if p is a prime number, then prove that rootp is irrational.
Answers
Let us assume, to the contrary, that √p is rational.
So, we can find coprime integers a and b(b ≠ 0) such that √p = a/b
=> √p b = a
=> pb2 = a2 ….(i) [Squaring both the sides]
=> a2 is divisible by p
=> a is divisible by p So, we can write a = pc for some integer c.
Therefore, a2 = p2c2 ….[Squaring both the sides]
=> pb2 = p2c2 ….[From (i)]
=> b2 = pc2
=> b2 is divisible by p
=> b is divisible by p
=> p divides both a and b.
=> a and b have at least p as a common factor.
But this contradicts the fact that a and b are coprime.
This contradiction arises because we have assumed that √p is rational.
Therefore, √p is irrational.
Step-by-step explanation:
given than p is prime number
let us assume that √p is a rational number
√p=a/b
where a and b are coprimes and b not equal to 0
now,
squaring both sides we get
p=a^2/b^2
now bring the b^2 to other side
=> pb^2=a^2--------------------(1)
this shows that p divides a^2
therefore, p divides a
=> a=p×c ----------------(2)
substitute (2) in (1) ,we get
pb^2=(pc)^2
=> pb^2=p^2×c^2
cancelling p we get
b^2= p×c^2
from this we can conclude that
p divides b^2
therefore p divides b
=> p divides both a and b. => a and b have at least p as a common factor. But this contradicts the fact that a and b are coprimes. This contradiction arises because we have assumed that √p is rational. Therefore, √p is irrational.
hence proved