Question

প্রমাণ কর যে √10 একটি অমূলদ সংখ্যা

Prove that √10 is an irrational number.

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Answer 1

Question:-

Prove that √10 is an irrational number.

Required Answer:-

We know that:-

$\sf{9 < 10 < 16}$

$\\ \sf{\rightarrow{ \sqrt{9} < \sqrt{10} < \sqrt{16} }}$

$\\ \sf{ \rightarrow{3 < \sqrt{10} < 4}}$

Therefore, √10 is greater than 3 and smaller than 4

∴ √10 is not a real number. It is a rational or irrational number.

Let us assume that:-

• √10 is an irrational number.

$\sf{ \therefore{ \sqrt{10} = \dfrac{p}{q} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \boxed{ \rm{where \: p \: and \: q \: are \: co \: primes \: and \: q > 1}}}}$

$\\ \sf{ \implies{10 = \dfrac{ {p}^{2} }{ {q}^{2}}} }$

$\\ \sf{ \implies{10 \times q = \frac{ {p}^{2} }{ {q}^{2} } \times q}}$

$\\ \sf{ \implies{10q = \dfrac{ {p}^{2} }{q} }}$

Here, 10q is a reall number but p^2/q is not a real number. Because p and q are co primes.

$\sf{ \therefore{10q \: \cancel{ = } \: \dfrac{ {p}^{2} }{q}}}$

∴ √10 can't be expressed as p/q

∴ √10 is not rational number.

Hence, √10 is irrational number.

Hence, Proved!

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