## 3.3 REAL NUMBERS

As regards the real numbers, there are two types of approximations: *fixed-point* and *floating-point* numeration systems. The fixed-point system is a simple extension of the integer representation system; it allows the representation of a relatively *reduced range* of numbers with some constant *absolute precision*. The floating point system allows the representation of a very large range of numbers, with some constant *relative precision*.

**Definitions 3.9**

- In a
*fixed-point numeration system*, the number represented in the formis

*x*/*B*, where^{p}*x*is the integer represented by the same sequence of digits without point. - Let
*x*_{min}and*x*_{max}be the minimum and maximum integers that can be represented with*n*digits, that is, x_{min}= 1 −*B*^{n−1}and*x*_{max}=*B*^{n−1}− 1 in sign-magnitude representation, and*x*_{min}= −*B*2 and^{n}/*x*_{max}=*B*2 − 1 in^{n}/*B*'s complement or excess-*B*/2 representation. Then, any real number^{n}*x*belonging to the intervalcan be represented in the form (3.21) with some

*error*equal to the absolute value of the difference between*x*and its representation. - The
*distance d*between exactly represented numbers is equal to the*unit in the least significant position*(*ulp*), that is,*B*, so that the^{−p}*maximum error*is equal to - The
*maximum relative error*is equal to then so that the maximum ...

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