A mathematical expression is said to be undefined if it is not possible to represent it with the number set being used. Consider two types of most common undefined expressions.

Division by Zero

Expressions similar to $\frac{a}{0},$ which show $a$ being divided by $0,$ are said to be undefined. This is because there exists no number $x$ (real or otherwise) for which the following equation is true assuming $a\ne 0.$ Rational expressions are also undefined for values that cause division by zero. For example, this rational expression is undefined when $x=5$, because the denominator then becomes $0.$

The Root of a Negative Number

Another common type of an undefined expression is the even root of a negative number. Such expressions are not defined when calculating using real numbers. This is because the function $f(x)=\sqrt{x}$ does not take any values for $x$ less than $0.$

However, note that imaginary numbers allow us to take even roots of negative numbers.