Infinitely many. And between any two of them you can insert infinitely many. And between ...
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There are infinitely many rational numbers between any two rational numbers. And the cardinality of irrational numbers between any two rational numbers is even greater.
There are countably infinite rational numbers between any two numbers.
There is an infinite number of them between any two rational numbers.
An infinite number
There are an infinite number of rational numbers between any two given numbers.