Integers and Rational Numbers

Date: 25 May 1995 20:04:26 -0400
From: Juan Sepulveda - Edwards y Sepulveda Ltda.
Subject: Integers and Rational Numbers/Set Theory

Can you explain the difference between SET Q and SET Z ?
This question was asked at school of a 14 year old boy.

Thanks

Date: 26 May 1995 15:45:58 -0400
From: Dr. Math
Subject: Integers and Rational Numbers/Set Theory

Hello there!

I assume that you mean by Q and Z the set of rational numbers Q and the set
of integers Z.  If I'm wrong, let me know.  Anyway, if you look at the set Q
of all rational numbers, all the numbers will be of the form m/n, where m
and n are integers and n isn't zero.  So it will contain the numbers 4/5, -8,
1.75 (which is 9/4), -97/3, and so on.  

Note that Q contains all the numbers ... -2/1, -1/1, 0/1, 1/1, 2/1, 3/1, ..., 

so it contains all the integers.  That means that there is a copy of the set
Z (the integers) inside of Q.  When this happens, we say that "Q contains Z"
or "Z is a subset of Q."  You can also say that "Q is a superset of Z,"
which means the same thing, but it's less common to say it that way.  Mostly
people talk about subsets, not supersets.  

Also, note that there are numbers in Q that aren't in Z.  When this happens,
we say that "Z is a proper subset of Q."  That rules out the possibility of
them actually being the same set.

If you've got any more specific questions about this, we'd be glad to take a
look at them!  Thanks.

-Dr. Ken
 Check out our web site!  http://mathforum.org/dr.math/