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### Closed Operations for Negative Irrationals

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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?

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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).

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Imaginary/Complex Numbers
High School Number Theory

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