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प्रश्न
Prove that the following number is irrational: √5 - 2
उत्तर
√5 - 2
Let √5 - 2 be a rational number.
⇒ √5 - 2 = x
Squaring on both the sides, we get
`( √5 - 2 )^2 = x^2`
⇒ 5 + 4 - 2 x 2 x √5 = x2
⇒ 9 - x2 = 4√5
⇒ √5 = `[ 9 - x^2 ]/4`
Here, x is a rational number.
⇒ x2 is a rational number.
⇒ 9 - x2 is a rational number.
⇒ `[ 9 - x^2 ]/4` is also a rational number.
⇒ √2 = `[ 9 - x^2 ]/4` is a rational number
But √2 is an irrational number.
⇒ √5 = `[ 9 - x^2 ]/4` is an irrational number.
⇒ 9 - 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 √5 - 2 is a rational number is wrong.
∴ √5 - 2 is an irrational number.
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