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 𝑎/𝑏
where a and b (b≠ 0) are co-prime (no common factor other than 1)
Hence,
√𝟓 = 𝒂/𝒃
√5 b = a
Squaring both sides
(√5b)2 = a2
5b2 = a2
𝒂^𝟐/𝟓 = b2
Hence, 5 divides a2
So, 5 shall divide a also
Hence, we can say
𝑎/5 = c where c is some integer
So, a = 5c
Now we know that
5b2 = a2
Putting a = 3c
5b2 = (5c)2
5b2 = 25c2
b2 = 1/5 × 25c2
b2 = 5c2
𝒃^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.

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