Yes, many people have asked the same question, and I have tried by best to get my head around it, but I still can't understand.
Every time I look at the proofs, they seem like a sleight of hand to me..
Kindly point out the faults in my arguments:
Argument - 1
To map every Real Number between 0 and 1, to a unique Natural Number, simply remove all trailing zeros, and then remove the decimal point!
0.1 maps to 1
0.254 maps to 254
0.2540000 maps to 254 (because we are removing the trailing zeros)
0.31415926... maps to 31415926... (if a real number can have infinite precision, then most probably a natural number can be infinitely large? Am I wrong?)
Argument - 2
Every pair of Natural Numbers can be mapped to a unique Natural Number. (for eg, Cantor's Pairing Function).
Every Rational Number 'r' can be mapped to a pair of Natural Numbers (p,q) such that
r = p/q
Since for every rational number 'r', we have an infinite number of such pairs
the cardinality of Natural Numbers must be greater than the cardinality of Rational Numbers.
So, Irrational Number can be mapped to Natural Numbers.
So, Real Numbers can be mapped to Natural Numbers.