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Yes.

There is an injective function from rational numbers to positive rational numbers*.

Every positive rational number can be written in lowest terms as a/b, so there is an injective function from positive rationals to pairs of positive integers.

The function f(a,b) = a^2 + 2ab + b^2 + a + 3b maps maps every pair of positive integers (a,b) to a unique integer.

So there is an injective function from rationals to integers.

Since every integer is rational, the identity function is an injective function from integers to rationals.

Then By the Cantor-Schroder-Bernstein theorem, there is a bijective function from rationals to integers, so the rationals are countably infinite.

*This is left as an exercise for the reader.

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Q: Is the cardinality of an infinitely countable set the same as the rational numbers?
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Is there rational numbers than irrational?

There are infinitely many rational numbers and irrational numbers but the cardinality of irrationals is larger by an order of magnitude.If the cardinality of the countably infinite rational numbers is represented by a, then the cardinality of irrationals is 2^a.


Can you make an operator that takes any given irrational number and maps it onto the rationals?

Of course not.Number if irrational numbers is larger than number of rational numbers.To be more exact: There is no one-to-one mapping of set of rational numbersto the set of irrational numbers. If there would be such a mapping, their cardinality(see Cardinality ) would be same.In reality, rational numbers are countable (cardinality alef0)real numbers, as well as irrational numbers are not countable (cardinality alef1).These are topics inwikipedia.org/wiki/Transfinite_numbertheory


Are the sets of rational and irrational numbers equal?

No. Although there are infinitely many of either, there are more irrational numbers than rational numbers. The cardinality of the set of rational numbers is À0 (Aleph-null) while the cardinality of the set of irrational numbers is 2À0.


What are the numbers between two rational number?

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.


How many irrational numbers are there between sqrt2 and sqrt3?

Infinitely many. In fact, the number of irrationals in any interval - no matter how small - is infinite. Furthermore, the cardinality of the set of irrationals in any interval is greater than the cardinality of rational numbers in total!


Is the set of all rational numbers continuous?

Continuity is a characteristic of functions not of sets.The set of rational number is infinitely dense. This means that between any two rational numbers, no matter how close together, there are infinitely many rational numbers. And then, between any two of them these is an infinte number of rational numbers, and so on.But, in case that gives you any wrong ideas, between any two rational numbers there is an even higher order of infinity of irrational numbers. In that respect the number of gaps in the set of rational numbers (where the irrational numbers would be) is greater than the cardinality of rational numbers.Continuity is a characteristic of functions not of sets.The set of rational number is infinitely dense. This means that between any two rational numbers, no matter how close together, there are infinitely many rational numbers. And then, between any two of them these is an infinte number of rational numbers, and so on.But, in case that gives you any wrong ideas, between any two rational numbers there is an even higher order of infinity of irrational numbers. In that respect the number of gaps in the set of rational numbers (where the irrational numbers would be) is greater than the cardinality of rational numbers.Continuity is a characteristic of functions not of sets.The set of rational number is infinitely dense. This means that between any two rational numbers, no matter how close together, there are infinitely many rational numbers. And then, between any two of them these is an infinte number of rational numbers, and so on.But, in case that gives you any wrong ideas, between any two rational numbers there is an even higher order of infinity of irrational numbers. In that respect the number of gaps in the set of rational numbers (where the irrational numbers would be) is greater than the cardinality of rational numbers.Continuity is a characteristic of functions not of sets.The set of rational number is infinitely dense. This means that between any two rational numbers, no matter how close together, there are infinitely many rational numbers. And then, between any two of them these is an infinte number of rational numbers, and so on.But, in case that gives you any wrong ideas, between any two rational numbers there is an even higher order of infinity of irrational numbers. In that respect the number of gaps in the set of rational numbers (where the irrational numbers would be) is greater than the cardinality of rational numbers.


What numbers are between 1 and 1.5?

There are infinitely many numbers between 1 and 1.5: in fact, there are infinitely many rational numbers. The cardinality of irrational numbers between 1 and 1.5 is even greater.Some examples:1.000000000000000000000000021.000000000000000000000000020011.000000000000000000000000020021.000000000000000000000000020031.0000000000000000000000000200307Hopefully, you get the idea.


Is There fewer rational numbers than irrational numbers.?

Yes. The infinity of rational numbers has the same size as the natural numbers, said to be "countable". The infinity of real numbers (and therefore, also of irrational numbers) is a larger infinity, said to be "uncountable".


How many irrational numbers are there between 1 and 2?

Infinitely many. In fact, there are more irrational numbers between 1 and 2 as there are rational numbers - in total. The cardinality of this set is Aleph-0ne.


Are whole numbers infinitely countable?

Yes.


How many unreal numbers are there?

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How many rational numbers are there?

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