Nowhere-zero flow

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 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 $x$.

Undefined terms in geometry

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 recognized that attempting to define every word inevitably led to circular definitions, 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 $a÷b=c$, then $b\times c=a$. But if $b=0$, then any multiple of $b$ is also $0$, and so if $a\ne 0$, no such $c$ exists. On the other hand, if $a$ and $b$ are both zero, then every real number $c$ satisfies $b\times c=a$. Either way, it is impossible to assign a particular real number to the quotient when the divisor is zero.

In calculus, $0/0$ 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 $x\ne 0$, then $x^0=1$. On the other hand, if $y$ is any positive number, $0^y=0$, while if $y$ is negative, $0^y$ leads to division by zero, which is undefined. Thus, to make the laws of exponents work in every case where exponents are defined, $0^0$ 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 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 $x$.

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".

The symbols of infinity

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

Arithmetic with the symbols $\pm \mathrm{\infty }$ is undefined. The following conventions of addition and multiplication are in common use:

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

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

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

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

• James R. Smart, Modern Geometries Third Edition, Brooks/Cole, 1988, ISBN 0-534-08310-2