|
|
| Line 1: |
Line 1: |
| In mathematics, the word '''''undefined''''' has several different meanings, depending on the context. In [[geometry]], simple words such as "point" and "line" are taken as undefined terms. In [[arithmetic]], some arithmetic operations are called "undefined". The most famous example is that [[division by zero]] is undefined. In [[algebra]], a [[function (mathematics)|function]] is said to be "undefined" at points not in its domain. For example, in the [[real number]] system, <math> f(x)=\sqrt{x} </math> is undefined for negative <math>x</math>.
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is what she does for a residing. I am truly fond of handwriting but I can't make it my profession really. Alaska is exactly where he's usually been residing.<br><br>Also visit my webpage ... psychic phone ([http://hknews.classicmall.com.hk/groups/some-simple-tips-for-personal-development-progress/ try what she says]) |
| | |
| ==Undefined terms in geometry==
| |
| | |
| In ancient times, geometers attempted to define every term. For example, [[Euclid of Alexandria|Euclid]] defined a point as "that which has no part". In modern times, mathematicians recognized that attempting to define every word inevitably led to [[circular definition]]s, and in geometry left some words, "point" for example, as undefined. See [[primitive notion]].
| |
| | |
| ==Undefined operations in arithmetic==
| |
| The reasoning behind leaving [[division by zero]] undefined is as follows. Division is the inverse of multiplication. If <math> a\div b=c</math>, then <math>b\times c=a</math>. But if <math> b=0 </math>, then any multiple of <math> b </math> is also <math> 0 </math>, and so if <math> a\ne 0 </math>, no such <math> c </math> exists. On the other hand, if <math> a </math> and <math> b </math> are both zero, then every real number <math> c </math> satisfies <math> b\times c=a </math>. Either way, it is impossible to assign a particular real number to the quotient when the divisor is zero.
| |
| | |
| In calculus, <math> 0/0 </math> is sometimes used as a symbol, and is called an [[indeterminate form]], but the symbol does not represent division in the sense the word is used in ordinary arithmetic.
| |
| | |
| Another common operation that is undefined is that of raising zero to the zero power. On the one hand, if <math> x\ne 0 </math>, then <math> x^{0}=1</math>. On the other hand, if <math> y </math> is any positive number, <math> 0^{y}=0 </math>, while if <math> y </math> is negative, <math> 0^y </math> leads to division by zero, which is undefined. Thus, to make the [[laws of exponents]] work in every case where exponents are defined, <math>0^0</math> is left undefined. That said, there are branches of higher mathematics where various definitions of zero to the zero power are given (see: [[Exponentiation]]).
| |
| | |
| ==Values for which functions are undefined==
| |
| The set of numbers for which a [[function (mathematics)|function]] is defined is called the ''domain'' of the function. If a number is not in the domain of a function, the function is said to be "undefined" for that number. Two common examples are <math> f(x)=\frac{1}{x}</math> which is undefined for <math>x=0</math>, and <math> f(x)=\sqrt{x}</math>, which is undefined (in the real number system) for negative <math> x </math>.
| |
| | |
| ==Notation using ↓ and ↑==
| |
| In [[computability theory (computer science)|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''".
| |
| | |
| ==The symbols of infinity==
| |
| | |
| In [[analysis]], [[measure theory]], and other mathematical disciplines, the symbol <math>\infty</math> is frequently used to denote an infinite pseudo-number, in real analysis alongside with its negative <math> -\infty</math>. The symbol has no well-defined meaning by itself, but an expression like <math>\left\{a_n\right\}\rightarrow\infty</math> is a shorthand for a [[divergent sequence]] which is eventually larger than any given real number.
| |
| | |
| Arithmetic with the symbols <math>\pm\infty</math> is undefined. The following conventions of addition and multiplication are in common use:
| |
| * <math>x+\infty=\infty</math> <math>\forall x\in\mathbb{R}\cup\{\infty\};-\infty+x=-\infty</math> <math>\forall x\in\mathbb{R}\cup\{-\infty\}</math>.
| |
| * <math>x\cdot\infty=\infty</math> <math>\forall x\in\mathbb{R}^{+}</math>.
| |
| | |
| No sensible extension of addition and multiplication with <math>\infty</math> exist in the following cases:
| |
| * <math>\infty-\infty</math>
| |
| * <math>0\cdot\infty</math> (although in [[measure theory]], this is often defined as <math>0</math>)
| |
| * <math>\frac{\infty}{\infty}</math>
| |
| | |
| See [[extended real number line]] for more information.
| |
| | |
| ==Singularities in complex analysis==
| |
| | |
| In [[complex analysis]], a point <math>z\in\mathbb{C}</math> where a [[holomorphic function]] is undefined is called a [[Mathematical singularity|singularity]]. One distinguishes between [[removable singularity|removable singularities]] (the function can be extended holomorphically to <math>z</math>, [[Pole (complex analysis)|poles]] (the function can be extended [[meromorphic function|meromorphically]] to <math>z</math>), and [[essential singularity|essential singularities]], where no meromorphic extension to <math>z</math> exists.
| |
| | |
| ==References==
| |
| | |
| * James R. Smart, ''Modern Geometries'' Third Edition, Brooks/Cole, 1988, ISBN 0-534-08310-2
| |
| | |
| [[Category:Mathematical terminology]]
| |
| [[Category:Calculus]]
| |
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Office supervising is what she does for a residing. I am truly fond of handwriting but I can't make it my profession really. Alaska is exactly where he's usually been residing.
Also visit my webpage ... psychic phone (try what she says)