prove root3 is irrational and hence 3/2root3 is irrational
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let √3 be rational number. Than,
√3=p/q (where p and q are coprimes)
squaring both the sides,
3=p^2/q^2
3q^2=p^2 ...(1)
this means that 3 is a factor of p^2.
=>3 is a factor of p.
so, 3m=p ...(2)
from eq. 1 and 2:
3q^2=(3m)^2
3q^2=9m^2
3q^2/9=m^2
q^2=3m^2
this means that 3 is a factor of q^2.
=>3 is a factor of q.
but this a contradiction since we assumed that p and q are coprimes.
so,our supposition is wrong.
hence,proved that √3 is irrational.
now,let 3/2√3be rational.
therefore,3/2√3=p/q (where p and q are coprimes)
√3/2=p/q
√3=2p/q
irrational=rational
but irrational is not equal to rational.
this is a contradiction ; so, our supposition is wrong.
hence, proved that 3/2√3 is irrational.
thank u, hope it helps!
√3=p/q (where p and q are coprimes)
squaring both the sides,
3=p^2/q^2
3q^2=p^2 ...(1)
this means that 3 is a factor of p^2.
=>3 is a factor of p.
so, 3m=p ...(2)
from eq. 1 and 2:
3q^2=(3m)^2
3q^2=9m^2
3q^2/9=m^2
q^2=3m^2
this means that 3 is a factor of q^2.
=>3 is a factor of q.
but this a contradiction since we assumed that p and q are coprimes.
so,our supposition is wrong.
hence,proved that √3 is irrational.
now,let 3/2√3be rational.
therefore,3/2√3=p/q (where p and q are coprimes)
√3/2=p/q
√3=2p/q
irrational=rational
but irrational is not equal to rational.
this is a contradiction ; so, our supposition is wrong.
hence, proved that 3/2√3 is irrational.
thank u, hope it helps!
HimanshiKankane:
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