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. Many widely used formulas involving natural-number exponents require 00 to be defined as 1. For example, the following three interpretations of b^0 make just as much sense for b = 0 as they do for positive integers b: The interpretation of b^0 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 0^0 = 1. When evaluating polynomials, it is convenient to define 00 as 1. A (real) polynomial is an expression of the form a0x0 + ⋅⋅⋅ + anxn, where x is an indeterminate, and the coefficients ai 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 evr : R[x] → R such that evr(x) = r. Because evr is unital, evr(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 = Σ xk 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.