Defined and undefined

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In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion.

The following expressions are undefined in all contexts, but remarks in the analysis section may apply.

$\frac{x}{0}$	(see also division by zero)
$\infty - \infty$
$(-1)^\infty$
$0 \cdot -\infty$
$\frac{\pm\infty}{\pm\infty}$

The following are defined in some, but not all contexts, as described in sections of this article.

$00$	zero to the zero power, analysis, and set theory
$\infty^0$	analysis and set theory
$1^\infty$	analysis and set theory
$0 \cdot \infty$	analysis, set theory, and measure theory

1 Zero to the zero power
2 Analysis
3 Set theory
4 Measure theory
5 Notation using ↓ and ↑
6 See also

[edit] Zero to the zero power

The question of $00$ may be the most common point on which branches of mathematics disagree. Here we note only two considerations, one from analysis and one from combinatorics, as an example of the way different approaches may yield different answers. A more complete discussion on $00$ is given at Zero to the zero power in the article on exponentiation.

In 1821, Cauchy also listed 0⁰ as undefined. The function 0^x (for x>0) is constantly 0, and the function x⁰ (for x>0) is constantly 1, so there seems to be no natural value for 0⁰. Indeed, for suitably chosen continuous functions f and g with whose limit as $x\to 0$ is 0 (with f taking positive values), the limit

$\lim_{x\to 0} f(x)^{g(x)}$

can be any nonnegative number.

Modern textbooks often define $00 = 1$ . For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics:

“

Some textbooks leave the quantity 0⁰ undefined, because the functions 0^x and x⁰ have different limiting values when x decreases to 0. But this is a mistake. We must define x⁰ = 1 for all x , if the binomial theorem is to be valid when x = 0 , y = 0, and/or x = −y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.

”

[edit] Analysis

In mathematical analysis the domain of a function is usually determined by the limit of the function, so as to make the function continuous. This definition makes all of the expressions undefined. In calculus, some of the expressions arise in intermediate calculations, where they are called indeterminate forms and dealt with using techniques such as L'Hôpital's rule.

[edit] Set theory

In set theory, X^Y is defined as the set of functions from Y to X, and for cardinal numbers, we take the cardinality of the set of functions, for arbitrary sets of cardinality Y and X. In this sense, $0^0 = \infty^0 = 1^\infty = 1$ . The set-theoretic product is similarly taken as the Cartesian product, and so $0\cdot\infty = 0$ in set theory.

[edit] Measure theory

In measure theory (which the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking $\infty$ as countable, $0 \cdot \infty = \sum_{n=0}^{\infty} 0 = 0$ .