Are there more rational than irrational numbers?

Answer 1

The answer requires a bit of mathematics, but goes like this:

The product of any 2 rational numbers is a rational number.

The product of any 2 irrational number is an irrational number.

The product of a rational and an irrational number is an irrational number!

Therefore simple logic tells us that there are more Irrational Numbers than rational numbers. There is a way to structure this mathematically, and I believe it is called an "Inductive Proof".

Interesting !

I'm going to say "No".

I reason thusly:

-- For every rational number 'N', you can multiply or divide it by 'e', add it to 'e',

or subtract it from 'e', and the result is irrational.

-- You can multiply or divide it by (pi), add it to (pi), or subtract it from (pi),

and the result is irrational.

-- You can take its square root, and more times than not, its square root is irrational.

There may be others that didn't occur to me just now. But even if there aren't,

here are a bunch of irrational numbers that you can make from every rational one.

This leads me to believe that there are more irrational numbers than rational ones.

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There are infinitely many more irrationals than rationals; this was proved by G. Cantor (born 1845, died 1918). His proof is basically:

The rational numbers can be listed by assigning to each of the counting numbers (1, 2, 3,...) one of the rational numbers in such a way that every rational number is assigned to at least one counting number;

If it is assumed that every irrational number can be assigned to at least one counting numbers (like the rationals), then with such a list it is possible to find an irrational number that is not on the list; so is it not possible as there are more irrationals than there are counting numbers, which has shown to be the same size as the rational numbers, thus showing that there are more irrationals than rationals.

Answer 2

Yes.

The rational numbers are countable, that is they can be put in a one-to-one correspondence with the counting numbers, but the irrational numbers cannot and so are not countable.

There is a higher infinity of irrational numbers than the infinity of rational numbers.

Answer 3

No. In fact the irrationals outnumber the rationals by an order of infinity. If the number of rationals is said to be countably infinite whereas that of the irrationals is uncountably infinite.

Answer 4

Yes.

There are countably infinite rational numbers. The cardinality of the set is Aleph-Null. The cardinality of the set of irrational numbers is 2^(Aleph-Null).

Answer 5

Yes! The set of rational numbers is countably infinite; the set of real numbers (and therefore, also the set of irrational numbers) is uncountably infinite.