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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Find the arithmetic average of the two rational numbers. It will be a rational number and will be between the two numbers.
For two rational numbers select any terminating or repeating decimal number which starts with 2.10 and for irrational numbers you require a non-terminating, non-repeating decimal which also starts with 2.10.
Rational zero test cannot be used to find irrational roots as well as rational roots.
Oh, dude, finding rational numbers between 0 and -1 is like trying to find a unicorn at a zoo. It's just not gonna happen. Rational numbers are all about fractions, and you can't have a fraction where the numerator is smaller than the denominator. So, in this case, there are no rational numbers between 0 and -1. It's a mathematical dead end, my friend.
Suppose the two rational numbers are x and y.Then (ax + by)/(a+b) where a and b are any positive numbers will be a number between x and y.