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If a and b are rational, with a < b, then a + (b-a) [sqrt(2)/ 2] is an irrational number between a and b.
This number is between a and b because sqrt(2)/2 is less than one and positive, so that a < a + (b-a) [sqrt(2)/3] < a + (b-a) [1] = b.
To prove that a + (b-a) [sqrt(2)/2] is not rational, suppose that
a + (b-a) [sqrt(2)/2] = p/q where p and q are integers.
Then, sqrt(2) = ( p/q -a ) 2/(b-a) which is rational since the rationals are a field, closed under arithmetical operation, but sqrt(2) not rational (Look up the elementary proof if you do not know it.)
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