Definition:Arbitrary Constant

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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 the constant of integration.


Linear Equation

Consider the equation:

$y = a x + b$


$x$ 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.