"floating point arithmetic" is a terminology that refer to the arithmetic perform over (finite) representation of real number. See the wikipedia article for more details.

In the formal specification of floating point arithmetic (that should be used by all the major programming languages), it is specified that a "Not A number" (NaN) value should be a number.

If we abstract the fact that floating point arithmetic care only of finitely many values and abstract a bit, we get a sort of arithmetic arithmetic over $\mathbb{R}\cup\{\textrm{NaN}\}$ with $\textrm{NaN}$ being a zero for any arithmetic operation. Formally, for all $x$,
$$x\times \textrm{NaN} = x + \textrm{NaN} = \frac{x}{0} = \frac{0}{0}= \cdots =\textrm{NaN}$$

Is there any algebraic structures $(E,+,\times)$ having axioms allowing this or is it just to arbitrary to have been introduced even in weird part of algebra?