prove that 2+√3.
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2+root3 irrational kar ke aa jai ga
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We can prove it by contradictory method..
We assume that 2 + √3 is a rational number.
=> 2 + √3 = p/q , where p & q are integers, ‘q’ not = 0.
=> √3 = (p/q) - 2
=> √3 = (p - 2q)/ q ………… (1)
=> here, LHS √3 is an irrational number.
But RHS is a rational number.. Reason- the difference of 2 integers is always an integer. So the numerator (p- 2q) is an integer.
& the denominator ‘q’ is an integer.&‘q’ not = 0
This way, all conditions of a rational number are satisfied.
=> RHS (p- 2q)/q is a rational number.
But , LHS is an irrational.
=> LHS of….. (1) is not = RHS.
=> Our assumption, that 2 + √3 is a rational number, is incorrect..
=> 2 + √3 is an irrational number
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