Zero to the power of zero
Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines  . In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also have differing ways of handling this expression. Discrete exponents
Many widely used formulas involving natural-number exponents require  to be defined as . For example, the following three interpretations of  make just as much sense for  as they do for positive integers :
 The interpretation of  as an empty product assigns it the value . The combinatorial interpretation of  is the number of 0-tuples of elements from a -element set; there is exactly one 0-tuple. The set-theoretic interpretation of  is the number of functions from the empty set to a -element set; there is exactly one such function, namely, the empty function. All three of these specialize to give . Polynomials and power series
When evaluating polynomials, it is convenient to define  as . A (real) polynomial is an expression of the form , where  is an indeterminate, and the coefficients  are real numbers. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. With these operations, polynomials form a ring . The multiplicative identity of  is the polynomial ; that is,  times any polynomial  is just . Also, polynomials can be evaluated by specializing  to a real number. More precisely, for any given real number , there is a unique unital -algebra homomorphism  such that . Because  is unital, . That is,  for each real number , including 0. The same argument applies with  replaced by any ring. Defining  is necessary for many polynomial identities. For example, the binomial theorem  holds for  only if . Similarly, rings of power series require  to be defined as 1 for all specializations of . For example, identities like  and  hold for  only if . In order for the polynomial  to define a continuous function , one must define . In calculus, the power rule  is valid for  at  only if . Continuous exponents

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. The expression  is an indeterminate form: Given real-valued functions  and  approaching  (as  approaches a real number or ) with , the limit of  can be any non-negative real number or , or it can diverge, depending on  and . For example, each limit below involves a function  with  as  (a one-sided limit), but their values are different:

Thus, the two-variable function , though continuous on the set , cannot be extended to a continuous function on , no matter how one chooses to define . On the other hand, if  and  are analytic functions on an open neighborhood of a number , then  as  approaches  from any side on which  is positive. This and more general results can be obtained by studying the limiting behavior of the function . Complex exponents
In the complex domain, the function  may be defined for nonzero  by choosing a branch of  and defining  as . This does not define  since there is no branch of  defined at , let alone in a neighborhood of . History

As a value
In 1752, Euler in Introductio in analysin infinitorum wrote that  and explicitly mentioned that . An annotation attributed to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis offered the "justification" 

as well as another more involved justification. In the 1830s, Libri published several further arguments attempting to justify the claim , though these were far from convincing, even by standards of rigor at the time. As a limiting form
Euler, when setting , mentioned that consequently the values of the function  take a "huge jump", from  for , to  at , to  for . In 1814, Pfaff used a squeeze theorem argument to prove that  as . On the other hand, in 1821 Cauchy explained why the limit of  as positive numbers  and  approach  while being constrained by some fixed relation could be made to assume any value between  and  by choosing the relation appropriately. He deduced that the limit of the full two-variable function  without a specified constraint is "indeterminate". With this justification, he listed  along with expressions like  in a table of indeterminate forms. Apparently unaware of Cauchy's work, Möbius in 1834, building on Pfaff's argument, claimed incorrectly that  whenever  as  approaches a number  (presumably  is assumed positive away from ). Möbius reduced to the case , but then made the mistake of assuming that each of  and  could be expressed in the form  for some continuous function  not vanishing at  and some nonnegative integer , which is true for analytic functions, but not in general. An anonymous commentator pointed out the unjustified step; then another commentator who signed his name simply as "S" provided the explicit counterexamples  and  as  and expressed the situation by writing that " can have many different values". Current situation
 Some authors define  as  because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define  is based on convenience, not on correctness. If we refrain from defining , then certain assertions become unnecessarily awkward. ... The consensus is to use the definition , although there are textbooks that refrain from defining ." Knuth (1992) contends more strongly that  "has to be "; he draws a distinction between the value , which should equal , and the limiting form  (an abbreviation for a limit of  where ), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side." Other authors leave  undefined because  is an indeterminate form:  does not imply . There do not seem to be any authors assigning  a specific value other than 1. 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: 
pown (whose exponent is an integer) treats  as ; see . pow (whose intent is to return a non-NaN result when the exponent is an integer, like pown) treats  as . powr treats  as NaN (Not-a-Number) due to the indeterminate form; see . The pow variant is inspired by the pow function from C99, mainly for compatibility. It is useful mostly for languages with a single power function. The pown and powr variants have been introduced due to conflicting usage of the power functions and the different points of view (as stated above). Programming languages
The C and C++ standards do not specify the result of  (a domain error may occur). But for C, as of C99, if the normative annex F is supported, the result for real floating-point types is required to be  because there are significant applications for which this value is more useful than NaN (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annex G is supported. The Java standard, the .NET Framework method System.Math.Pow, Julia, and Python also treat  as . Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua and Perl's ** operator (where it is explicitly mentioned that the result of 0**0 is platform-dependent). Mathematical and scientific software
APL, R, Stata, SageMath, Matlab, Magma, GAP, Singular, PARI/GP, and GNU Octave evaluate  to . Mathematica and Macsyma simplify  to  even if no constraints are placed on ; however, if  is entered directly, it is treated as an error or indeterminate. SageMath does not simplify . Maple, Mathematica and PARI/GP further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a  of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error. See also
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References

External links

 sci.math FAQ: What is ? What does  (zero to the zeroth power) equal? on AskAMathematician.com

Exponentials
Mathematical analysis
Computer arithmetic
Computer errors
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Software anomalies
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