Question

[The Blade Dancer]

How does this highlighted part work out? What does 100 have to do with anything?

[debbie]

@The-Blade-Dancer Thank you for writing and for being willing to share your comments. Looking back on this explanation, I sort of want to smack my head ... You're right, it is a bit obtuse. (Okay not a bit, but very!)

The question is asking whether $a + b\sqrt{c} = x + y + 2\sqrt{xy}$ for integers $a, b,$ and $c$ implies that $x$ and $y$ are always rational. (Rational means that the number is of the form $\frac{m}{n}$ where $m$ and $n$ are both integers.) Hmmm... it seems like this statement can't be true, just from using some common sense, because how can we be so lucky as to always get $x$ and $y$ rational... but how do we prove this?

Luckily, with a question of the form "It's always true that.... ", all we have to do is find a single counterexample where it isn't true.

I've updated the explanation to hopefully be a bit more simple and easy to understand. The counterexample I'm using is a case where $x$ and $y$ are both imaginary numbers but $a,$ $b,$ and $c$ are integers.