Stating that more precisely, if you pick a real number uniformly from the range [0, 1), the probability that it's rational is 0.
One way to see this is to imagine a procedure for picking the number:
- Start with "0."
- Roll a D10 and append the digit.
- Repeat an infinite number of times.
- In the unlikely event that you wrote down a number that's not in standard format, like "0.1499999..." (which should instead be written "0.15"), toss it out and start again.
The digits of every rational number eventually repeat forever. For example, 1/7 is "0. 142857 142857 ...". So what's the probability that your sequence of rolls settles on a pattern and then repeats it forever, without once deviating in an infinite number of rolls? Pretty clearly zero.
One way to see this is to imagine a procedure for picking the number:
- Start with "0."
- Roll a D10 and append the digit.
- Repeat an infinite number of times.
- In the unlikely event that you wrote down a number that's not in standard format, like "0.1499999..." (which should instead be written "0.15"), toss it out and start again.
The digits of every rational number eventually repeat forever. For example, 1/7 is "0. 142857 142857 ...". So what's the probability that your sequence of rolls settles on a pattern and then repeats it forever, without once deviating in an infinite number of rolls? Pretty clearly zero.