If z_0 and z_1 are real irrational numbers I write q=z0+z_1sqrt 1 Surely q is just a complex number. Under what condition will the number |q| be an integer ?

BenguelaktR

BenguelaktR

Answered question

2022-11-27

If z 0 and z 1 are real irrational numbers
q = z 0 + z 1 1
Surely q is just a complex number. Under what condition will the number | q | be an integer ?

Answer & Explanation

Lucas Contreras

Lucas Contreras

Beginner2022-11-28Added 11 answers

Suppose z 0 2 + z 1 2 is a natural number n. Then ( z 0 , z 1 ) corresponds to the following points on the unit circle: ( ± z 0 / n , ± z 1 / n ) and vice versa.
From here, the set of points ( z 0 , z 1 )consists of exactly all of the unit circle's points with both irrational coordinates scaled by its square, that is,
E = { n 2 q  q  S 1  where  q  has irrational coordinates and  n  is a positive natural number } .

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