Zero to the power of zero

There are many widely used formulas having terms involving natural-number exponents that require 0⁰ to be evaluated to 1. For example, regarding b⁰ as an empty product assigns it the value 1, even when b = 0. Alternatively, the combinatorial interpretation of b⁰ is the number of empty tuples of elements from a set with b elements; there is exactly one empty tuple, even if b = 0. Equivalently, the set-theoretic interpretation of 0⁰ is the number of functions from the empty set to the empty set; there is exactly one such function, the empty function.[1]

Likewise, when working with polynomials, it is convenient to define 0⁰ as having the value 1. A polynomial is an expression of the form ${\displaystyle a_{0}x^{0}+\cdots +a_{n}x^{n}}$ where x is an indeterminate, and the coefficients ${\displaystyle a_{n}}$ are real numbers (or, more generally, elements of some ring). The set of all real polynomials in x is denoted by ${\displaystyle \mathbb {R} [x]}$. Polynomials are added termwise, and multiplied by the applying the usual rules for exponents in the indeterminate x (see Cauchy product). With these algebraic rules for manipulation, polynomials form a polynomial ring. The polynomial ${\displaystyle x^{0}}$ is the identity element of the polynomial ring, meaning that it is the (unique) element such that the product of ${\displaystyle x^{0}}$ with any polynomial ${\displaystyle p(x)}$ is just ${\displaystyle p(x)}$.[2] Polynomials can be evaluated by specializing the indeterminate x to be a real number. More precisely, for any given real number ${\displaystyle x_{0}}$ there is a unique unital ring homomorphism ${\displaystyle \operatorname {ev} _{x_{0}}:\mathbb {R} [x]\to \mathbb {R} }$ such that ${\displaystyle \operatorname {ev} _{x_{0}}(x^{1})=x_{0}}$.[3] This is called the evaluation homomorphism. Because it is a unital homomorphism, we have ${\displaystyle \operatorname {ev} _{x_{0}}(x^{0})=1.}$ That is, ${\displaystyle x^{0}=1}$ for all specializations of x to a real number (including zero).

This perspective is significant for many polynomial identities appearing in combinatorics. For example, the binomial theorem ${\displaystyle (1+x)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}}$ is not valid for x = 0 unless 0⁰ = 1.[4] Similarly, rings of power series require ${\displaystyle x^{0}=1}$ to be true for all specializations of x. Thus identities like ${\displaystyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}}$ and ${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$ are only true as functional identities (including at x = 0) if 0⁰ = 1.

In differential calculus, the power rule ${\displaystyle {\frac {d}{dx}}x^{n}=nx^{n-1}}$ is not valid for n = 1 at x = 0 unless 0⁰ = 1.

Plot of z = x^y. The red curves (with z constant) yield different limits as (x, y) approaches (0, 0). The green curves (of finite constant slope, y = ax) all yield a limit of 1.

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.[5] In fact, when f(t) and g(t) are real-valued functions both approaching 0 (as t approaches a real number or ±∞), with f(t) > 0, the function f(t)^g(t) need not approach 1; depending on f and g, the limit of f(t)^g(t) can be any non-negative real number or +∞, or it can diverge. For example, the functions below are of the form f(t)^g(t) with f(t), g(t) → 0 as t → 0⁺ (a one-sided limit), but the limits are different:

${\displaystyle \lim _{t\to 0^{+}}{t}^{t}=1,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t^{2}}}}\right)^{t}=0,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t^{2}}}}\right)^{-t}=+\infty ,\quad \lim _{t\to 0^{+}}\left(e^{-{\frac {1}{t}}}\right)^{at}=e^{-a}}$.

Thus, the two-variable function x^y, though continuous on the set {(x, y) : x > 0}, cannot be extended to a continuous function on ${\displaystyle \{(x,y):x>0\}\cup \{(0,0)\}}$, no matter how one chooses to define 0⁰.[6] However, under certain conditions, such as when f and g are both analytic functions at zero and f is positive on the open interval (0, b) for some positive b, the limit approaching from the right is always 1.[7][8][9]

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.[10][11][12]

The debate over the definition of ${\displaystyle 0^{0}}$ has been going on at least since the early 19th century. At that time, most mathematicians agreed that ${\displaystyle 0^{0}=1}$, until in 1821 Cauchy[13] listed ${\displaystyle 0^{0}}$ along with expressions like ${\displaystyle \textstyle {\frac {0}{0}}}$ in a table of indeterminate forms. In the 1830s Guglielmo Libri Carucci dalla Sommaja[14][15] published an unconvincing argument for ${\displaystyle 0^{0}=1}$, and Möbius[16] sided with him, erroneously claiming that ${\displaystyle \textstyle \lim _{t\to 0^{+}}f(t)^{g(t)}\;=\;1}$ whenever ${\displaystyle \textstyle \lim _{t\to 0^{+}}f(t)\;=\;\lim _{t\to 0^{+}}g(t)\;=\;0}$. A commentator who signed his name simply as "S" provided the counterexample of ${\displaystyle \textstyle (e^{-1/t})^{t}}$, and this quieted the debate for some time. More historical details can be found in Knuth (1992).[17]

More recent authors interpret the situation above in different ways:

• Some argue that the best value for ${\displaystyle 0^{0}}$ depends on context, and hence that defining it once and for all is problematic.[18] According to Benson (1999), "The choice whether to define ${\displaystyle 0^{0}}$ is based on convenience, not on correctness. If we refrain from defining ${\displaystyle 0^{0}}$, then certain assertions become unnecessarily awkward. [...] The consensus is to use the definition ${\displaystyle 0^{0}=1}$, although there are textbooks that refrain from defining ${\displaystyle 0^{0}}$."[19]
• Others argue that ${\displaystyle 0^{0}}$ should be defined as 1. Knuth (1992) contends strongly that ${\displaystyle 0^{0}}$ "has to be 1", drawing a distinction between the value ${\displaystyle 0^{0}}$, which should equal 1 as advocated by Libri, and the limiting form ${\displaystyle 0^{0}}$ (an abbreviation for a limit of ${\displaystyle \textstyle f(x)^{g(x)}}$ where ${\displaystyle \textstyle f(x),g(x)\to 0}$), which is necessarily an indeterminate form as listed by Cauchy: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why truth was on their side."[17] Vaughn gives several other examples of theorems whose (simplest) statements require 0⁰ = 1 as a convention.[20]

• sci.math FAQ: What is 0⁰?
• What does 0^0 (zero to the zeroth power) equal? on AskAMathematician.com