Question

Prove that 5 + √3 is irrational.​

Answer 1

Step-by-step explanation:

Let us assume that 5 - √3 is a rational We can find co prime a & b ( b≠ 0 )such that 5 - √3 = a/b Therefore 5 - a/b = √3 So we get 5b -a/b = √3 Since a & b are integers, we get 5b -a/b is rational, and so √3 is rational. But √3 is an irrational number Let us assume that 5 - √3 is a rational We can find co prime a & b ( b≠ 0 )such that ∴ 5 - √3 = √3 = a/b Therefore 5 - a/b = √3 So we get 5b -a/b = √3 Since a & b are integers, we get 5b -a/b is rational, and so √3 is rational. But √3 is an irrational number Which contradicts our statement ∴ 5 - √3 is irrational

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Answer 2

Let us assume the given number be rational and we will write the given number in p/q form

⇒

$5 - \sqrt{3 = \frac{p}{q} }$

$\sqrt{3 = \frac{5q - p}{q} }$

We observe that LHS is irrational and RHS is rational, which is not possible.

This is contradiction.

Hence our assumption that given number is rational is false

⇒

$5 - \sqrt{3} \: is \: irrational$