Zero to the power of zero

Zero to the power of zero, denoted as 00, is a mathematical expression that can take different values depending on the context.

In certain areas of mathematics, such as combinatorics and algebra, 00 is conventionally defined as 1 because this assignment simplifies many formulas and ensures consistency in operations involving exponents.

For instance, in combinatorics, defining 00 = 1 aligns with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions.

However, in other contexts, particularly in mathematical analysis, 00 is often considered an indeterminate form.

This is because the value of xy as both x and y approach zero can lead to different results based on the limiting process.

The expression arises in limit problems and may result in a range of values or diverge to infinity, making it difficult to assign a single consistent value in these cases.

The treatment of 00 also varies across different computer programming languages and software.

While many follow the convention of assigning 00 = 1 for practical reasons, others leave it undefined or return errors depending on the context of use, reflecting the ambiguity of the expression in mathematical analysis.

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: All three of these specialize to give 00 = 1.

Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents.

[4] Similarly, rings of power series require x0 to be defined as 1 for all specializations of x.

[7] 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.

[8] This and more general results can be obtained by studying the limiting behavior of the function

[11][12][13] In 1752, Euler in Introductio in analysin infinitorum wrote that a0 = 1[14] and explicitly mentioned that 00 = 1.

[15] An annotation attributed[16] to Mascheroni in a 1787 edition of Euler's book Institutiones calculi differentialis[17] offered the "justification"

In the 1830s, Libri[18][16] 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.

[19] Euler, when setting 00 = 1, mentioned that consequently the values of the function 0x take a "huge jump", from ∞ for x < 0, to 1 at x = 0, to 0 for x > 0.

[14] In 1814, Pfaff used a squeeze theorem argument to prove that xx → 1 as x → 0+.

[8] On the other hand, in 1821 Cauchy[20] explained why the limit of xy 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 xy 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[8] 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;[21] 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".

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).

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[28] (for instance, with discrete exponents); the result on complex types is not specified, even if the informative annex G is supported.

The Java standard,[29] the .NET Framework method System.Math.Pow,[30] Julia, and Python[31][32] also treat 00 as 1.

Mathematica[38] 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[38] and PARI/GP[37][39] 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.