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प्रश्न
Prove that the following number is irrational: 3 - √2
उत्तर
3 - √2
Let 3 - √2 be a rational number.
⇒ 3 - √2 = x
Squaring on both the sides, we get
( 3 - √2 )2 = x2
⇒ 9 + 2 - 2 x 3 x √2 = x2
⇒ 11 - x2 = 6√2
⇒ √2 = `[ 11 - x^2 ]/6`
Here, x is a rational number.
⇒ x2 is a rational number.
⇒ 11 - x2 is a rational number.
⇒ `[ 11 - x^2 ]/6` is also a rational number.
⇒ `sqrt2 = [ 11 - x^2 ]/6` is a rational number.
But √2 is an irrational number.
⇒ `[ 11 - x^2 ]/6 = sqrt2` is an irrational number.
⇒ 11 - x2 is an irrational number.
⇒ x2 is an irrational number.
⇒ x is an irrational number.
But we have assume that x is a rational number.
∴ we arrive at a contradiction.
So, our assumption that 3 - √2 is a rational number is wrong.
∴ 3 - √2 is an irrational number.
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