Does it make sense to say that there are more irrational numbers than rational?
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Yes, in the sense of cardinality. In the same sense that one can say that there are more real numbers than there are rationals (or integers; or natural numbers).
The rational numbers are countable, but the irrationals are uncountable.
Never forget that irrational numbers are the result of our choice of unit length, indicating the inherent limitations of a given number system. An irrational number in one number system can be rational number in other number system.