Types of Real Numbers

Date: 8/16/96 at 21:1:48
From: Louie M
Subject: The Types of Real Numbers

I would like to know all the types of real numbers.
Thanks!
Chris

Date: 8/30/96 at 16:56:13
From: Doctor James
Subject: Re: The Types of Real Numbers

There are many, many types of real numbers. I'll tell you about a few
here.

The most general division of the real numbers is the division between
the rational numbers and the irrational numbers. A rational number is
defined to be any number that can be expressed as p/q, where p and q
are both integers (we'll come to them in a second) and q does not 
equal 0. You can recognize a rational number because it either has 
only a finite number of decimal places (or no decimal places at all!), 
or the decimal places repeat themselves. That is, the following are 
all rational numbers:

      348100
      78.33954001
      3.333333333...
      .0000123232323232323...

But you can make some numbers that don't follow those rules. That is,
you could have a number like 1.01001000100001000001... which goes on 
forever, but never repeats itself. These are called irrational 
numbers, and most aren't that pretty. Two of the more famous ones are

     pi = 3.14159...
     e  = 2.718...

You can subdivide the rationals further, in many ways. Here are some 
ways: 

     negative numbers, 0, and positive numbers
     integers and non-integers.  

Integers are, roughly speaking, numbers with no decimal points, like 
4 or 10 or -3788882 or 0. I say roughly speaking, because 9/9 is an 
integer, and so is 3.00. In these cases, it's important to be equal to 
something without a decimal point. Non-integers are everything else 
(irrational numbers are also non-integers).

Then there are whole numbers and natural numbers (also called counting
numbers). Natural numbers are positive integers, like 1, 2, 3, 4... 
Whole numbers are natural numbers and 0.

There are also something called imaginary or complex numbers, which 
are outside of the real numbers. These come about when you ask 
questions like, 'what is the square root of -4?' The fundamental unit 
of the imaginary numbers is i, which is defined to be the number such 
that i^2 = -1. Thus, the square root of -4 is 2i (can you see why?). 
Imaginary numbers are a very interesting subject, which if you keep on 
in math you'll learn about soon enough.

Hope this helps!

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

PS. In a branch of math called number theory you learn a lot about 
various types of numbers, such as prime numbers and perfect numbers 
and square numbers and triangular numbers...