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Zero to the power of zero

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From Wikipedia, the free encyclopedia

Mathematical expression with disputed status

Zero to the power of zero , denoted as 00, is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, 00 is defined as 1 because this simplifies many formulas and ensures consistency in operations involving exponents. For instance, defining 00 = 1 aligns with the interpretation of choosing 0 elements from a set and simplifies polynomial and binomial expansions.

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 0 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 varies across different computer programming languages and software. While many follow the convention 00 = 1, others leave it undefined or return errors, depending on the context.

Discrete exponents<br>[edit]

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 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.[1]

All three of these specialize to give 00 = 1.

Polynomials and power series<br>[edit]

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).[2] 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.[3]

Defining 00 = 1 is necessary for many polynomial identities. For example, the binomial theorem

{\textstyle (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

{\textstyle {\frac {1}{1-x}}=\sum _{n=0}^{\infty }x^{n}}

and

{\textstyle 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

{\textstyle {\frac {d}{dx}}x^{n}=nx^{n-1}}

is valid for n = 1 at x = 0 only if 00 = 1.

Continuous exponents<br>[edit]

Plot of z = xy. 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.<br>Limits involving algebraic operations can often be evaluated by replacing subexpressions with 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:

lim

lim

lim

lim

{\displaystyle {\begin{aligned}\lim _{t\to 0^{+}}{t}^{t}&=1,\\\lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{t}&=0,\\\lim _{t\to 0^{+}}\left(e^{-1/t^{2}}\right)^{-t}&=+\infty ,\\\lim _{t\to 0^{+}}\left(a^{-1/t}\right)^{-t}&=a.\end{aligned}}}

Thus, the two-variable function xy, 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...