User:Johnwbyrd/Undefined

In mathematics, the term undefined refers to a value, function, or other property that cannot be assigned a meaning within a specific formal system. Attempting to assign or use an undefined value within a particular formal system, can produce contradictory or meaningless results within that system.

In practice, mathematicians often use the term undefined to warn that a particular calculation or property can produce mathematically inconsistent results, and therefore should be avoided. Caution must be taken to avoid the use of undefined values in a deduction or proof.

Whether a particular function or value is undefined, depends on the rules of the formal system in which it is used. For example, the imaginary number ${\sqrt {-1}}$ is undefined on the real number plane. So it is meaningless to reason about the value, solely within the discourse of real numbers. However, defining the imaginary value i to be equal to ${\sqrt {-1}}$ , allows there to be an internally consistent set of mathematics referred to as the complex plane. Therefore, within the discourse of complex numbers, ${\sqrt {-1}}$ is in fact defined.

Many new fields of mathematics have been created, by taking previously undefined functions and values, and assigning them new meanings. Most mathematicians generally consider these innovations significant, to the extent that they are both internally consistent and practically useful. For example, although Ramanujan summation may appear to be unintuitive, and its results may appear to be contradictory, it is useful for modelling a number of phenomena in string theory.

Common examples of undefined functions

Division by zero

In algebra, division by zero is undefined. That is to say, use of a division by zero in an algebraic calculation or proof, can produce absurd or meaningless results.

Assuming that division by zero exists, can produce inconsistent logical results, such as the following fallacious "proof" that 1 is equal to 2:

$a=ba^{2}=b^{2}$

If we assume that a non-zero answer n exists when some non-zero number k is divided by zero, then that would imply that k * 0 = n. But there exists no number that, when multiplied by zero, produces a number that is not zero. Therefore, our assumption is incorrect.

A function may be said to be undefined, outside of its domain. As one example, $f(x)={\frac {1}{x}}$ is undefined when $x=0$ . As division by zero is illegal in algebra, $x=0$ is not part of the domain of $f(x)$ .

Other uses of the term

Contrast the term undefined in computer science, in which the term indicates that a function may produce or return any result, which may or may not be correct. In mathematics, undefined means that no value can be reasonably assigned, within the particular field of discourse.

Related terms

Indeterminate

The term undefined should be contrasted with the term indeterminate. In the first case, undefined generally indicates that a value or property can have no meaningful definition. In the second case, indeterminate generally indicates that a value or property can have many meaningful definitions. Additionally, it seems to be generally accepted that undefined values may not be safely used within a particular formal system, while indeterminate values might be, depending on the relevant rules of the particular formal system.

Division by zero

In mathematics, the term undefined is used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values).^[1] The term can take on several different meanings depending on the context. For example:

In various branches of mathematics, certain concepts are introduced as primitive notions (e.g., the terms "point", "line" and "plane" in geometry). As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms".
A function is said to be "undefined" at points outside of its domain – for example, the real-valued function $f(x)={\frac {1}{x}}$ is undefined for $x=0$ .
In algebra, some arithmetic operations may not assign a meaning to certain values of its operands (e.g., division by zero). In which case, the expressions involving such operands are termed "undefined".^[2]

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 (such as "point") undefined (see primitive notion for more).

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", which are again primitive, undefined terms. This allows such abstract mathematical theories to be applied to very diverse concrete situations.

In arithmetic

The expression ${\frac {n}{0}},n\neq 0$ is undefined in arithmetic, as explained in division by zero (the ${\frac {0}{0}}$ 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.

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 ${\textstyle f(x)={\frac {1}{x}}}$ , 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, for all $n\in \mathbb {Z}$ , the functions $\tan \theta$ and $\sec \theta$ are undefined for all ${\textstyle \theta =\pi \left(n-{\frac {1}{2}}\right)}$ , while the functions $\cot \theta$ and $\csc \theta$ are undefined for all $\theta =\pi n$ .

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 (i.e., the function can be extended holomorphically to $z$ ), poles (i.e., the function can be extended meromorphically to $z$ ), and essential singularities (i.e., no meromorphic extension to $z$ can exist).

In computability theory

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 $f(a)\downarrow$ , and is read as "f(a) is defined."^[3]

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

It is important to distinguish "logic of existence"(the standard one) and "logic of definiteness". Both arrows are not well-defined as predicates in logic of existence, which normally uses the semantics of total functions. Term f(x) is a term and it has some value for example $\emptyset$ , but in the same time $\emptyset$ can be a legitimate value of a function. Therefore the predicate "defined" doesn't respect equality, therefore it is not well-defined.

The logic of definiteness has different predicate calculus, for example specialization of a formula with universal quantifier requires the term to be well-defined. $(\forall x:\varphi )\&(t\downarrow )\Longrightarrow \varphi [t/x]$ Moreover, it requires introduction of a quasi-equality notion, which makes necessary the reformulation of the axioms. ^[4]

The symbols of infinity

In analysis, measure theory and other mathematical disciplines, the symbol $\infty$ is frequently used to denote an infinite pseudo-number, 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:

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

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 $0$ )
${\frac {\infty }{\infty }}$
$-\infty [{1(0)}]$

For more detail, see extended real number line.

References

^ Weisstein, Eric W. "Undefined". mathworld.wolfram.com. Retrieved 2019-12-15.
^ Bogomolny, Alexander. "Undefined vs Indeterminate in Mathematics". Cut-the-Knot. Retrieved 2019-12-15.
^ Enderton, Herbert B. (2011). Computability: An Introduction to Recursion Theory. Elseveier. pp. 3–6. ISBN 978-0-12-384958-8.
^ Farmer, William M.; Guttman, Joshua D. (October 2000). "A Set Theory with Support for Partial Functions" (PDF). Studia Logica. 66 (1, Partiality and Modality): 59–78. doi:10.1023/A:1026744827863.

User:Johnwbyrd/Undefined

Contents

Common examples of undefined functions

Division by zero

Other uses of the term

Related terms

Indeterminate

Undefined terms

In arithmetic

Values for which functions are undefined

In trigonometry

In complex analysis

In computability theory

Notation using ↓ and ↑

The symbols of infinity

References

Further reading

See also