# Damn irrationals

You have the 3 numbers $$\big\{4,\ 1+2\sqrt{2},\ \sqrt{2}\big\}$$ on a blackboard. You are permitted to perform the following operation whenever you want and as many times as you want:

• You can choose 2 numbers $$a$$ and $$b$$ and replace them with $$\frac{a+b}{\sqrt{2}}$$ and $$\frac{a-b}{\sqrt{2}}$$.

Is it possible to attain the 3 numbers $$\big\{3+\sqrt{2},\ 2\sqrt{2},\ \sqrt{2}-1\big\}$$ after a finite number of operations?

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