Definition:Independent Variable

Definition

Real Function

Let $f: \R \to \R$ be a real function.

Let $\map f x = y$.

Then $x$ is referred to as an independent variable.

Complex Function

Let $f: \C \to \C$ be a complex function.

Let $\map f z = w$.

Then $z$ is referred to as an independent variable (of $f$).

Also known as

An independent variable can also be referred to as an argument.

Also see

• Definition:Dependent Variable

The terms independent variable and dependent variable arise from the idea that it is usual to consider that $x$ can be chosen independently of $y$, but having chosen $x$, the value of $y$ then depends on the value of $x$.

Sources

• 1914: G.W. Caunt: Introduction to Infinitesimal Calculus ... (previous) ... (next): Chapter $\text I$: Functions and their Graphs: $2$. Functions
• 1952: H.T.H. Piaggio: An Elementary Treatise on Differential Equations and their Applications (revised ed.) ... (previous) ... (next): Chapter $\text I$: Introduction and Definitions. Elimination. Graphical Representation: $3$. Definitions (footnote $*$)
• 1956: E.L. Ince: Integration of Ordinary Differential Equations (7th ed.) ... (next): Chapter $\text {I}$: Equations of the First Order and Degree: $1$. Definitions
• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 1$. Introduction
• 1968: G. Stephenson: An Introduction to Partial Differential Equations for Science Students ... (previous) ... (next): Chapter $1$ Basic Concepts: $1.1$ Introduction
• 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Functions
• 1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $1$. Functions: $1.1$ The Mathematical Concept of Functions: $1.1.1$ Introduction
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): variable: 1.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): argument: 3.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): variable: 1.