Question

Prove that √2 is irrational​

Answer 1

ANSWER

Let us assume on the contrary that

2

is a rational number. Then, there exist positive integers a and b such that

2

=

b

a

where, a and b, are co-prime i.e. their HCF is 1

⇒(

2

)

2

=(

b

a

)

2

⇒2=

b

2

a

2

⇒2b

2

=a

2

⇒2∣a

2

[∵2∣2b

2

and 2b

2

=a

2

]

⇒2∣a...(i)

⇒a=2c for some integer c

⇒a

2

=4c

2

⇒2b

2

=4c

2

[∵2b

2

=a

2

]

⇒b

2

=2c

2

⇒2∣b

2

[∵2∣2c

2

]

⇒2∣b...(ii)

From (i) and (ii), we obtain that 2 is a common factor of a and b. But, this contradicts the fact that a and b have no common factor other than 1. This means that our supposition is wrong.

Hence,

2

is an irrational number.

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Answer 2

Answer:

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