The Real Number System in a Venn Diagram

Date: 09/08/98 at 22:53:02
From: Amy
Subject: The real number system

My PreCal teacher gave us an assignment on the first day to construct 
a Venn diagram that illustrates the real number system. I have no 
idea where to begin and was hoping you could at least identify all the 
parts of the real number system for me and how they relate to each 
other so I can at least have some reference.  

Thank you.

Date: 09/08/98 at 23:17:47
From: Doctor Margaret
Subject: Re: The real number system

Hi Amy,

Thanks for writing to Dr. Math. I'll give you an explanation of the 
real numbers first:

Real numbers are in a set made up of all different kinds of numbers.

First we have the natural numbers, or counting numbers, such as:

   1, 2, 3, ...

Then we have numbers where addition and multiplication will always 
result in a natural number, but these numbers alone are not sufficient 
for subtraction, as in the case of 3 - 9, or 1 - 1. So we have to 
add zero and negative whole numbers to our collection, and they are 
called integers:

   ..., -3, -2, -1, 0, 1, 2, 3, ...

Now we have enough numbers for addition, multiplication, and 
subtraction, but not for division, so we need other numbers called 
ratios of integers, better known as fractions:

   1/2, 2/3, 34/37, etc.

All these numbers together, the natural numbers, integers, and ratios 
of integers are known as rational numbers. 

Rational numbers generally produce other rational numbers when they are 
added, multiplied, subtracted, or divided, but not always. For example, 
in the equation x^2 - 2 = 0, when we solve for x, we get x^2 = sqrt 2. 
This is irrational because it doesn't work out to something neat 
like x^2 = sqrt 9, which is x = 3. Sqrt 2 is a non-repeating, 
non-terminating decimal. So are numbers like Pi and e. We call all 
numbers of this type irrational.

So real numbers are made up of all rational and irrational numbers. 
No number can be both rational and irrational, so you will have at 
least two separate circles for your diagram. There is a really good 
description of how to do a Venn diagram at:

   Venn Diagrams
   http://mathforum.org/library/drmath/view/52420.html   

Good luck!

- Doctor Margaret, The Math Forum
Check out our web site! http://mathforum.org/dr.math/