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Zero to the power of zero

Zero to the power of zero, denoted by 0⁰, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 0⁰ = 1. 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 0⁰ to be defined as 1. For example, the following three interpretations of b⁰ make just as much sense for b = 0 as they do for positive integers b:

The interpretation of b⁰ as an empty product assigns it the value 1.
The combinatorial interpretation of b⁰ is the number of 0-tuples of elements from a b-element set; there is exactly one 0-tuple.
The set-theoretic interpretation of b⁰ is the number of functions from the empty set to a b-element set; there is exactly one such function, namely, the empty function.

All three of these specialize to give 0⁰ = 1.

Polynomials and power series

When evaluating polynomials, it is convenient to define 0⁰ as 1. A (real) polynomial is an expression of the form a₀x⁰ + ⋅⋅⋅ + a_nxⁿ, where x is an indeterminate, and the coefficients a_i 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 R[x]. The multiplicative identity of R[x] is the polynomial x⁰; that is, x⁰ times any polynomial p(x) is just p(x). Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism ev_r : R[x] → R such that ev_r(x) = r. Because ev_r is unital, ev_r(x⁰) = 1. That is, r⁰ = 1 for each real number r, including 0. The same argument applies with R replaced by any ring.

Defining 0⁰ = 1 is necessary for many polynomial identities. For example, the binomial theorem $(1+x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}$ holds for x = 0 only if 0⁰ = 1.

Similarly, rings of power series require x⁰ to be defined as 1 for all specializations of x. For example, identities like ${\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}$ and $e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}$ hold for x = 0 only if 0⁰ = 1.

In order for the polynomial x⁰ to define a continuous function R → R, one must define 0⁰ = 1.

In calculus, the power rule ${\frac {d}{dx}}x^{n}=nx^{n-1}$ is valid for n = 1 at x = 0 only if 0⁰ = 1.

Continuous exponents

Limits involving algebraic operations can often be evaluated by replacing subexpressions with their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. The expression 0⁰ is an indeterminate form: Given real-valued functions f(t) and g(t) approaching 0 (as t approaches a real number or ±∞) with f(t) > 0, the limit of f(t)^g(t) can be any non-negative real number or +∞, or it can diverge, depending on f and g. For example, each limit below involves a function f(t)^g(t) with f(t), g(t) → 0 as t → 0⁺ (a one-sided limit), but their values are different:

Thus, the two-variable function x^y, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on {(x, y) : x > 0} ∪ {(0, 0)}, no matter how one chooses to define 0⁰.

On the other hand, if f and g are analytic functions on an open neighborhood of a number c, then f(t)^g(t) → 1 as t approaches c from any side on which f is positive. This and more general results can be obtained by studying the limiting behavior of the function $\log(f(t)^{g(t)})=g(t)\log f(t)$ .

Complex exponents

In the complex domain, the function z^w may be defined for nonzero z by choosing a branch of log z and defining z^w as e^{w log z}. This does not define 0^w since there is no branch of log z defined at z = 0, let alone in a neighborhood of 0.

History

As a value

In 1752, Euler in Introductio in analysin infinitorum wrote that a⁰ = 1 and explicitly mentioned that 0⁰ = 1. 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 0⁰ = 1, though these were far from convincing, even by standards of rigor at the time.

As a limiting form

Euler, when setting 0⁰ = 1, mentioned that consequently the values of the function 0^x take a "huge jump", from ∞ for x < 0, to 1 at x = 0, to 0 for x > 0. In 1814, Pfaff used a squeeze theorem argument to prove that x^x → 1 as x → 0⁺.

On the other hand, in 1821 Cauchy explained why the limit of x^y as positive numbers x and y approach 0 while being constrained by some fixed relation could be made to assume any value between 0 and ∞ by choosing the relation appropriately. He deduced that the limit of the full two-variable function x^y without a specified constraint is "indeterminate". With this justification, he listed 0⁰ along with expressions like 0/0 in a table of indeterminate forms.

Apparently unaware of Cauchy's work, Möbius in 1834, building on Pfaff's argument, claimed incorrectly that f(x)^g(x) → 1 whenever f(x),g(x) → 0 as x approaches a number c (presumably f is assumed positive away from c). Möbius reduced to the case c = 0, but then made the mistake of assuming that each of f and g could be expressed in the form Pxⁿ for some continuous function P not vanishing at 0 and some nonnegative integer n, 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 (e^−1/x)^x → e⁻¹ and (e^−1/x)^2x → e⁻² as x → 0⁺ and expressed the situation by writing that "0⁰ can have many different values".

Current situation

Some authors define 0⁰ as 1 because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define 0⁰ is based on convenience, not on correctness. If we refrain from defining 0⁰, then certain assertions become unnecessarily awkward. ... The consensus is to use the definition 0⁰ = 1, although there are textbooks that refrain from defining 0⁰." Knuth (1992) contends more strongly that 0⁰ "has to be 1"; he draws a distinction between the value 0⁰, which should equal 1, and the limiting form 0⁰ (an abbreviation for a limit of f(t)^g(t) where f(t), g(t) → 0), 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 0⁰ undefined because 0⁰ is an indeterminate form: f(t), g(t) → 0 does not imply f(t)^g(t) → 1.

There do not seem to be any authors assigning 0⁰ 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 0⁰ as 1; see § Discrete exponents.
pow (whose intent is to return a non-NaN result when the exponent is an integer, like pown) treats 0⁰ as 1.
powr treats 0⁰ as NaN (Not-a-Number) due to the indeterminate form; see § Continuous exponents.

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 0⁰ (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 1 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 0⁰ as 1. Some languages document that their exponentiation operation corresponds to the pow function from the C mathematical library; this is the case with Lua's ^ operator and Perl's ** operator (where it is explicitly mentioned that the result of 0**0 is platform-dependent).

Mathematical and scientific software

R, SageMath, and PARI/GP evaluate x⁰ to 1. Mathematica simplifies x⁰ to 1 even if no constraints are placed on x; however, if 0⁰ is entered directly, it is treated as an error or indeterminate. Mathematica and PARI/GP further distinguish between integer and floating-point values: If the exponent is a zero of integer type, they return a 1 of the type of the base; exponentiation with a floating-point exponent of value zero is treated as undefined, indeterminate or error.