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You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
by daef 3y ago
You can uniquely map all rationals onto the natural numbers, thus they are of the same quantity. That doesn't work for all real numbers thou.
- warent 3y agoOh right this is only true for irrational and transcendental numbers
- aimor 3y agoMaybe this is misguided cheat, but couldn't you map any real number (between 0 and 1) to a natural number by mirroring the decimal digits across the decimal point. So 0.123 -> 321, but also sqrt(2)/2 -> ?601707 where ? is the rest of the decimal representation. This creates infinitely large numbers, but it's still a 1-to-1 mapping.
- dangond 3y agoUnfortunately, numbers with infinitely many digits are not natural numbers. You cannot count to ?601707, even with an infinite amount of time.