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  • Prove that  is an irrational number, given that  i

Prove that (5-3\sqrt{2}) is an irrational number, given that \sqrt{2} is irrational number.

 

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Given \rightarrow \sqrt{2} is irrational

To prove \rightarrow 5-3\sqrt{2} is irrational

Proof \rightarrow Lets assume 5-3\sqrt{2} is rational

5 is a rational number.

We know that difference of two rational is rational

\Rightarrow 5-3\sqrt{2}-5=-3\sqrt{2}

This means 3\sqrt{2} is rational

Now we know the product of two rational number is rational

\Rightarrow 3\sqrt{2}\times \frac{1}{3}\; \; \; \; (\therefore \frac{1}{3}\; is\; rational\; no)

This implies \Rightarrow \sqrt{2} is also rational.

But it is contradictory because we have given that \sqrt{2} is irrational.

Hence by contradiction \sqrt{2} is irrational.

Hence Proved

Posted by

Safeer PP

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