N=0 x n / n! hold for x = 0 only if 0 0 = 1. For example, identities like 1 / 1− x = Σ ∞ Similarly, rings of power series require x 0 to be defined as 1 for all specializations of x. For example, the binomial theorem (1 + x) n = Σ n ĭefining 0 0 = 1 is necessary for many polynomial identities. The same argument applies with R replaced by any ring. That is, r 0 = 1 for each real number r, including 0.

More precisely, for any given real number r, there is a unique unital R-algebra homomorphism ev r : R → R such that ev r( x) = r. Also, polynomials can be evaluated by specializing x to a real number. The multiplicative identity of R is the polynomial x 0 that is, x 0 times any polynomial p( x) is just p( x). With these operations, polynomials form a ring R. Polynomials are added termwise, and multiplied by applying the distributive law and the usual rules for exponents. A (real) polynomial is an expression of the form a 0 x 0 + ⋅⋅⋅ + a n x n, where x is an indeterminate, and the coefficients a i are real numbers. When evaluating polynomials, it is convenient to define 0 0 as 1. Īll three of these specialize to give 0 0 = 1.

The set-theoretic interpretation of b 0 is the number of functions from the empty set to a b-element set there is exactly one such function, namely, the empty function.

The combinatorial interpretation of b 0 is the number of 0-tuples of elements from a b-element set there is exactly one 0-tuple.

The interpretation of b 0 as an empty product assigns it the value 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: Many widely used formulas involving natural-number exponents require 0 0 to be defined as 1. Computer programming languages and software also have differing ways of handling this expression. In mathematical analysis, the expression is sometimes left undefined. In algebra and combinatorics, one typically defines 0 0 = 1. Zero to the power of zero, denoted by 0 0, is a mathematical expression that is either defined as 1 or left undefined, depending on context.

Mathematical expression with disputed status

Zero to One free