Question

prove that 5 - 2 √3 is an irrational number

Answer 1

To prove:

5 - 2√3  is irrational.

Assumption:

Let us assume 5 - 2√3 to be rational.

Proof,

As, 5 - 2√3 Is rational it can be written in the form of p/q where q ≠ 0.

(p , q are coprime [i.e no common factor other than 1] ) 

Then,

$\tt{5-2 \sqrt{3} = \dfrac{p}{q}}$  

$-\tt{2 \sqrt{3} = \dfrac{p}{q} - 5 }$

$\tt{- 2 \sqrt{3} = \dfrac{p-5q}{q}}$

$\tt{\sqrt{3} = - ( \dfrac{p-5q}{2q}) }$

We know that √3   is irrational and,   $\tt{- ( \dfrac{p-5q}{2q})}$     is rational and we arrive at the point where they are equal.

But, Irrational  ≠  rational.

So, we contradict the statement that  5 - 2 √ 3  is rational.

Therefore 5 - 2√3 is an irrational number

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Hope my answer is helpful to  you.

Answer 2

Answer:

Step-by-step explanation:

Solution :-

Let us assume that 5 - 2 √3 is a rational number.

So, 5 - 2 √3 may be written as

5 - 2 √3 = p/q, where p and q are integers, having no common factor except 1 and q ≠ 0.

⇒ 5 - p/q = 2 √3

⇒ √3 = 5q - p/2q

Since, 5q - p/2q is a rational number as p and q are integers.

Therefore, √3 is also a rational number, which contradicts our assumption.

Thus, Our supposition is wrong.

Hence, 5 - 2 √3 is an irrational number.