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
  empty function.^[1]

Defining 0⁰ = 1 is necessary for many polynomial identities. For example, the binomial theorem ${\displaystyle (1+x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}}$ holds for x = 0 only if 0⁰ = 1.^[4]

Similarly, rings of power series require x⁰ to be defined as 1 for all specializations of x. For example, identities like ${\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}}$ and ${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$ hold for x = 0 only if 0⁰ = 1.^[5]

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

In calculus, the power rule ${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}$ 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 by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form.^[6] 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:

$${\displaystyle \lim _{t\to 0^{+}}{t}^{t}=1,}$$

$${\displaystyle \lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{t}=0,}$$

$${\displaystyle \lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{-t}=+\infty ,}$$

$${\displaystyle \lim _{t\to 0^{+}}\left(e^{-1/t}\right)^{at}=e^{-a}.}$$

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.^[7]

On the other hand, if f and g are

This and more general results can be obtained by studying the limiting behavior of the function ${\displaystyle \log(f(t)^{g(t)})=g(t)\log f(t)}$.

Complex exponents

In the

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"

$${\displaystyle 0^{0}=(a-a)^{n-n}={\frac {(a-a)^{n}}{(a-a)^{n}}}=1}$$

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.^[14] In 1814, Pfaff used a squeeze theorem argument to prove that x^x → 1 as x → 0⁺.^[8]

On the other hand, in 1821

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".^[21]

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."^[22] 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."^[19]
• Other authors leave 0⁰ undefined because 0⁰ is an indeterminate form: f(t), g(t) → 0 does not imply f(t)^g(t) → 1.^[23][24]

There do not seem to be any authors assigning 0⁰ a specific value other than 1.^[22]

Treatment on computers

IEEE floating-point standard

The

• 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.^[26] 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).^[27]

Programming languages

The C and C++ standards do not specify the result of 0⁰ (a domain error may occur). But for C, as of