Question

Prove that $2{\sqrt{3}}-1$ is an irrational number

Answer 1

SOLUTION :  

Let us assume , to the contrary ,that  2√3 - 1 is rational. Then,it will be of the form a/b where a, b are co primes integers and b ≠0.

2√3 - 1 = a/b

2√3 = a/b + 1

2√3 =[ (a + b)/b]  

√3 = [ (a + b)/b] /2

√3 =  (a + b)/2b

since, a & b is an integer so, (a + b)/2b is a rational number.  

∴ √3 is rational  

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

Hence,  2√3 - 1  is an irrational .

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Answer 2

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