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Question

Prove that $$\sqrt {5}$$ is an irrational number.


Solution

Let us assume that $$√5$$ is a rational number.

we know that the rational numbers are in the form of p/q form where p,q are coprime numbers.

so, $$\sqrt5 = \dfrac{p}{q}$$
    $$ p = \sqrt(5)q$$

we know that 'p' is a rational number. so $$\sqrt(5)$$q must be rational since it equals to p

but it doesnt occurs with $$\sqrt(5)q$$ since its not an integer
therefore, p is not equal to $$\sqrt(5)q$$

this contradicts the fact that $$\sqrt(5)$$ is an irrational number

hence our assumption is wrong and $$\sqrt(5)$$ is an irrational number

Mathematics

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