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प्रश्न
Show that the following numbers are irrational.
उत्तर
(i) Let us assume that `1/sqrt2` is rational .Then , there exist positive co primes a and b such that
`1/sqrt2`=`a/b`
`1/sqrt2`=`(a/b)^2`
`⇒ 1/sqrt2 = a^2/b^2`
`⇒ b^2 = 2a^2`
`⇒ 2 | b^2 (because 2|2a^2)`
`⇒ 2|b`
`⇒ b= 2c \text{for some posibive integer c}`
`⇒ 2a^2=b^2`
`⇒ 2a^2=4c^2(because a= pc)`
`⇒ a^2 = 2c^2`
`⇒ 2|a ^2 (2|2c^2)`
`⇒ 2|a`
`⇒ 2|a and 2|b`
Hence `1/sqrt2` is irrational
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