Question

prove that (2✓3-1) is an irrational number..

Answer 1

2√3 = a / b + 1 

2√3 = ( a + b ) / b

√3 = ( a + b ) / 2b

Since a and b are integers , we get 

( a + b ) / 2b is rational , and so √3 is

rational .

But this contradicts the fact that √3 

is irrational.

This contradiction has arisen 

because of our incorrect assumption

that 2√3 - 1 is rational.

So , we conclude that 2√3 -1 is 

irrational.

Answer 2

Answer :

2√3-1 is an irrational number.

Proved.

Step-by-step explanation :

Given :

$\sf 2 \sqrt{3-1}$

To Prove :

$\sf {2 \sqrt{3-1}\;is\;an\;irrational\;number.$

Solution :

Assume that :

2√3-1 be a rational number.

As we know :

$\bigstar$ A irrational number cannot be written in the form of p/q where p,q are integers.

So,

$\sf \\\implies 2\sqrt3-1=p/q\\ \\\implies 2\sqrt3=p/q+1\\ \\\implies 2\sqrt3=(p+q)/q\\ \\\implies \sqrt3=(p+q)/2q$

Note :

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

It implies that :

√3 is also a rational number.

But we know,

√3 is an irrational number.

Therefore,

Our assumption goes wrong.

$\bigstar\;\sf Hence,\\2\sqrt3-1\;is\;an\;irrational\;number.$

$\rule{300}{1.5}$

$\bigstar$ Extra Information :

Rational Numbers  :

A Rational Number can be written as a Ratio of two integers as a simple fraction.

Example :

7 is rational, because it can be written as the ratio 7/1

Irrrational Numbers  :

The numbers cannot be written as a ratio of two integers they are called Irrational Numbers.

Example :

√2  cannot be written in as a Ratio of two integers as a simple fraction.