Question

Prove that √3 is irrational and hence prove that √3 − 5/6 is irrational

Answer 1

after proving root 3 is irrational

for second part of question take5/6 other side and divide it by p/q and after that write as root3 is irrational therefore root 3-5/6 is also irrational

Answer 2

Step-by-step explanation:

let root 3 be rational and let it's simplest form be a/b.

Then a and b are the integer having no common factor other than 1 , and b not equal to 0.

Now, root 3 =a/b=》3=a square /b square [ on squaring both sides ]

=》3b square = a square _: (¡)

=》3 divides a square [since 3 divides 3a square ]

=》3 divides a [ since 3 is a prime and 3 divides a square =》3 divides a ]

Let a=3c for some integer c

Putting a=3c in equation (i)

3b square = 9c square =》be square =3c square

=》3 divides b square [since 3 divide 3c square ]

=》3 divides b [ since 3 is a prime and 3 divides b square =》3 divides b]

Thus , 3 is a common factor of a and b .

But , this contradict the fact that a and b have no common factor other than 1 .

The contradiction arises by assuming that root 3 is a ration.

Hence, root 3 is irrational.