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Name of Property of Fractions not Held by Integers

 
 
Reply Fri 3 Jun, 2011 03:58 pm
Integers and fractions have the same cardinality (countably infinite) but are different
in that between any two distinct fractions is another fraction (actually an infinite
number of them) but this property does not hold for integers. I am looking for a
mathematical term that captures this property of fractions that integers do not have.
Note that this property also holds for the set of strings (lexicographically sorted);
that is there are an infinite number of strings between the strings 'abc' and 'defg'.
We could also say this property holds for reals though, of course, reals are
uncountably infinite.
Another example having this property would be decimal numbers.

Regards and thanks

Ralph Boland

P.S. In case anyone is interested I need this term to document a computer
data structure I am implementing which allows ranges of values to be stored.
It will be open sourced of course.
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raprap
 
  1  
Reply Fri 3 Jun, 2011 04:42 pm
@RalphBoland,
Not fractions, Rational numbers-all rational numbers can be expressed as a ratio of integers-consequently integers are rational numbers.

Let p and q be any two integers, any rational number then can be expressed as p/q where q<>0. If p/q is an integer then p=n*q, if not then p=n*q+s where s is an integer <q --this is the basis for Euclid's algorithm for finging the greatest common divisor of any two integers http://en.wikipedia.org/wiki/Euclidean_algorithm.

For a better discussion than I'm able to do here I would recommend a picking up an elementary (sophomore or junior year) college text on discrete mathematics. http://en.wikipedia.org/wiki/Discrete_mathematics or enter discrete mathematics into Planet Math http://planetmath.org/.

BTW Discrete Math also includes Combinations and Permutations which would give some indication on the number of vaiations possible with a string of n characters.

Rap
raprap
 
  1  
Reply Fri 3 Jun, 2011 06:35 pm
@raprap,
Consider the relation

p=n*q+s where p,n,q,s are all integers, q<>0, and s<q

Then any rational number p/q can be expressed as n+s/q. An integer would be when s=o.

Rap
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Oylok
 
  1  
Reply Fri 3 Jun, 2011 09:22 pm
@RalphBoland,
RalphBoland wrote:

Integers and fractions have the same cardinality (countably infinite) but are different
in that between any two distinct fractions is another fraction (actually an infinite
number of them) but this property does not hold for integers. I am looking for a
mathematical term that captures this property of fractions that integers do not have.


We say that the order relation "<" is dense on the rationals, but not on the integers.

http://en.wikipedia.org/wiki/Dense_order
http://en.wikipedia.org/wiki/Dense_relation

So, I guess you could say that the rationals are dense as a partially ordered set under the ordinary ordering "<", whereas the integers are not.

(Some readers may be more familiar with the idea that a metric space can be a "dense" subset of another metric space. For example, the rationals are a "dense" subset of the reals, whereas the integers are not. Similar concept. Slightly different use of the same word.)

Quote:
Note that this property also holds for the set of strings (lexicographically sorted);
that is there are an infinite number of strings between the strings 'abc' and 'defg'.


In other words, the set of all lexicographically sorted character-strings is dense as a partially ordered set.
raprap
 
  1  
Reply Sat 4 Jun, 2011 12:10 pm
@Oylok,
I've always been curious about this relation between rationals and real numbers--you'll have to forgive me because I'm not formally a mathematician just a simple engineer that considers math to be the biggest hammer in my tool box--but from my understanding there are an infinite number of irrational numbers between any two rational ones (Cantor's dust if my poor memory serves me) so this term 'dense order and dense set' seem to be somewhat subjective. Yes, there are an infinite number of rationals between any two integers, and integers and rationals are the same cardinality (there exists a function and inverse that provides a one to one mapping ) but there still remains a lot of numbers between each rational number and the next one. That (to me) means that density is a relative term that only provides a qualitative ranking.

Is my conjecture correct?

Rap
Oylok
 
  1  
Reply Sat 18 Jun, 2011 10:34 am
@raprap,
raprap wrote:

so this term 'dense order and dense set' seem to be somewhat subjective.


Well, I don't know that we can read too much into what mathematicians decided to call things. The people who wrote the Math books may have just as well called the rationals "inwardly bountiful", and in light of your remarks about Cantor's dust, that would probably have been a better phrase than "dense". In any case, you've demonstrated that you know as much as I do about this branch of Math, so, um, "job done" I suppose? Smile
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