prove that √3 is irrational number
Answers
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/q ²(Squaring on both the sides)
⇒ 3q² = p²………………………………..(1)
This means that 3 divides p2. This means that 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.
Answer:
Let assume the contantry
(i.e) root 3 is rational number
Root 3 =p/q
p=root3q
p square = 3q square - 1
p square /3=q2