Prove that √3 is an irraional number
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Answer:
Step-by-step explanation:
Assume to the contrary that √3 is rational.
then
- √3 is of form p/q
- p, q are co-prime
- q≠0
√3=p/q ⇒ √3 q=p
Squaring on both the sides,
(√3 q)² = p²
3 q² = p²------------------------------------------------------------------------------------------------1
q²=p²/3 ⇒ p/3 = c(c is a constant)
p=3c-----------------------------------------------------------------------------------------------------2
Substitute 2 in 1
3 q² = (3c)²
q²=9c² / 3
q² = 3c²
q²/3 = c² ⇒ q/3 = d
But this is contrary to the condition that p,q are co-prime.
∴ our assumption is wrong.
∴√3 is irrational
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