Question

prove that root 2,root 3,root 5 and root 7 are irrational numbers

Answer 1

Answer:

Step-by-step explanation:

First We prove is irrational numbers.

Suppose is rational.That mean it can be written as a ratio of two integers p and q.

Where we may assume that p and q have no common factors.(If there are any common factor we cancel them in numerator and denominator.) Squaring in (1) on both sides gives

$2=\frac{p^{2} }{q^{2} }...(2)$

Which implies

Thus is even. The only way this can be true is that p itself is even.But is actually divisible by 4.Hence and therefore q must be even. So p and q are both even which is contradiction to our assumption that they have no common factors.The square root of 2 can not be rational.

So $\sqrt{2}$ is irrational numbers.

By this method you can prove that $\sqrt{3} ,\sqrt{5} ,\sqrt{7}$ are irrational numbers.

Hope this help you.Thanks