Rrational numbers are fractions where both numerator and denominator are integers. Irrational numbers are conversely, any number that cannot be written as a fraction with one integer divided by another.
Suppose then we want to prove thatis irrational. We can prove this by contradiction.
Suppose thatwhere both
and
are numbers and suppose that
and
have no factors in common, so cannot be simplified. If
could be simplified then
could written
where
and
and relabelled
Square both sides ofto give
and multiply both sides by
to obtain
(1)
This means thatmust be even, since
is a whole number so
is even.
This means we can writewhere
is an integer, so that (1) becomes
Cancelling 2 from both sides gives
(2)
This means thatmust be even, since
is an integer so
is even.
Writeand subsitute into (2) to give
Cancelling 2 from both sides gives
henceand
this is a contradiction since it is an assumption thatcannot be simplified, so that
is irrational.