prove that root 3 is irrational number
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prove that root 3 is irrational number
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Answer:
we have to prove that √3 is irrational
let us assume √3 is rational
then
√3 can written in the form of a/b
√3=a/b
√3b=a
S.O.B.S
squaring on both sides
(√3b)²=a²
3b²=a²
b²=a²/3
hence
3 divides a²
theorem: if p is a prime number and p divides a²
then, where 'a' is a positive number
then, 3 shall also divide 'a'. __________(1)
hence we can say
a/3=c
a=3c
now we know that
3b²=a²
substituting a=3c
3b²=(3c)²
3b²=9c²
b²=9c²/3
b²=3c²
b²/3=c²
hence
3 divides b²
and by using the same theorem
so,
3 divides also b ____________(2)
by (1)&(2)
3 divides both a & b
hence
3 is a factor of a & b
so,
a & b have factor 3
therefore
hence
a & b are not co-prime
assuming it's wrong
by
contradiction
√3 is irrational