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Prove that √3is irrational

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Prove that √3is irrational

Posted by Dilraj Kâlêsh 5 years, 2 months ago

CBSE > Class 10 > Mathematics

1 answers

Gaurav Seth 5 years, 2 months ago

Let us assume to the contrary that √3 is a rational number.

It can be expressed in the form of p/q

where p and q are co-primes and q≠ 0.

⇒ √3 = p/q

⇒ 3 = p²/q² (Squaring on both the sides)

⇒ 3q² = p²………………………………..(1)

It means that 3 divides p²and also 3 divides p because each factor should appear two times for the square to exist.

So we have p = 3r

where r is some integer.

⇒ p² = 9r²………………………………..(2)

from equation (1) and (2)

⇒ 3q² = 9r²

⇒ q² = 3r²

Where q² is multiply of 3 and also q is multiple of 3.

Then p, q have a common factor of 3. This runs contrary to their being co-primes. Consequently, p / q is not a rational number. This demonstrates that √3 is an irrational number.

2Thank You

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