Types of Numbers

Date: 10/30/98 at 10:54:13
From: Pegi Kustera
Subject: Types of Numbers

I am currently taking a introductory math education course. It's been 
quite a while since I've taken any math courses, and I am confused.

I don't understand the different types of numbers - counting, whole, 
rational, irrational, integers, prime, cardinal, natural, real, etc. 
I just can't seem to put it all in proper perspective. There are sets 
that are inclusive of some sets but exclusive of others. Is there are a 
set of numbers that is totally inclusive of all other sets? What is 
that set of numbers called? 

Date: 10/30/98 at 12:27:01
From: Doctor Rob
Subject: Re: Types of Numbers

Here is a diagram:

   Complex = (Algebraic U Transcendental)
     |            |                 |
     |            |                 |
   Algebraic    Transcendental    (Real U Imaginary)
     |                              |            |
     |                              |            |
     |        Real = (Rational U Irrational)   Imaginary
     |          |                        |
     |          |                        |
   Rational = (Fractional U Integer)   Irrational
     |              |
     |              |
   Fractional     Whole = Integer     Cardinal
                    |                   |
                    |                   |
           Natural = Counting = (Prime U Composite U {1})
                       |        |           |
                       |        |           |
                     Prime    Composite    {1}

Explanation: When a vertical line connects two sets, the upper one
properly contains the lower one.  A U B is the *disjoint* union of 
sets, so A and B are mutually exclusive.

There is no overriding set containing everything. The closest is the
Complex Numbers. The only numbers not contained in the Complex Numbers
are the infinite ones of the Cardinal Numbers.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   
    

Date: 05/08/2007 at 12:27:44
From: Bob
Subject: Re: Types of Numbers

Dr. Rob's identification of the integers and whole numbers isn't universally
accepted, nor does there appear to be universal agreement on the meanings of 
'natural', 'counting', and 'whole' as names of subsets of the integers. 

Those interested in alternative definitions should see Eric Weisstein's 
MathWorld entries on natural numbers, counting numbers, and whole numbers 
for a succinct and interesting discussion of this confusing state of affairs:

  http://mathworld.wolfram.com/NaturalNumber.html
  http://mathworld.wolfram.com/WholeNumber.html
  http://mathworld.wolfram.com/CountingNumber.html

I have looked at many web pages, and none gives a clearer discussion than 
his site.

-Bob