perplexus.info :: Numbers : Sufficient or not?

Sufficient or not? (Posted on 2018-08-29)

M & N are distinct real numbers.
M+N is a rational number.
M^2+N^2 is a rational number.
M*N is rational as well.

Provided all the above is true must each of M, N be rational?

No Solution Yet Submitted by Ady TZIDON

Rating: 5.0000 (1 votes)

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Proof (spoiler)

| Comment 1 of 6

No, it is not sufficient.

Let M = sqrt(2), N = -sqrt(2)

M+N = 0

M^2 + N^2 = 4

M*N = -2

More generally, (a + sqrt(b)) and (a-sqrt(b)) are also counterexamples if a and b are rational and sqrt(b) is not

Posted by Steve Herman on 2018-08-29 08:04:14

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