Real Number Terminology

Date: 12/04/96 at 20:25:04
From: Philip Silver
Subject: Not existent over the reels 

What does it mean to be non-existent over the reals?  Can you explain 
this so I can explain it to my 7th grade daughter?  
Thank you. 

Date: 12/10/96 at 11:56:38
From: Doctor Rob
Subject: Re: Not existent over the reels 

This is a difficult idea for youngsters (or oldsters!) to grasp.

The real number system, sometimes called "the reals" for short, 
consists of: all the whole numbers (0, 7, -3, and so on), all the 
fractions (1/2, -22/7, and so on), and all the decimal fractions, 
even with infinitely many digits (0.72, 3.333333333..., -24.380783... 
and the like, including Pi = 3.14159265358979...).  They are called 
"real" numbers because they are used to measure real things like 
lengths, speeds, volumes, areas, forces, and so on.

Often we are faced with the task of finding a real number that is a
solution of an equation, such as the solution x of the equation
7*x - 13 = 8, or the solution A of the equation A^2 = 81.  Students 
are taught how to deal with these equations in algebra class.

Sometimes, however, there is no real number which works for a given
equation.  The usual example of this is x^2 = -1 (x^2 here means "x
squared" or x times x).  Why is there no real number x which works in
this equation?  It is because if x is positive, x^2 will be positive, 
but if x is negative, x^2 will still be positive, and if x is zero, 
x^2 will be zero.  This covers every possible real number, since each
one is either positive, negative, or zero.  So x^2 is positive or 
zero, but -1 is negative, so the two can never be equal.  

The phrase that is used to express this situation is that a solution x 
does not exist in the real number system (or "over the reals").

Sometimes, however, we really do want to have a solution to x^2 = -1
that we can work with.  This can be very useful in higher mathematics,
for reasons I can't go into here.  Since no real number will do, we
have to invent a new kind of number, called an "imaginary" number.  
This name was chosen just to contrast them with the real numbers.

Another case where there is no solution over the reals is an equation
like 2^x = -4, since powers of 2 are all positive.

The imaginary number which is the solution of x^2 = -1 is called "i",
and it is also called the square root of -1.  The creation of this new
kind of number allows us to expand our number system by looking at all
numbers of the form a + b*i, where a and b are real numbers, and i is
as above.  This system of numbers is called the complex number system.
The real numbers are part of this system, being the complex numbers 
for which b is zero, that is a + 0*i = a.  In the complex number 
system, every number has a square root, which isn't true in the real 
number system (as we saw above).

If you want to learn more about all the different types of real 
numbers, take a look at this Web page:

  http://mathforum.org/dr.math/problems/chris.8.16.96.html   

I hope this helps.  If I can explain further, please write back.

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