The paper is intended to show that floating-point arithmetic can be implemented in a way which leads to reasonable mathematical structures as described in chapters 5 and 6. It turns out for instance that all the rules of the minus-operator of the real numbers can be saved and that with respect to ≦ and ≧ inequalities can be manipulated as if they were real inequalities. These structures also occur in other fields of mathematics. They allow among others many theoretical considerations with floating-point arithmetics. Theorem 5.1 is the main result for the implementation. It reduces the structures to special properties of the rounding function. In chapter 3 these properties are derived as necessary conditions for an algebraic and order homomorphism between the real numbers and a floating-point system.