For example, NaNs can represent infinity, result of division by zero, missing value, or the square root of a negative (which is imaginary, whereas a floating-point number is real).

Practically speaking, if I divide two variables in a JavaScript program, the result may be NaN, which is predefined in JavaScript as "undefined". Hence this division may break the program. Now, if this computation was a small part of a much larger algorithm, it would be really painful to figure out where the error actually occurs. Fortunately, since the result will be NaN and I know my divisor may turn out to be 0, I can set up testing conditions that prevent any such computations in the first place or notify me of where they happen.