Rational numbers are real numbers that can be expressed in the form a/b where b is not zero.
Irrational numbers are real numbers that cannot be simplified into a/b and exhibit a non-terminating nor repeating decimal.
Let us call the "both are the same" option (3rd one) as the default that does not require proof. If you do pick the other two, let's see if you can come up with a reason why you picked your answer (because mathematics is all about trying to prove why a statement is true).
[Edited by Neo7, 10/10/2011 6:54:40 PM]
Actually a "non-real number" is known as an imaginary number and revolves around the square root of -1 and it being raised to various powers.
A Complex Number is a number containing a real component and an imaginary component. It is expressed as the form a+bi where a is any real number and bi is a scalar (b) times the square root of -1 (i).
[Edited by Neo7, 10/10/2011 7:18:25 PM]
There is a proof, called Cantor's diagonalization argument, which shows that the set of irrational numbers (non-terminating, non-repeating decimals) is uncountably infinite (while rational numbers are countably infinite).
I think in this case, you need the bijection function (not the surjection function). You can represent an infinite amount of irrational numbers between 0 and 1 by using infinite strings of just 1s and 0s while you can represent the natural numbers as 1,2,3,~
Using Cantor's diagonalization, you will generate a particular number that will not be able to pair up with another rational number so long as they go out infinitely long (which makes creating a bijection impossible)
Day probably explained it much better here: Link
[Edited by Neo7, 10/11/2011 11:14:13 AM]
Cantor's argument contradicts epsilon-delta proof.
According to the epsilon-delta proof, width/length of the list converges to zero.
When we increase digits of the list, width/length monotonically decreases.
So, we can easily accept the result.
However, Cantor's argument is as follows.
When the number of digits reaches the actual infinity, suddenly width/length becomes 1.