# Definition:Variable

## Definition

A **variable** is a label which is used to refer to an unspecified object.

A **variable** can be identified by means of a symbol, for example:

- $x, y, z, A, B, C, \phi, \psi, \aleph$

It is often convenient to append a subscript letter or number to distinguish between different objects of a similar type:

- $a_0, a_1, a_2, \ldots, a_n; S_\phi, S_{\phi_x}, \ldots$

The type of symbol used to define a **variable** is purely conventional.

Particular types of object, as they are introduced, frequently have a particular range of symbols specified to define them, but there are no strict rules on the subject.

## Domain

The collection of all possible objects that a variable may refer to has to be specified.

This collection is the **domain** of the variable.

### Real Variable

A **real variable** is a symbol which can stand for any one of a set of real numbers.

### Complex Variable

A **complex variable** is a symbol which can stand for any one of a set of complex numbers.

## Satisfaction

Let $\map P x$ be a propositional function such that $x$ is a variable with a given domain $S$.

Let a specific element $a$ of $S$ be substituted for $x$ in $\map P x$ such that $\map P a$ is true.

Then $a$ is said to **satisfy** the propositional function $P$.

## Propositional Logic

A **statement variable** is a variable which is used to stand for an arbitrary and unspecified statement.

For a **statement variable**, a lowercase letter is usually used, for example:

- $p, q, r, \ldots{}$, and so on

or lowercase Greek letters, for example:

- $\phi, \psi, \chi$ and so on.

The citing of a **statement variable** can be interpreted as an assertion that the statement represented by that symbol is true.

That is:

**$p$**

means

**$p \text { is true}$**

## Predicate Logic

In the context of predicate logic, a variable is often called an **object variable** or **arbitrary name**.

As such, it is a symbol which is assigned to an *arbitrarily selected* object from a given universe of discourse.

The understanding is that (during the scope of the argument to which it is relevant) the arbitrary name could apply equally well to *any* of the objects in that universe.

## Descriptive Statistics

A **variable** is a characteristic property of all individuals in a population or sample.

It is a categorization of the population such that each individual can be unambiguously described with respect to said variable.

### Quantitative Variable

A **quantitative variable** is a variable such that:

- The variable can be described by numbers
- The performing of arithmetical operations on the data is meaningful.

### Qualitative Variable

A **qualitative variable** is a variable such that:

- The variable is not a quantitative variable
- The variable describes each individual as either having, or not having, some specific property.

## Value

A variable $x$ may be (temporarily, conceptually) identified with a particular object.

If so, then that object is called the **value** of $x$.

## Restricted Variable

A **restricted variable** is a variable whose values are confined to only some of those of which it is capable.

## Unrestricted Variable

An **unrestricted variable** is a variable whose values are not confined in any way to some only of those of which it is capable.

## Also known as

When it occurs in a mathematical equation, a **variable** is often referred to as an **unknown**.

In the specific context of elementary algebra, the ugly misnomer **pronumeral** is frequently found in Australia.

This was introduced by extension of the concept of a pronoun: a symbol that **stands in** for a **numeral**, by which the term number is actually meant.

Thankfully the term appears not to have caught on in general.

## Historical Note

The term **variable**, as opposed to a constant, was introduced by Gottfried Wilhelm von Leibniz.

However, the concept had already been established by Aristotle in his work on logic.

## Also see

- Results about
**variables**can be found**here**.

## Sources

- 1910: Alfred North Whitehead and Bertrand Russell:
*Principia Mathematica: Volume $\text { 1 }$*... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations - 1914: G.W. Caunt:
*Introduction to Infinitesimal Calculus*... (previous) ... (next): Chapter $\text I$: Functions and their Graphs: 1. Constants and Variables - 1919: Horace Lamb:
*An Elementary Course of Infinitesimal Calculus*(3rd ed.) ... (previous) ... (next): Chapter $\text I$. Continuity: $1$. Continuous Variation - 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{VI}$: On the Seashore - 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S 1.1$: Constants and variables - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): Chapter $\text I$ Introductory: $1$. Symbolic Logic and Classical Logic - 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (next): Chapter $1$: Basic Concepts: Lesson $1$: How Differential Equations Originate - 1968: Nicolas Bourbaki:
*Theory of Sets*... (previous) ... (next): Chapter $\text I$: Description of Formal Mathematics: $1$. Terms and Relations: $1$. Signs and Assemblies - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Discrete and Continuous Variables - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**variable**:**1.** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S S 1.1.1$: 'You talking to me?': Definition $1.1.4$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**variable**:**1.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**variable**