In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Whether an expression has a meaningful value depends on the context of the expression. For example the value of 4-5 is undefined if an positive integer result is required.
Sometimes an expression is not well defined but the degree of undefinedness can be qualified in a useful way. There may be more than one possible value, or extra information might help to determine the value for instance.
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Arithmetic forms
Undefined forms
The following expressions are undefined in all contexts[citation needed], but remarks in the analysis section may apply.
The following form is normally undefined
- for x≠0, but see division by zero on the real projective line
Indeterminate forms
They are defined in some, but not all contexts, as described in sections of this article. They are also indeterminate forms, if they occur as the limit of an expression it may be possible to find the value of the limit using other means.
| Expression | Context in which it is used |
|---|---|
| See division by zero. | |
| 00 | zero to the zero power, analysis, and set theory |
| analysis and set theory | |
| analysis and set theory | |
| set theory, and measure theory | |
| analysis | |
| analysis |
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 is the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking as countable, .
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
| Look up defined or undefined in Wiktionary, the free dictionary. |
- Bottom type
- Expression (mathematics)
- Indeterminate form
- Indefinite
- L'Hôpital's rule
- Mathematical singularity
- Well-defined
Categories: Mathematical terminology
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