Undefined (mathematics)

In mathematics, undefined has several different meanings, depending on the context. Such contexts include the following, among others:

In various branches of mathematics, certain concepts are introduced as primitive notions; for example, in geometry these might be given the names "point", "line", and "angle". As these terms are not defined in terms of other concepts, such terms may be called "undefined terms".
A function is said to be "undefined" at points not in its domain – for example, in the real number system, $f(x)={\sqrt {x}}$ is undefined for negative $x$ , i.e., function $f$ assigns no value to negative arguments.
As a special case of the latter, some arithmetic operations may not assign a meaning to certain values of its operands, such as happens with division by zero. In such a case an expression involving such an operation is called "undefined".

Undefined terms

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

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

In arithmetic

The expression 0/0 is undefined in arithmetic, as explained in division by zero (the expression is used in calculus to represent an indeterminate form).

Mathematicians have different opinions as to whether 0⁰ should be defined to equal 1, or be left undefined; see Zero to the power of zero for details.

Values for which functions are undefined

The set of numbers for which a 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 $f(x)={\frac {1}{x}}$ , which is undefined for $x=0$ , and $f(x)={\sqrt {x}}$ , which is undefined (in the real number system) for negative $x$ .

In trigonometry

In trigonometry, $\tan \theta$ and $\sec \theta$ are undefined for all $\theta =180^{\circ }(n-{\frac {1}{2}})$ , and $\cot \theta$ and $\csc \theta$ are undefined for all $\theta =180^{\circ }(n)$ .

In computer science

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 as "f(a) is defined." ^[1]

If a is not in the domain of f, then this is written as f(a)↑ and is read as "f(a) is undefined".

The symbols of infinity

In analysis, measure theory, and other mathematical disciplines, the symbol $\infty$ is frequently used to denote an infinite pseudo-number in real analysis, along with its negative, $-\infty$ . The symbol has no well-defined meaning by itself, but an expression like $\left\{a_{n}\right\}\rightarrow \infty$ is shorthand for a divergent sequence, which at some point is eventually larger than any given real number.

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

$x+\infty =\infty$ $\forall x\in {\mathbb {R}}\cup \{\infty \}$ .
$-\infty +x=-\infty$ $\forall x\in {\mathbb {R}}\cup \{-\infty \}$ .
$x\cdot \infty =\infty$ $\forall x\in {\mathbb {R}}^{{+}}$ .

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

$\infty -\infty$
$0\cdot \infty$ (although in measure theory, this is often defined as $0$ )
${\frac {\infty }{\infty }}$

See extended real number line for more information.

Singularities in complex analysis

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

References

↑ Enderton, Herbert B. Computability: An Introduction to Recursion Theory. Elseveier, 2011, pp. 3-6, ISBN 978-0-12-384958-8