10.7 Defined and Undefined Evaluation Results

Defined and Undefined Evaluation Results distinguishes the cases in which substituting values into an expression produces a legitimate single number from the cases in which the substitution produces no meaningful value at all, such as division by a zero created through substitution or a square root of a negative radicand.

When Evaluation Succeeds

Defined Numerical Result

An evaluation produces a defined numerical result when every operation in the expression can be carried out on the substituted values without encountering an undefined operation such as division by zero or an even root of a negative number, yielding a single specific number as the expression's value.

Zero Expression Value

An expression's evaluated result may legitimately equal zero, which is itself a perfectly defined numerical result, and must be distinguished from an expression that is undefined; zero is a valid outcome of evaluation, not a failure of it.

When Evaluation Fails

Zero Denominator Created by Substitution

An evaluation becomes undefined when the substituted values cause a denominator within the expression to evaluate to zero, since division by zero has no numerical meaning, even though the expression's structure was perfectly valid before the specific values were substituted in.

Negative Square-Root Radicand in the Real Numbers

An evaluation becomes undefined when the substituted values cause the radicand beneath a square root, or any other even-indexed root, to evaluate to a negative number, since no real number multiplied by itself an even number of times can produce a negative result.

Knowing Which Values Are Allowed

Allowed and Disallowed Assigned Values

For any given expression, the allowed values of a variable are those that avoid creating a zero denominator or a negative radicand under an even root anywhere in the expression, while disallowed values are those that trigger one of these conditions; identifying this distinction ahead of substitution prevents wasted effort attempting to evaluate an expression at a value where it is undefined.

Distinguishing Undefined from Incomplete

Incomplete Assignment without a Numerical Result

An expression may also fail to produce a numerical result simply because one or more of its variables has not yet been assigned a value at all, which is a different situation from a defined but not-yet-completed assignment being undefined due to a zero denominator or negative radicand; here the expression is not undefined, it is merely not yet fully evaluated.

Precision of a Defined Result

Exact Result and Decimal Approximation Distinction

Even when an evaluation is defined, the resulting value may be exact, such as a whole number or a simple fraction, or it may require a decimal approximation, such as when the evaluation involves an irrational root; both are legitimate defined results, differing only in whether the value can be written completely or must be rounded.

Communicating an Undefined Outcome

Undefined Result Reporting

When substitution produces an undefined result, this outcome should be stated explicitly as "undefined" for the given values, rather than reporting an arbitrary number, an error symbol without explanation, or silently skipping the evaluation, since correctly reporting undefined results is itself a necessary and meaningful part of a complete evaluation process.