Prove that sqrt3 is an irrational number.

tisanurnr9c

tisanurnr9c

Answered question

2023-02-24

Prove that 3 is an irrational number.

Answer & Explanation

Jayden Landry

Jayden Landry

Beginner2023-02-25Added 7 answers

Let us suppose that 3 is a rational number.
Then there are positive integers a and b such that 3 = a b , where a and b are co-prime, meaning their HCF is 1.
3=ab3=a2b23b2=a23dividesa2[3divides3b2]3dividesa.....................(i)a=3cforsomeintegerca2=9c23b2=9c2[a2=3b2]b2=3c23dividesb2[3divides3c2]3dividesb..............................(ii)
We can see that a and b share at least 3 as a common factor from (1) and (2).
The fact that a and b are co-prime, on the other hand, contradicts this and indicates that our hypothesis is incorrect.
Therefore, 3 ​ is an irrational number.

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