Some languages document that their exponentiation operation corresponds to the <code>pow</code> function from the [[C mathematical functions|C mathematical library]]; this is the case of [[Lua (programming language)|Lua]]<ref>{{cite web |url=https://www.lua.org/manual/5.3/manual.html |title=Lua 5.3 Reference Manual |access-date=2019-05-27}}</ref> and [[Perl]]'s <code>**</code> operator<ref>{{cite web |url=https://perldoc.perl.org/perlop.html#Exponentiation |title=perlop – Exponentiation |access-date=2019-05-27}}</ref> (where it is explicitly mentioned that the result of <code>0**0</code> is platform-dependent).

===Mathematical and scientific software===
*[[SageMath]] simplifies ''b''<sup>0</sup> to 1,  even if no constraints are placed on ''b''. It takes 0<sup>0</sup> to be 1, but does not simplify 0<sup>''x''</sup> for other ''x''.{{cn|date=June 2018}}
*[[Maple (software)|Maple]] distinguishes between ''integers'' 0, 1, ... and the corresponding ''[[floating point|floats]]'' 0.0, 1.0, ... (usually denoted 0., 1., ...). If ''x'' does not evaluate to a number, then ''x''<sup>0</sup> and ''x''<sup>0.0</sup> are respectively evaluated to 1 (''integer'') and 1.0 (''float''); on the other hand, 0<sup>''x''</sup> is evaluated to the integer 0, while 0.0<sup>''x''</sup> is evaluated as 0.<sup>''x''</sup>. If both the base and the exponent are zero (or are evaluated to zero), the result is ''Float(undefined)'' if the exponent is the ''float'' 0.0; with an integer as exponent, the evaluation of 0<sup>0</sup> results in the ''integer'' 1, while that of 0.<sup>0</sup> results in the ''float'' 1.0.{{cn|date=June 2018}}
*[[Macsyma]] also simplifies ''b''<sup>0</sup> to 1 even if no constraints are placed on ''b'', but issues an error for 0<sup>0</sup>. For ''x'' > 0, it simplifies 0<sup>''x''</sup> to 0. {{Citation needed|date=July 2010}}
*[[Mathematica]] distinguishes between integers as exact, and floating point numbers as approximate. It simplifies ''b''<sup>0</sup> to 1, even if no constraints are placed on ''b'', does not simplify 0<sup>''x''</sup>, and takes 0<sup>0</sup> to be "Indeterminate".<ref>{{cite web |url=http://reference.wolfram.com/language/ref/Power.html |title=Wolfram Language & System Documentation: Power  |publisher=Wolfram| access-date =August 2, 2018}}</ref>
*[[PARI/GP]] also distinguishes between integer and floating-point types: When the exponent 0 is of integer type, the expression, such as 0^0 or 0.^0, is simplified to 1.<ref>{{cite web |url=http://pari.math.u-bordeaux.fr/cgi-bin/gitweb.cgi?p=pari.git;a=commitdiff;h=c2fac9a15 |title=pari.git / commitdiff – 10- x ^ t_FRAC: return an exact result if possible; e.g. 4^(1/2) is now 2 |access-date=September 10, 2018}}</ref> When the exponent is not of integer type, PARI/GP treats ''x''^''n'' as the transcendental function <math>e^{n \log x}</math>, so that 0^0. yields a domain error.<ref>{{cite web |url=https://pari.math.u-bordeaux.fr/pub/pari/manuals/2.11.0/users.pdf |title=Users' Guide to PARI/GP (version 2.11.0) |author=The PARI Group |format=PDF |pages=10,122 |year=2018 |quote="There is also the exponentiation operator ^, when the exponent is of type integer; otherwise, it is considered as a transcendental function. [...] If the exponent ''n'' is an integer, then exact operations are performed using binary (left-shift) powering techniques. [...] If the exponent ''n'' is not an integer, powering is treated as the transcendental function exp(''n'' log ''x'')." |access-date=September 4, 2018}}</ref>
*[[APL (programming language)|APL]],<ref>{{cite web|url=http://help.dyalog.com/latest/#Language/Primitive%20Functions/Power.htm|title=Language Reference Guide|publisher=Dyalog Ltd|quote="If ''Y'' is zero, ''R'' is defined to be 1."}}</ref> [[MATLAB|Matlab]], [[Python (programming language)|Python]], [[Magma computer algebra system|Magma]], [[GAP computer algebra system|GAP]], [[SINGULAR|singular]], and [[GNU Octave]] evaluate 0<sup>0</sup> as 1.{{cn|date=June 2018}}
*[[Microsoft Excel]] treats =0^0 as #NUM!.<ref>Microsoft Docs, [https://docs.microsoft.com/en-us/openspecs/office_standards/ms-oe376/11087d65-0fe1-44c6-bbeb-61380bc34bad "{{bracket|MS-OE376}}: Part 4 Section 3.17.7.252, POWER"]</ref>

==References==

{{reflist}}

==External links==
* [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

[[Category:Exponentials]]
[[Category:Mathematical analysis]]
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