Real and Other Numbers

Date: 08/29/2001 at 16:38:17
From: Nick Bogan
Subject: Real numbers

What is a real number? What are the different kinds of real numbers?
Can you tell me about each kind of real number? Are there any more 
numbers besides real numbers?

Date: 08/30/2001 at 10:35:07
From: Doctor Ian
Subject: Re: Real numbers

Hi Nick,

Real numbers include: 

  counting numbers (1, 2, 3, ...)
  whole numbers (0, 1, 2, 3, ...)
  integers (..., -2, -1, 0, 1, 2, ...)
  rational numbers (any integer divided by any non-zero integer)
  irrational numbers (any real number that isn't rational)

Note that every counting number is also a whole number; every whole 
number is also an integer; and every integer is also a rational 
number.  

One way to think about the real numbers is this: Imagine a line that 
extends in each direction forever, with two marks on it:

      ----------------|---|------------------
                      0   1

What we'd like to do is figure out how to get from 0 to any other 
point on the line, using a 'yardstick' that is the same length as the 
distance between these two marks. 

One obvious way is to start laying the yardstick down and moving to 
the end of it:

      ----------------|---|---|---|---|------
                      0   1   2   3   4

We can use this method to get to all of the whole numbers. If we move 
in both directions, 

      --------|---|---|---|---|---|---|------
             -2  -1   0   1   2   3   4

we can get to all of the integers. To get to the rational numbers, we 
have to be a little more clever, by introducing the notion of 
'division'.  

Basically, division works this way. Suppose we use our yardstick to 
make a bigger yardstick, whose length is some integer. (We can do this 
by using our original yardstick to move to any integer, and then 
making a new yardstick that extends from 0 to there.)

    0                      5
    +----------------------+
    |                      |
    +----------------------+

Now suppose we fold this yardstick into halves, or thirds, or sixths:

    0                       5
    +-----------------------+
    |   |   |   |   |   |   |
    +-----------------------+
        ^
        |
    1/6 of 5 = 5/6

If we cut it at one of the folds, we get a new yardstick, which we can 
use to move to a new set of locations:  5/6, 10/6, 15/6, and so on.

The places we can get to using yardsticks like this are called 
rational numbers, because we use ratios of integers to construct our 
yardsticks. 

So now we arrive at the question: Can we get to _every_ location on 
the line in this way? For example, consider the number pi, which is 
the ratio of the circumference of any circle to its diameter. Can we 
get to pi using a yardstick constructed this way?

The answer is no, although we can get as close as we want. Since there 
is no way to get to pi using rational yardsticks, we say that pi is 
'not rational', or 'irrational'.  

There are lots of other numbers like this - for example, the square 
root of any number that isn't a perfect square is irrational. Much of 
advanced mathematics consists of finding new ways to 'move' to these 
irrational numbers. 

To answer your final question, there _are_ other kinds of numbers, 
including complex (or 'imaginary') numbers, 

  http://mathforum.org/dr.math/faq/faq.imag.num.html   

hyper-real numbers, 

  http://mathforum.org/dr.math/faq/analysis_hyperreals.html     

and transfinite numbers,

  http://mathforum.org/dr.math/faq/faq.large.numbers.html   

There is quite a lot more that can be said about the real numbers, but 
to cover it all would require me to write a book, and of course there 
are already lots of books like that available. But please feel free to 
write back with specific questions that you'd like answered.  

I hope this helps. 

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