Question

Show that 5-2√3 is irrational

Answer 1

Here you go...

Proving:

5 - 2√3 is irrational.

Assumption:

Let us assume 5 - 2√3 to be " a " and it is 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)

Then,

5-2 \sqrt{3} = \frac{p}{q}

→ -2 \sqrt{3} = \frac{p}{q} - 5

→ - 2 \sqrt{3} = \frac{p-5q}{2}

→ \sqrt{3} = - ( \frac{p-5q}{2q})

We know that ,

√3 is irrational .

And, - ( \frac{p-5q}{2q}) is rational.

We know that ,

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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