1.

Prove that 5-2√3 is a irrational number

Answer»

Step-by-step EXPLANATION:

Here is your answer:

To prove:

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}5−2

3

=

q

p

→ -2 \sqrt{3} = \frac{p}{q} - 5−2

3

=

q

p

−5

→ - 2 \sqrt{3} = \frac{p-5q}{2}−2

3

=

2

p−5q

→ \sqrt{3} = - ( \frac{p-5q}{2q})

3

=−(

2q

p−5q

)

We KNOW that ,

√3 is irrational .

And, - ( \frac{p-5q}{2q})−(

2q

p−5q

) 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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