Math CentralQuandaries & Queries


Question from Anagh:

If x and y are rational numbers and
(x+y) + (x - 2y)√2 = 2x -y + (x-y-1) √5

Then find the value of x and y.


Let's rewrite this as

\[(x + y) - (2x - y) = (x - y - 1)\sqrt{5} - (x-2y) \sqrt{2}.\]

If $x$ and $y$ are rational then the left side of the above equation is rational and so are $x - y - 1$ and $x - 2y.$ But if $a$ and $b$ are rational then

\[a\sqrt{5} + b \sqrt{2}\]

is rational only if $a = b = 0.$


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