Ex 1.2 , 1
Prove that 5 is irrational.
We have to prove 5 is irrational
Let us assume the opposite,
i.e., 5 is rational
Hence, 5 can be written in the form /
where a and b (b 0) are co-prime (no common factor other than 1)
Hence, 5 = /
5b = a
Squaring both sides
( 5b)2 = a2
5b2 = a2
^2/5 = b2
Hence, 5 divides a2
So, 5 shall divide a also
Hence, we can say
/5 = c where c is some integer
So, a = 5c
Now we know that
5b2 = a2
Putting a = 5c
5b2 = (5c)2
5b2 = 25c2
5b2 = 25c2
b2 = 1/5 25c2
b2 = 5c2
^2/5 = c2
Hence 5 divides b2
So, 5 divides b also
By (1) and (2)
5 divides both a & b
Hence 5 is a factor of a and b
So, a & b have a factor 5
Therefore, a & b are not co-prime.Hence, our assumption is wrong
By contradiction,
5 is irrational

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

Hi, it looks like you're using AdBlock :(

Displaying ads are our only source of revenue. To help Teachoo create more content, and view the ad-free version of Teachooo... please purchase Teachoo Black subscription.

Please login to view more pages. It's free :)

Teachoo gives you a better experience when you're logged in. Please login :)

Solve all your doubts with Teachoo Black!

Teachoo answers all your questions if you are a Black user!