edwards@houxa.UUCP (D.LEWAN) (02/06/86)
I am currently considering the real numbers, some of their constructive aspects and specific uses, and different encodings for them for digital computers. This article is a request for encodings that may not be well known as well (but are indeed useful), as particular uses for those encodings. Such a use is probably one for which 1. that encoding is convenient and 2. another encoding would be difficult to use. A few philosophical notes should be made before I go too far. The act of encoding is that of assigning, to a *meaningful* object, a name from which the object (or its meaning, anyway) can be recreated. For example, to the object "fifteen" we may assign the name "15". Now, from "15" we can reconstruct "fifteen" by interpreting it as "10 + 5", and the objects named by "10", "+" and "5" are all well defined and understood. I cannot name all the real numbers since there are only a countable number of names available, and there are an uncountable number of reals. But I can select subsets of the reals tailored to my particular needs and for some of those subsets I can find worthwhile encodings. The standard exponent/base approach to real number representation is a style that in some sense has the best and worst of all worlds. Its precision is generally good enough for most arithmetic applications, but absolute precision waxes for large numbers, where the type integer might in fact be more precise. It also doesn't guarantee, for example, that n-th roots, when raised to the n-th power, yield the original numbers, i.e. that n ---- n \/ q = q in all cases. (Imagine moving your PC's best approximation of the square root of 2 to a CRAY II and then performing arithmetic.) There are representations that do preserve such algebraic properties, unfortunately they lose meaningful precision. If one only knows that the square of a certain number always yields 2 one does not necessarily know if it is greater than or less than 0. (High school math conventions do no good here.) Now, I know that preservation of algebraic properties is just one of many that might be of interest to a particular application. I am sending this plea to the net to find a wealth of applications that might run better with less well known encodings for specific real numbers. Great thanks go to anyone who can help out. Doug (...!ihnp4!houxa!edwards)
mac@uvacs.UUCP (Alex Colvin) (02/11/86)
> request for encodings that may not be well known as well
Continued fractions are a number system without unique representation,
allowing arithmetic to be pipelined digit by digit. Results can be
computed to the required precision.
There is a description of this by R. W. Gosper in MIT AI memo 239,
item#101. He refers to Knuth, V. 2, p. 316.