9.2 Algebraic Expression Construction

Algebraic Expression Construction describes how expressions are built up from the smallest indivisible pieces — numbers, variables, and named constants — through operations, grouping, fractions, powers, and roots, into compound expressions of arbitrary complexity.

Atomic Expressions

Number as an Atomic Expression

A single number, with no operation applied to it, is itself a complete expression in its own right, representing a fixed quantity that requires no further construction to be evaluated.

Variable as an Atomic Expression

A single variable, with no operation applied to it, is also a complete expression, representing a quantity that depends entirely on whatever value is assigned to that variable.

Named Constant as an Atomic Expression

A named mathematical constant, standing alone with no operation applied to it, is likewise a complete atomic expression, representing its own fixed value even though that value is not written out as an ordinary number.

Building Expressions from Operations

Unary Operation Construction

A new expression can be constructed by applying a single operation to one existing expression, such as attaching a negative sign to a variable or applying a root to a number, producing a new expression whose value depends on the original expression's value combined with that one operation.

Binary Operation Construction

A new expression can be constructed by combining two existing expressions with an operation such as addition, subtraction, multiplication, or division, producing a compound expression whose value depends on both original expressions together with the operation joining them.

Product Construction by Juxtaposition

Multiplication between a number and a variable, or between two variables, is commonly constructed without writing an explicit multiplication symbol, simply by placing the two expressions directly next to each other, with adjacency itself indicating that the two quantities are multiplied.

Building Expressions with Grouping

Grouped Expression Construction

An existing expression, however complex, can be enclosed in parentheses to form a new expression that is treated as a single unit within any larger expression it becomes part of, allowing an entire sum or difference to be used as one factor of a product or as the base of a power.

Fractional Expression Construction

Two existing expressions can be combined into a new expression by placing one over the other as numerator and denominator of a fraction, forming a compound expression whose value is the result of dividing the numerator expression by the denominator expression once both are evaluated.

Building Expressions with Powers and Roots

Power Expression Construction

An existing expression can be raised to an exponent to construct a new expression representing repeated multiplication of the original expression by itself, with the original expression serving as the base and the grouping symbols, if any, determining exactly what is included in that base.

Radical Expression Construction

An existing expression can be placed beneath a radical bar, with a chosen index, to construct a new expression representing a root of the original expression, with the radical bar acting as a grouping symbol around everything beneath it until the root is applied.

Combining Constructions

Compound Expression Construction

Repeated application of these basic constructions — combining atomic expressions with operations, enclosing sub-expressions in groups, forming fractions, and applying powers or roots — allows expressions of arbitrary complexity to be built up piece by piece, with each larger expression fully determined by the smaller expressions and operations it is constructed from.