# Definition:Arbitrary Constant

## Definition

An **arbitrary constant** is a symbol used to represent an object which is neither a specific number nor a variable.

It is used to represent a general object (usually a number, but not necessarily) whose value can be assigned when the expression is instantiated.

### In the context of Calculus

From the language in which it is couched, it is apparent that the primitive of a function may not be unique, otherwise we would be referring to $F$ as ** the primitive** of $f$.

This point is made apparent in Primitives which Differ by Constant: if a function has a primitive, there is an infinite number of them, all differing by a constant.

That is, if $F$ is a primitive for $f$, then so is $F + C$, where $C$ is a constant.

This constant is known as an **arbitrary constant**.

## Examples

### Linear Equation

Consider the equation:

- $y = a x + b$

where:

- $x$ is an independent variable
- $y$ is a dependent variable.

The symbols $a$ and $b$ stand for **arbitrary constants**.

## Also see

- Results about
**arbitrary constants**can be found**here**.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**arbitrary constant** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**arbitrary constant**