Find an example where the expression is rational, or give a proof of how it has to be irrational?

Annie French

Annie French

Answered question

2022-11-13

Find an example where the expression is rational, or give a proof of how it has to be irrational?

Answer & Explanation

Hailee West

Hailee West

Beginner2022-11-14Added 15 answers

Note that x + y + x y = ( 1 + x ) ( 1 + y ) 1. Hence, if x + y + x y is irrational, then so is ( 1 + x ) ( 1 + y ). Furthermore, x and y are assumed irrational, so 1 + x , 1 + y are already irrational. Hence, 1 + x , 1 + y are irrational numbers whose product is irrational.
This , as you know, need not happen all the time. For example, using the very common example ( 2 ) 2 = 2, one sees that x = y = 2 1 would do the job here. Indeed, x + y + x y = 1 in this case.

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