Question

prove that if x is irrational , then 1/x is irrational ​

Answer 1

Answer:

Let

x+1x−1=r∈Q

Thus, r≠1,

x=r+1r−1,

which is a contradiction.

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answered

Oct 21 '18 at 11:35

Michael Rozenberg

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Oct 21 '18 at 11:37

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If y is irrational then y+q,q⋅y,qy are irrational, if q∈Q∖{0}.

We have

x+1x−1=(x−1)+2x−1=1+2x−1.

From x irrational follows x−1 irrational, therefore is 2x−1 irrational and so 2x−1+1 is irrational.