Question

prove that 1/2 -√5/3 is irrational number​

Answer 1

Answer:

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Answer 2

Answer:

If possible let x be a rational number equal to 1/2 -√5/3

$x = \frac{1}{2} - \frac{ \sqrt{5} }{3} \\ squaring \: both \: sides \\ {x}^{2} = (\frac{1}{2} - \frac{ \sqrt{5} }{3} ) {}^{2} \\ {x}^{2} = ( \frac{1}{2} ) {}^{2} + ( \frac{ \sqrt{5} }{3} ) {}^{2} - 2( \frac{1}{2} )( \frac{ \sqrt{5} }{3} ) \\ {x}^{2} = \frac{1}{4} + \ \frac{5}{9} - \frac{ \sqrt{5} }{3} \\ {x}^{2} - \frac{1}{4} - \frac{5}{9} = \frac{ - \sqrt{5} }{3} \\ {x}^{2} - \frac{21}{36} = \frac{ - \sqrt{5} }{3} \\ 3( {x}^{2} - \frac{21}{36}) = - \sqrt{5} \\ {3x}^{2} - \frac{21}{12} = - \sqrt{5}$

Now‚

x² is a rational no.

-21/36 is a rational no.

3x² is also a rational no.

-21/12 is also an irrational no.

-√5 is a rational no.

But,

-√5 is an irrational no.

Thus , we arrive at a contradiction . So , our supposition that 1/2 -√5/3 is a rational no. is wrong Hence 1/2 -√5/3 is an irrational no.

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