Question

Prove that 1/(2+3^1/2)Is an irrational number

Answer 1

Answer:

CASE I -

Let us assume that √3 is a rational number.

Then, as we know a rational number should be in the form of p/q

where p and q are co- prime number.

So,

√3 = p/q { where p and q are co- prime}

√3q = p

Now, by squaring both the side

we get,

(√3q)² = p²

3q² = p² ........ ( i )

So,

if 3 is the factor of p²

then, 3 is also a factor of p ..... ( ii )

=> Let p = 3m { where m is any integer }

squaring both sides

p² = (3m)²

p² = 9m²

putting the value of p² in equation ( i )

3q² = p²

3q² = 9m²

q² = 3m²

So,

if 3 is factor of q²

then, 3 is also factor of q

Since

3 is factor of p & q both

So, our assumption that p & q are co- prime is wrong

Hence,. √3 is an irrational number

CASE II -

Let 1/2+√3 be a rational number.

A rational number can be written in the form of p/q where p,q are integers.

1/(2+√3) = p/q

√3 = p/q - 1/2

√3 = (2p-q)/2q

p, q are integers then (2p-q)/2q is a rational number.

Then √3 is also a rational number.

But this contradicts the fact that √3 is an irrational number.

So,our supposition is false.

Therefore,1/(2+√3) is an irrational number