Undefined — Definition, Formula & Examples

Undefined means a mathematical expression has no valid or meaningful value. The most common example is division by zero — any number divided by zero is undefined because no answer exists that makes the equation true.

A mathematical expression is undefined when it does not correspond to any value within the relevant number system. An operation is undefined at a point when no consistent result can be assigned to it without violating the axioms of arithmetic. For instance, $a \div 0$ is undefined for all real numbers $a$ because there is no real number $q$ such that $0 \times q = a$ (when $a \neq 0$ ).

How It Works

When you see a fraction or division problem, check the denominator. If the denominator equals zero, the expression is undefined. In algebra, this often happens with variables: the expression

\frac{5}{x - 3}

is undefined when

x = 3

because the denominator becomes zero. Identifying undefined values is essential when stating the domain of a function — you must exclude any input that makes the expression undefined.

Worked Example

Problem: For what value of x is the expression 10 / (x + 2) undefined?

Set the denominator equal to zero: An expression is undefined when its denominator is zero.

x + 2 = 0

Solve for x: Subtract 2 from both sides.

x = -2

Verify: Substituting gives 10 / 0, which has no valid result.

\frac{10}{-2 + 2} = \frac{10}{0} \; \text{(undefined)}

Answer: The expression is undefined when x = −2.

Why It Matters

Recognizing undefined expressions is critical when you graph rational functions — vertical asymptotes occur exactly where the function is undefined. In any science or engineering formula, plugging in a value that makes a denominator zero signals a physical limit or boundary condition that needs special attention.

Common Mistakes

Mistake: Saying that a number divided by zero equals zero or infinity.

Correction: Division by zero is not zero and is not infinity — it is simply undefined. No real number satisfies the equation 0 × q = a (for nonzero a), so no answer can be assigned.