Closed Operations for Negative Irrationals

Date: 04/28/2001 at 13:44:23
From: Lisa
Subject: Negative Irrational Numbers

In my Algebra 1 class we were discussing negative irrational numbers 
and what the set of closed operations was for them. Our book said 
there are none, but we don't understand why addition isn't closed.

So the question is, what set of operations is closed under negative 
irrational numbers?

Thank you for your time.

Date: 04/28/2001 at 14:29:15
From: Doctor Douglas
Subject: Re: Negative Irrational Numbers

Hi Lisa, and thanks for writing.

The set of negative irrationals is not closed under any of the usual 
elementary operations (+,-,*,/). For example, let's take for granted 
that sqrt(2) is irrational, and that -5-sqrt(2) and -6-sqrt(2) and 
-6+sqrt(2) also irrational. The proof that these last three numbers 
are indeed irrational involves a simple "proof by contradiction."

Then we see that:

     [-5-sqrt(2)] + [-6+sqrt(2)] = -11, which is not irrational

     [-5-sqrt(2)] - [-6-sqrt(2)] = +1, neither irrational nor negative

     [-sqrt(2)] * [-sqrt(2)] = +2, neither irrational nor negative

     [-sqrt(2)] / [-sqrt(2)] = +1, neither irrational nor negative

So in the addition case, we can find two negative irrationals whose 
sum is rational (even though it is negative).

I hope this answers your question. Please write back if you have 
further questions about this.

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