There are infinitely many rational numbers.
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Infinitely many. Between any two different real numbers (not necessarily rational) there are infinitely many rational numbers, and infinitely many irrational numbers.
Infinitely many. In fact, between any two different real numbers, there are infinitely many rational numbers, and infinitely many irrational numbers. (More precisely, beth-zero rational numbers, and beth-one irrational numbers - that is, there are more irrational numbers than rational numbers in any such interval.)
A mixed number is a rational number. Mixed numbers are not a rational number but many of them.
There are infinitely many rational numbers between any two rational numbers - no matter how close together they are.
No, not all negative numbers are rational. There are many negative numbers that are irrational, just like the positive numbers.