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Is there a standard notation to handle the effective representation of some real number $x$ on a finite machine ? I have in mind some kind of braces, but I am not sure it is appropriate. Let me try to precise my question. Fractional part function is handled with $\{\cdot\}$, the floor function with $\lfloor\cdot\rfloor$, the ceil function with $\lceil\cdot\rceil$, and so forth. I am looking something similar for effective representations.

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  • $\begingroup$ Are you asking about notations for a particular representation, or are you interested in different representations? $\endgroup$ – Chris Culter Aug 14 '15 at 0:58
  • $\begingroup$ Have you tried caunchy sequences? $\endgroup$ – PyRulez Aug 14 '15 at 2:58
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Scientific notation in base-2 is probably most appropriate for a standard computer, because it is exact with respect to how a modern computer natively represents a floating point number. For example, you could have 4 represented as $1.00 \ldots 0 \, \times 2^2$ in binary scientific notation . Here your mantissa (everything after the 1 in the first part of the expression ) will be some fixed number of binary (base-2) digits, depending on what the floating point convention dictates for the number of mantissa bits. The exponential you multiply this number between $1$ and $2$ by will always be $2^k$ for some integer $k$, and the range of possible $k$ will depend on how many bits are used for the exponent in your floating point convention (system/machine dependent).

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  • $\begingroup$ That's only done of the rational numbers. Almost no reals are rational. $\endgroup$ – PyRulez Aug 14 '15 at 2:59

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