In K-12 mathematics you have encountered several number systems. Since naming conventions for some of them are not agreed on and for others much more care is taken in higher level mathematics it is desirable to specify standard names for certain sets of numbers:
Note that these are sometimes called whole numbers (when the naturals are thought of as starting at 1). To a certain extent you can tell whether the mathematician you are talking to is a logician or an analyst by asking if 0 is a natural number. I'm a logician, so I use that convention. Restricting to the positive natural numbers is done by
are obtained by closing the natural numbers under additive inverse. The use of a Z comes from the German term Zahlen for this set.
are those which can be written as a quotient 
with 
and (p,q)=1. The Q comes from thinking of rationals as those which can be thought of as quotients. This system results when we close 
under multiplicative inverses. All our exact computation is carried out in 
with reasonably small denominators.
result when we fill in all of the holes in the line left when we try to use rationals to measure distances. They also can be represented as arbitrary decimals. Approximation is very important when using reals, since many of them cannot be exactly specified in a finite number of symbols.
become important when we want to solve algebraic equations. Complex numbers can be represented in the form a+bi where 
and 
.
Notice that 
and that in each case the inclusion is proper (it cannot be reversed).