Let's prove
that
is
the irrational number. Assume the opposite:
is
the rational number, then according to the definition of
rational number we can write:
=
m
/
n ,
then: 2 =
m^{2}
/
n^{2},
hence,
m^{2}
= 2
n^{2},
that is m^{2}
is divisible by 2, hence,
m is divisible by 2
and it is possible to write:
m
= 2
k, then
m^{2}
= 4
k^{2}
or 4
k^{2
}= 2
n^{2},
hence, n^{2}
= 2
k^{2},
that is n^{2}
is divisible by 2, then n
is divisible by 2, hence, m
è n have the common factor 2,
what contradicts to
the definition of rational number (see above). So, we
have proved that
is
the irrational number.
