Question

Prove that  1/(2+√3)  is an irrational number.​

Answer 1

Answer:

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

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Step-by-step explanation:

Answer 2

Answer:

Let √3 be rational

So, √3 = a/b       where 'a' and 'b' are co-primes

Squaring both side , we get

3 = $\frac{a^{2}}{b^{2} }$

$=> b^{2} =\frac{a^{2} }{3}$

If 3 divides $a^{2}$, then 3 divides a

Let a = 3c

So,

$b^{2} = \frac{(3c)^{2} }{3}\\b^{2} = \frac{9c^{2} }{3}\\ b^{2} = 3c^{2}\\\frac{b^{2} }{3} =c^{2}$

If 3 divides $b^{2}$, then 3 divides b

But 'a' and 'b' are co-primes

Hence, our supposition is wrong

Then, √3 is irrational

Now, let us suppose  1/(2+√3) is rational

So, $\frac{1}{2+\sqrt{3} } = \frac{a}{b}$       where 'a' and 'b' are co-primes

$=> 1 . \frac{b}{a} = 2+\sqrt{3}\\ => \frac{b}{a} - 2 = \sqrt{3}\\ => \frac{b - 2a}{a} = \sqrt{3}$

This shows that  √3 is rational

But √3 is irrational

Hence, this contradicts our supposition

Then,   1/(2+√3)  is an irrational number.​

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