Question

prove that 3+2✓3 is an irrational number​

Answer 1

Answer:

Prove :

Let 3+√2 is an rational number.. such that

3+√2 = a/b ,where a and b are integers and b is not equal to zero ..

therefore,

3 + √2 = a/b

√2 = a/b -3

√2 = (3b-a) /b

therefore, √2 = (3b - a)/b is rational as a, b and 3 are integers..

It means that √2 is rational....

But this contradicts the fact that √2 is irrational..

So, it concludes that 3+√2 is irrational..

hence proved..

l hope it helped u..

Answer 2

ANSWER:

• (3+2√3) is an irrational number.

GIVEN:

• Number = 3+2√3

TO PROVE:

• (3+2√3) is an irrational number.

SOLUTION:

Let (3+2√3) be a rational number which can be expressed in the form of p/q where p and q have no other common factor than 1.

$\implies \: 3 + 2 \sqrt{3} = \dfrac{p}{q} \\ \\ \implies \: 2 \sqrt{3} = \frac{p}{q} - 3 \\ \\ \implies \: 2 \sqrt{3} = \frac{p - 3q}{q} \\ \\ \implies \: \sqrt{3} = \frac{p - 3q}{2q}$

• (p-3q)/2q is rational but √3 is irrational.
• Thus our contradiction is wrong.
• So (3+2√3) is an irrational number.

NOTE:

• This method of proving an irrational number is called contradiction method.
• In this method we first Contradict a fact and than we prove the contradiction wrong.