Advertisements
Advertisements
Question
Show that `sqrt (2)/3 `is irrational.
Solution
Let `sqrt (2)/3 ` is a rational number.
∴`sqrt (2)/3 ` = `p/q` where p and q are some integers and HCF(p,q) = 1 …..(1)
⇒`sqrt(2) q = 3p`
⇒(`sqrt(2)q) ^2 = (3p)^2`
⇒`2q^2 = 9p^2`
⇒`p^2` is divisible by 2
⇒p is divisible by 2 ….(2)
Let p = 2m, where m is some integer.
∴`sqrt (2)`q = 3p
⇒`sqrt (2)`q = 3(2m)
⇒(`sqrt (2)q)^2 = [ 3(2m) ]^2`
⇒ `2q^2 = 4 (9p^2)`
⇒ `q^2 = 2 (9p^2)`
⇒ `q^2 `is divisible by 2
⇒ q is divisible by 2 …(3)
From (2) and (3), 2 is a common factor of both p and q, which contradicts (1).
Hence, our assumption is wrong.
Thus, `sqrt (2)/3` is irrational.
APPEARS IN
RELATED QUESTIONS
State whether the given statement is true or false:
1 . The product of two rationals is always rational
Express 0.`bar (23)` as a rational number in simplest form.
Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.
Write the condition to be satisfied by q so that a rational number\[\frac{p}{q}\]has a terminating decimal expansions.
Write the condition to be satisfied by q so that a rational number\[\frac{p}{q}\]has a terminating decimal expansion.
If p and q are two prime number, then what is their HCF?
State whether the following rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion:
`35/50`
Write down the decimal expansion of the following number which have terminating decimal expansion.
`23/(2^3xx5^2)`
m2 – 1 is divisible by 8, if m is ______.
