define

Dictionary: de·fine (dĭ-fīn') pronunciation

v., -fined, -fin·ing, -fines.

v.tr.

1. To state the precise meaning of (a word or sense of a word, for example).
2. To describe the nature or basic qualities of; explain: define the properties of a new drug; a study that defines people according to their median incomes.
1. To delineate the outline or form of: gentle hills that were defined against the sky.
2. To specify distinctly: define the weapons to be used in limited warfare.
To give form or meaning to: “For him, a life is defined by action” (Jay Parini).

v.intr.

To make or write a definition.

[Middle English definen, diffinen, from Old French definir, diffiner, from Latin dēfīnīre, to limit, determine : dē-, intensive pref.; see de– + fīnis, boundary, limit.]

definability de·fin'a·bil'i·ty n.
definable de·fin'a·ble adj.
definably de·fin'a·bly adv.
definement de·fine'ment n.
definer de·fin'er n.

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Antonyms: define

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Definition: give description
Antonyms: confuse, distort, tangle, twist

v

Definition: outline
Antonyms: conceal, obscure

Word Tutor: define

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IN BRIEF: To explain the meaning of a word or phrase.

Accept no one's definition of your life; define yourself.

Wikipedia: Defined and undefined

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In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Not all branches of mathematics come to the same conclusion.

1 Examples and workarounds
2 Notation using ↓ and ↑
3 See also

Examples and workarounds

The following expressions are undefined in all contexts^{[citation needed]}, but remarks in the analysis section may apply.

$\infty - \infty$
$(-1)^{\pm\infty}$
$\frac{\pm\infty}{\pm\infty}$

The following are defined in some, but not all contexts, as described in sections of this article.

Expression	Context in which it is used
$\frac{x}{0}$	See division by zero.
$00$	zero to the zero power, analysis, and set theory
$\infty^0$	analysis and set theory
$1^\infty$	analysis and set theory
$0 \cdot \infty$	analysis, set theory, and measure theory

Zero to the zero power

Further information: Exponentiation: Zero to the zero power

The question of 0⁰ may be the most common point on which branches of mathematics disagree^{[citation needed]}. Here we note only two considerations, one from analysis and one from combinatorics, as an example of the way different approaches may yield different answers.

In 1821, Cauchy also listed 0⁰ as undefined. The function 0^x (for x > 0) is constantly 0, and the function x⁰ (for x > 0) is constantly 1, so there seems to be no natural value for 0⁰. Indeed, for suitably chosen continuous functions ƒ and g with whose limit as x → 0< is 0 (with ƒ taking positive values), the limit

$\lim_{x\to 0} f(x)^{g(x)}$

can be any nonnegative number, or infinity, or fail to exist.

Modern textbooks often define 0⁰ = 1. For example, Ronald Graham, Donald Knuth and Oren Patashnik argue in their book Concrete Mathematics:

“

Some textbooks leave the quantity 0⁰ undefined, because the functions 0^x and x⁰ have different limiting values when x decreases to 0. But this is a mistake. We must define x⁰ = 1 for all x, if the binomial theorem is to be valid when x = 0 , y = 0, and/or x = −y . The theorem is too important to be arbitrarily restricted! By contrast, the function 0^x is quite unimportant.

”

Analysis

In mathematical analysis the domain of a function is usually determined by the limit of the function, so as to make the function continuous. This definition makes all of the expressions undefined. In calculus, some of the expressions arise in intermediate calculations, where they are called indeterminate forms and dealt with using techniques such as L'Hôpital's rule.

Measure theory

In measure theory (which is the common way of treating probability theory in mathematics), measures are preserved under countable addition. Taking $\infty$ as countable, $0 \cdot \infty = \sum_{n=0}^{\infty} 0 = 0$ .

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" .

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