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From: c60b-4ah@e260-2b.berkeley.edu (Phantom)
Newsgroups: comp.sys.mac.programmer
Subject: Re: Why can't the Mac add?
Message-ID: <1990Sep24.014125.25065@agate.berkeley.edu>
Date: 24 Sep 90 01:41:25 GMT
References: <45060@apple.Apple.COM> <1990Sep24.010249.16647@agate.berkeley.edu> <8614@jarthur.Claremont.EDU>
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(As a result of some distraction, I have accidentally 
 mismanipulated the cardinality of my last  posting.  
 I hope that that has been corrected and the old article has been engulfed
 by oblivion.  The following is the normalized version.  I sincerely apologize 
 for the anarchy it has caused in the set of all sets.)

With all respect, I was somewhat surprised to learn that some person 
from Apple should not know this.  I shall refer anyone who is intrigued by 
this to the SANE manual published by Addison-Wesley (Someone in Apple wrote 
it!)  But I will discuss it briefly here.

No matter what base is chosen to represent even ration number, there are 
countablely infinite fractional rational numbers that cannot be represented 
in finite digits.  Suppose the base used is N, which is divisible by the 
prime numbers p1, p2, p3, ... pk.  Suppose further that fractional part of 
the number you want to represent (call it X) in that system can be written 
in the form

				Q
			---------------------       
				R			,
where both Q and R are integers, whose Greatest Common Divider is 1.  Let 
s1, s2, ... sl be all the prime facotors of R, and t1, t2, ... tm be all the 
prime factors of N, then the necessary and sufficient condition for X to be 
representable in finit number of digits in base N 
is that the set { s1, s2, ... sl } is a subset of 
the set { t1, t2, ... tm }.  In you example, the number 0.2=1/5 cannot be 
exactly represented in a finite sequence of 0's and 1's because 2 <> 5.  
In fact, 0.5 in binary notation is 0.0011 0011 0011 0011 0011 0011 0011 
0011 0011 0011 0011 0011 0011 0011 ...