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Marshall wrote:
> On Mar 25, 1:30 pm, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
>
>>Marshall wrote: >> >> >>>So what if we had an internal representation for >>>integer similar to java.math.BigInteger, and an >>>internal representation for rational that was a pair >>>of integers. We can define *exact* operators for >>>these types for basic arithmetic functions. >> >>I disagree. Unless one has infinite precision, rational is >>always an approximation.
Precise for integers but rationals are not integers.
> In my mind, there is a significant difference between
> operations that will either produce precise and exact
> answers or else fail, having consumed all the machine's
> resources, and operations that will return results that
> are only guaranteed up to some degree of precision.
Do you prefer square root to fail having consumed all resources?
> Now admittedly the integers limited to four billion digits
> is not the whole of the integers, but it's a big step up
> from 32 bit integers. In fact, I will make a perhaps
> reckless claim and assert it is enough for all possible
> future applications.
However, that is not relevant to my point regarding rationals.
For example, supposing we might
> one day need a computer to operate on the number
> of hydrogen atoms in the universe, one hundred digits
> is more than enough. This leads me to suspect that
> one billion digits is at least ten million times more
> digits than anyone will ever need.
I still don't see the point in what you wrote. A billion digit integer doesn't make a whit of difference to what I wrote earlier: One still has a maximum representable integer, and a maximum representable integer defines a minimum range for the representation of each value.
>>>This differs from the cartesian/polar case, in that >>>every integer values is *exactly* representable as >>>a rational. >> >>But that doesn't change the fact that a rational is always a small >>interval in the computational model. Thus, when treated as an integer, 1 >>is 1. When treated as a rational, 1 is some subrange within >>[(maxint-1)/maxint,maxint/(maxint-1)] or perhaps within >>[maxnegint/(maxnegint+1),(maxnegint+1)/maxnegint]
Nevertheless, that is the best we can do. If we want to pretend something is a rational, we must acknowledge that it necessarily stands for any rational within some range regardless how large or small the range.
If you have a rational one and a rational three
> and you divide, you get the exact rational one-third.
Except that the represention of 1 actually represents a range of rational values and the representation of 3 actually represents a range of rational values before we even do the divide.
> It is more useful to say that the operators, rather
> than the operands, are approximations. Floating
> point operations return approximate results in
> the worst case, but may return precise results in
> the easier cases.
Except that the operands themselves are approximations representing ranges. When the rational add operation adds 1 to 1, it doesn't know whether 1 is 1 or 1+epsilon/2 or 1-epsilon/5 etc. It uses the same representation for all three and for an infinite number of other rationals.
For example, 80 bit floating point
> add, subtract, and multiply on numbers with less than
> 32 bits of mantissa are always exact. Floating point
> sqrt(2) is an approximation; sqrt(4) is not.
Except that 4 itself is an approximation covering the range [4-epsilon,4+epsilon] and the result of the operation represents the range [2-epsilon,2+epsilon].
>>>It also occurs to me that an integer is a rational >>>where the denominator = 1. This starts to sound >>>like specialization by constraint. >> >>You assume one always normalizes the rational. An integer is a rational >>where the numerator modulo the denominator is always zero, which is >>trivially true when the denominator is 1.
Physical independence means allowing either without affecting the logical model. Normalization of rationals has a cost, and performance requirements may not tolerate that cost or may require it.
>>>Crazier still, the fact that some rationals are integers >>>and some aren't starts to remind me of parsing a >>>string as a number. A possrep for an integer could >>>be a sequence of ascii digit characters. Then >>>constructing (selecting) an integer from a string >>>(succeeding only if the string represents an integer) >>>is isomorphic to parsing the string into a number, >>>a la parseInt or atoi or whatever. Integer isn't a >>>subset of String, but the Integer possrep sequence- >>>of-digit-chars *is* a subset of the String possrep >>>sequence-of-chars. >> >>An optional sign and a string of digits is a possible representation for >>an integer.
I thought I was agreeing not correcting. Were you objecting to an optional sign and a string of digits as a possible representation for an integer?
>>>Anyway, this post is long and confusing enough >>>for now. I think I'll go reread the relevant sections >>>of TTM now. Again. I really must buy the 3rd ed. >> >>What did you find out?
No biggie. Now that you have had it off the shelf for a while longer, what did you find out? Received on Sun Mar 25 2007 - 21:33:56 CDT