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Real Numbers


Date: 08/08/97 at 14:36:30
From: Hari Seldon
Subject: Real numbers

What exactly is a real number? When I first started algebra, I used 
them all the time, because they were the only numbers I knew. Then I 
learned about complex numbers, and how they could be used in various 
ways. Then I learned about things like "Complex conjugates" and other 
things which basically break down a complex number Z into the sum of a 
real and an imaginary, x + y*i, where x and y are real. But What Is a 
Real Number?  I've never seen a definition of it.  Am I supposed to 
assume that it's an intuitive idea, or is it something that can be 
defined? 


Date: 08/14/97 at 16:12:05
From: Doctor Rob
Subject: Re: Real numbers

See   http://www.shu.edu/~wachsmut/reals/history/     and keep pressing 
"Next Chapter."

This will tell you how to define the real numbers formally in several
steps:

1. Start with the Peano Axioms to define the Natural Numbers, and
   define addition, subtraction, multiplication, and division of
   natural numbers.
2. Define the Integers as equivalence classes of pairs of natural
   numbers.  [a-b <--> (a,b) ~ (a+c,b+c) for all natural a,b,c.]
3. Define the Rational Numbers as equivalence classes of pairs of
   integers.  [a/b <--> (a,b) ~ (a*c,b*c) for all integer a,b,c, b 
   and c nonzero.]
4. Define the Real Numbers as equivalence classes of Cauchy sequences
   of rational numbers.  [lim a(n) <--> {a(n)} ~ {a(n)+b(n)} for all
   rational Cauchy sequences a and b such that b(n) approaches 0 as 
   n gets large.]
5. Define the Complex Numbers as pairs of real numbers.
   [a+b*i <--> (a,b) for all real a,b.]

At each step, one has to verify that the previous set can be naturally
imbedded in the new one, that arithmetic operations can be defined,
and that they are consistent with their definitions in the previous
set.

Of course this is the formal definition.  We virtually never compute
with equivalence classes of Cauchy sequences of rational numbers!
Instead we compute with rational approximations (such as 31416/10000
for Pi) or decimal expansions (such as 3.1416 for Pi) or symbolic
expressions (such as Sqrt[31] or e^2 or log_10(7)) for irrational
numbers. If we were pressed, however, we would be able to produce
several equivalent Cauchy sequences for each of these irrational
numbers.

Because the functional definition just preceding differs from the
formal definition, you probably have not seen the formal one.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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