Re: Rational/Irrational Numbers

The mathematical proof which I know goes like this:

(1) Rational numbers are countable: by enumerating all rationals like
this:

1 2 3 4 5 6.......
2 3 4 5 6 7...
3 4 5 6 7 8..
4 5 6 7 8 9...
........................
........................

(2) Then there is proof that real numbers are uncountable by supposing
that a list of real numbers between 0 and 1 (in binary format) exists
and then using Cantor's diagonalization argument to show that it is
impossible.

(3) Construction of real number by Dedekind's cut. From that cut we can
see that
- between two rationals infinite irrational numbers exist. -----(a)
- Between two irrational infinite rational numbers exist. ------(b)

But just by looking at the conclusion (a), and (b) I cannot understand
that there are much more irrational numbers than rational numbers.
Because if I see any two irrational numbers, which may be arbitrarily
close, there are still many(infinite) rational numbers between them.
Then how irrationals are more in number. It looks contradictory to
me...

Thanks for your help...

Alok Bakshi

.

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