Real NumbersDate: 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/ |
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