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/
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