Question

Prove that √3 is an irrational mumber. hence Show that (9- 34/3) is an irrational number ​

Answer 1

Answer:

$Let us assume on the contrary that 3 is a rational number. </p><p></p><p></p><p>Then, there exist positive integers a and b such that</p><p>3=ba where, a and b, are co-prime i.e. their HCF is 1</p><p>Now,</p><p>3=ba</p><p>⇒3=b2a2 </p><p>⇒3b2=a2 </p><p>⇒3 divides a2[∵3 divides 3b2] </p><p>⇒3 divides a...(i) </p><p>⇒a=3c for some integer c</p><p>⇒a2=9c2 </p><p>⇒3b2=9c2[∵a2=3b2] </p><p>⇒b2=3c2 </p><p></p><p>⇒3 divides b2[∵3 divides 3c2] </p><p></p><p>⇒3 divides b...</p><p></p><p>$

$From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.</p><p></p><p></p><p>Hence, 3 is an irrational number.</p><p></p><p></p><p>$

Step-by-step explanation:

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