What Makes An Expression Undefined

Expression which is non assigned an interpretation

In mathematics, the term undefined is oftentimes used to refer to an expression which is not assigned an interpretation or a value (such equally an indeterminate class, which has the propensity of assuming different values).^[ane] The term can take on several different meanings depending on the context. For example:

Undefined terms [edit]

In ancient times, geometers attempted to define every term. For example, Euclid divers a betoken every bit "that which has no function". In modern times, mathematicians recognize that attempting to define every word inevitably leads to circular definitions, and therefore leave some terms (such as "point") undefined (see primitive notion for more than).

This more abstruse approach allows for fruitful generalizations. In topology, a topological space may be divers as a set up of points endowed with certain backdrop, only in the general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.

In arithmetics [edit]

The expression 0 / 0 is undefined in arithmetic, as explained in division by zero (the same expression is used in calculus to stand for an indeterminate course).

Mathematicians have different opinions equally to whether 0⁰ should be defined to equal one, or be left undefined.

Values for which functions are undefined [edit]

The ready of numbers for which a office is defined is called the domain of the function. If a number is not in the domain of a function, the office is said to be "undefined" for that number. Two mutual examples are ${\textstyle f(x)={\frac {ane}{ten}}}$ , which is undefined for ${\displaystyle x=0}$ , and ${\displaystyle f(x)={\sqrt {x}}}$ , which is undefined (in the real number system) for negative ${\displaystyle x}$ .

In trigonometry [edit]

In trigonometry, the functions ${\displaystyle \tan \theta }$ and ${\displaystyle \sec \theta }$ are undefined for all ${\textstyle \theta =180^{\circ }\left(due north-{\frac {one}{2}}\right)}$ , while the functions ${\displaystyle \cot \theta }$ and ${\displaystyle \csc \theta }$ are undefined for all ${\displaystyle \theta =180^{\circ }(n)}$ .

In information science [edit]

Notation using ↓ and ↑ [edit]

In computability theory, if ${\displaystyle f}$ is a partial function on ${\displaystyle S}$ and ${\displaystyle a}$ is an element of ${\displaystyle S}$ , and so this is written as ${\displaystyle f(a)\downarrow }$ , and is read every bit "f(a) is divers."^[three]

If ${\displaystyle a}$ is not in the domain of ${\displaystyle f}$ , and so this is written as ${\displaystyle f(a)\uparrow }$ , and is read every bit " ${\displaystyle f(a)}$ is undefined".

The symbols of infinity [edit]

In assay, mensurate theory and other mathematical disciplines, the symbol ${\displaystyle \infty }$ is frequently used to denote an infinite pseudo-number, along with its negative, ${\displaystyle -\infty }$ . The symbol has no well-divers meaning past itself, but an expression similar ${\displaystyle \left\{a_{n}\correct\}\rightarrow \infty }$ is autograph for a divergent sequence, which at some point is somewhen larger than whatever given real number.

Performing standard arithmetic operations with the symbols ${\displaystyle \pm \infty }$ is undefined. Some extensions, though, ascertain the following conventions of addition and multiplication:

No sensible extension of addition and multiplication with ${\displaystyle \infty }$ exists in the following cases:

For more detail, see extended existent number line.

Singularities in complex assay [edit]

In complex analysis, a signal ${\displaystyle z\in \mathbb {C} }$ where a holomorphic part is undefined is called a singularity. One distinguishes between removable singularities (i.east., the function tin exist extended holomorphically to ${\displaystyle z}$ ), poles (i.due east., the part can be extended meromorphically to ${\displaystyle z}$ ), and essential singularities (i.due east., no meromorphic extension to ${\displaystyle z}$ can exist).

References [edit]

1. ^ Weisstein, Eric W. "Undefined". mathworld.wolfram.com . Retrieved 2019-12-15 .
2. ^ "Undefined vs Indeterminate in Mathematics". www.cut-the-knot.org . Retrieved 2019-12-15 .
3. ^ Enderton, Herbert B. (2011). Computability: An Introduction to Recursion Theory. Elseveier. pp. 3–half-dozen. ISBN978-0-12-384958-8.

Further reading [edit]

• Smart, James R. (1988). Modern Geometries (Third ed.). Brooks/Cole. ISBN0-534-08310-2.

What Makes An Expression Undefined,

Source: https://en.wikipedia.org/wiki/Undefined_(mathematics)

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