If √2 is given as an irrational number, then prove that

Q) If √2 is given as an irrational number, then prove that (5 – 2√2) is an irrational number.

Ans:

Let’s start by considering 5 – 2 √2 is a rational number.

∴ 5 – 2 √2 = $\frac{p}{q}$ (here p and q are integers and q ≠ 0)

∴ – 2 √2 = $\frac{p}{q}$ – 5 = $\frac{p - 5 q}{q}$

∴ √2 = $\frac{p - 5 q}{- 2 q}$

Since p and q are integers, so, $\frac{p - 5 q}{- 2 q}$ is a rational number.

If RHS is a rational number, then LHS will also be a rational

Therefore √2 is a rational number.

But it contradicts the given condition (given that √2 is an irrational number)

Therefore, it is confirmed that 5 – 2 √2 is an irrational number.

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