If you calculate those square roots, you can count the integers in between pretty fast.
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There are infinitely many numbers. Amongst them are4.980000000000000000001, 4.980000000000000000001200000003, sqrt(24.8005).
There are no rational numbers between sqrt(-26) and sqrt(-15). The interval comprises purely imaginary numbers.
There are infinitely many such numbers. One such is sqrt(1000) = approx 31.62
No, some numbers, eg sqrt 2, are irrational, which means they cannot be written as a ratio between two whole numbers. Some numbers are also transcendental, but you don't want to go there!
The difference can be rational or irrational.5 + sqrt(3) and 2 + sqrt(3) are both irrational numbers but their difference is[5 + sqrt(3)] - [2 + sqrt(3)] = 3, which is rational.