8.2 Constants and Fixed Values

Constants and Fixed Values concerns quantities within an algebraic expression, equation, or formula that do not change value across the context in which they appear, in contrast to variables, and the several distinct forms a constant can take depending on its role.

Basic Numerical Constants

Numerical Constant

A numerical constant is any specific number appearing in an expression or equation that is not attached to a variable and does not change value within the given context. It contributes a fixed amount to the value of the expression regardless of what any variable in the same expression is set to.

Positive Constant

A positive constant is a fixed numerical value greater than zero, contributing a fixed positive amount wherever it appears in an expression, unaffected by any sign changes elsewhere in the expression unless directly combined with another signed term.

Negative Constant

A negative constant is a fixed numerical value less than zero, contributing a fixed negative amount wherever it appears, and must be tracked carefully through operations exactly as any negative number would be, since its sign remains fixed as part of its identity as a constant.

Zero as a Constant

Zero is itself a constant, representing the fixed absence of quantity, and it behaves distinctly from other constants in that adding it to any expression leaves the expression's value unchanged, while its presence as a coefficient eliminates the term it multiplies entirely.

Named Constants and Fixed Quantities in Formulas

Named Mathematical Constant

A named mathematical constant is a fixed numerical value that recurs across many different formulas and contexts and is given its own symbol because its exact value is either irrational or otherwise inconvenient to write out in full, such as the constant representing the ratio of a circle's circumference to its diameter.

Fixed Quantity within a Formula

A fixed quantity within a formula is a value that does not change across every application of that formula, distinguishing it from the variables in the same formula, which do take on different values for different instances. In a formula for the area of a circle, the constant relating circumference to diameter is fixed, while the radius varies from one circle to the next.

Constants within the Structure of an Expression

Constant Term

A constant term is a term in an expression that consists of a number alone, with no variable factor attached to it, and its value therefore does not depend on the value assigned to any variable elsewhere in the expression.

Constant Factor

A constant factor is a fixed number multiplying a variable or expression within a term, distinguishing the fixed multiplicative contribution from the variable part of the same term, which is otherwise known as the coefficient of that term.

Constant Term and Coefficient Distinction

A constant term must be distinguished from a coefficient: a constant term stands entirely on its own, unattached to any variable, while a coefficient is a constant factor that multiplies a variable and only ever appears together with that variable, changing the expression's value indirectly through whatever value the variable takes.

Recognizing a Constant's Role

Context-Dependent Constant Role

Whether a given symbol or number is treated as a constant depends entirely on the context in which the expression is used: a letter representing a fixed physical value in one formula is a constant there, but the same letter could represent a variable quantity in a different formula, so identifying a constant requires knowing what stays fixed and what is permitted to change within the specific problem at hand.