Defined and undefined: Difference between revisions
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===Undefined forms=== |
===Undefined forms=== |
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The following expressions are undefined in all contexts{{Citation needed|date=March 2008}}, but remarks in the [[defined and undefined#Analysis|analysis]] section may apply. |
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* <math>\infty - \infty</math> |
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* <math>(-1)^{\pm\infty}</math> |
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* <math>\frac{\pm\infty}{\pm\infty}</math> |
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The following form is normally undefined |
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* <math>\frac{x}{0}</math> for x≠0, but see [[Division_by_zero#Real_projective_line|division by zero on the real projective line]] |
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===Indeterminate forms=== |
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They are defined in some, but not all contexts, as described in sections of this article. They are also [[indeterminate form]]s, if they occur as the limit of an expression it may be possible to find the value of the limit using other means. |
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{|class="wikitable" |
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!Expression !! Context in which it is used |
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| <math>\frac{0}{0}</math> |
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| See [[division by zero]]. |
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|- |
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| <math>0^0</math> |
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| [[Exponentiation#Zero to the zero power|zero to the zero power]], [[Defined and undefined#Analysis|analysis]], and [[Defined and undefined#Set theory|set theory]] |
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|- |
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| <math>\infty^0</math> |
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| [[Defined and undefined#Analysis|analysis]] and [[set theory]] |
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|- |
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| <math>1^\infty</math> |
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| [[Defined and undefined#Analysis|analysis]] and [[set theory]] |
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|- |
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| <math>0 \cdot \infty</math> |
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| [[Defined and undefined#Set theory|set theory]], and [[Defined and undefined#Measure theory|measure theory]] |
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|- |
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| <math>\infty - \infty</math> |
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|[[Defined and undefined#Analysis|analysis]] |
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|- |
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| <math>\frac{\infty}{\infty}</math> |
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|[[Defined and undefined#Analysis|analysis]] |
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|} |
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===Analysis=== |
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In [[analysis (mathematics)|mathematical analysis]] the domain of a [[function (mathematics)|function]] is usually determined by the [[limit of a function|limit]] of the function, so as to make the function [[continuous (mathematics)|continuous]]. This definition makes all of the expressions undefined. In [[calculus]], some of the expressions arise in intermediate calculations, where they are called [[indeterminate form]]s and dealt with using techniques such as [[L'Hôpital's rule]]. |
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===Measure theory=== |
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In [[measure theory]] (which is the common way of treating [[probability theory]] in mathematics), measures are preserved under [[countable]] addition. Taking <math>\infty</math> as countable, |
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<math>0 \cdot \infty = \sum_{n=0}^{\infty} 0 = 0</math>. |
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==Notation using ↓ and ↑== |
==Notation using ↓ and ↑== |
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Revision as of 22:17, 15 May 2010
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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 a 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.
Arithmetic forms
Undefined forms
sup bitch how bout u suck my cock nigga
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" .
