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Let $I\textit{I \kern-.55em R}$ be the set of closed and bounded intervals of real numbers. Arithmetic in $I\textit{I \kern-.55em R}$ can be defined via the power set $\textit{I \kern-.54em P}\textit{I \kern-.55em R}$ of real numbers. If divisors containing zero are excluded, arithmetic in $I\textit{I \kern-.55em R}$ is an algebraically closed subset of the arithmetic in $\textit{I \kern-.54em P}\textit{I \kern-.55em R}$, i.e., an operation in $I\textit{I \kern-.55em R}$ performed in $\textit{I \kern-.54em P}\textit{I \kern-.55em R}$ gives a result that is in $I\textit{I \kern-.55em R}$. Arithmetic in $\textit{I \kern-.54em P}\textit{I \kern-.55em R}$ also allows division by an interval that contains zero. Such division results in closed intervals of real numbers which, however, are no longer bounded. The union of the set $I\textit{I \kern-.55em R}$ with these new intervals is denoted by $(I\textit{I \kern-.55em R})$. This paper shows that arithmetic operations can be extended to all elements of the set $(I\textit{I \kern-.55em R})$. Let $F \subset \textit{I \kern-.55em R}$ denote the set of floating-point numbers. On the computer, arithmetic in $(I\textit{I \kern-.55em R})$ is approximated by arithmetic in the subset (IF ) of closed intervals with floating-point bounds. The usual exceptions of floating-point arithmetic like underflow, overflow, division by zero, or invalid operation do not occur in (IF ).