0 to the power of 0

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:
$\lim_ ^ = 1 ,$
$\lim_ \left(e^\right)^t = 0,$
$\lim_ \left(e^\right)^ = +\infty,$
$\lim_ \left(e^\right)^ = e^ .$

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"
$0^0 = (a-a)^ = \frac = 1$
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.

References

External links

sci.math FAQ: What is ?

What does {{math, 0⁰ (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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