# Show that 2sqrt(7) is Irrational. - Mathematics

Show that 2sqrt(7) is irrational.

#### Solution

2/sqrt(7) = 2/sqrt(7) xx sqrt(7)/sqrt(7) = 2/7 sqrt(7)

Let 2/7 sqrt (7) is a rational number.

∴ 2/7 sqrt (7) = p/q, where p and q are some integers and HCF(p,q) = 1     ….(1)

⇒2sqrt(7)q = 7p

⇒(2sqrt(7) q) ^ 2 = (7p)^ 2
⇒7(4q^2) = 49p^2
⇒4q^2 = 7p^2
⇒ q^2 is divisible by 7
⇒ q is divisible by 7        …..(2)
Let q = 7m, where m is some integer.
∴2sqrt(7) q = 7p
⇒ [2sqrt(7) (7m)]^2 = (7p)^2
⇒343(4m^2) = 49p^2
⇒ 7(4m^2) = p^2
⇒ p^2 is divisible by 7
⇒ p is divisible by 7             ….(3)
From (2) and (3), 7 is a common factor of both p and q, which contradicts (1).
Hence, our assumption is wrong.
Thus, 2 sqrt(7) is irrational.

Concept: Concept of Irrational Numbers
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