Discover

(Redirected from Defined)
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.

Contents

Examples and workarounds

Zero to the zero power

Analysis

Measure theory

Notation using ↓ and ↑

See also

Examples and workarounds

The following expressions are undefined in all contexts, but remarks in the analysis section may apply.
{|border="1" style="text-align:center" cell-padding="3"
|
| (see also division by zero)
|-
| $infty\; -\; infty$
|-
| $(-1)^infty$
|-
| $0\; cdot\; -infty$
|-
|
|}
The following are defined in some, but not all contexts, as described in sections of this article.
{|border="1" style="text-align:center" cell-padding="3"
| $0^0$
| 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
|}

Zero to the zero power

The question of $0^0$ 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.
In 1821, Cauchy also listed 0⁰ as undefined. The function 0^''x'' (for ''x''>0) is constantly 0, and the function ''x''⁰ (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 $x\; o\; 0$ is 0 (with ''f'' taking positive values), the limit
: $lim\_\{x\; o\; 0\}\; f(x)^\{g(x)\}$
can be any nonnegative number, or infinity, or fail to exist.
Modern textbooks often define $0^0=1$. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete mathematics:

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.

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,
$0\; cdot\; infty\; =\; sum\_\{n=0\}^\{infty\}\; 0\; =\; 0$.

Notation using ↓ and ↑

In computability theory, if ''f'' is a partial function on ''S'' and ''a'' is an element of ''S'', then this is written as ''f''(''a'')↓ and is read "''f''(''a'') is ''defined''."
If ''a'' is not in the domain of ''f'', then ''f''(''a'')↑ is written and is read as "''f''(''a'') is ''undefined''" .

See also

★ Well-defined

★ Indeterminate

★ Mathematical singularity

★ L'Hôpital's rule

★ Bottom type