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Carly Cannon

Carly Cannon

Answered question

2022-07-04

For every irrational number i 1 [ 0 , 1 ], there is only one irrational number i 2 [ 0 , 1 ] such that i 1 i 2 Q ?

Answer & Explanation

Mateo Carson

Mateo Carson

Beginner2022-07-05Added 15 answers

No, the statement you are trying to prove is false. If i 1 is irrational, then i 1 ( i 1 + q ) is rational for any rational q, while i 1 + q is irrational.

Now for any i 1 [ 0 , 1 ], we may find infinitely many q Q such that i 1 + q is also in [ 0 , 1 ] (exercise); so in fact there are infinitely many i 2 for every i 1 .

Indeed, of the uncountably many -equivalence classes, most (= all but countably many) contain only irrationals.

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