Exactly right. You can pick and use real numbers, as long as they are only queried to finite precision. There are lots of super cool algorithms for doing this!
Kind of, but you're not just picking rationals, you're picking rationals that are known to converge to a real number with some continuous property.
You might be interested in this paper [1] which builds on top of this approach to simulate arbitrarily precise samples from the continuous normal distribution.
Not really. You can simulate a probability of 1/x by expanding 1/x in binary and flipping a coin repeatedly, once for each digit, until the coin matches the digit (assign heads and tails to 0 and 1 consistently). If the match happened on 1, then it's a positive result, otherwise negative. This only requires arbitrary but finite precision but the probability is exactly equal to 1/x which isn't rational.
Exactly right. You can pick and use real numbers, as long as they are only queried to finite precision. There are lots of super cool algorithms for doing this!
That's just saying that you can pick and use rational numbers (which are a subset of the reals.)
Kind of, but you're not just picking rationals, you're picking rationals that are known to converge to a real number with some continuous property.
You might be interested in this paper [1] which builds on top of this approach to simulate arbitrarily precise samples from the continuous normal distribution.
[1] https://dl.acm.org/doi/10.1145/2710016
Not really. You can simulate a probability of 1/x by expanding 1/x in binary and flipping a coin repeatedly, once for each digit, until the coin matches the digit (assign heads and tails to 0 and 1 consistently). If the match happened on 1, then it's a positive result, otherwise negative. This only requires arbitrary but finite precision but the probability is exactly equal to 1/x which isn't rational.
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At no point will your number be transcendental (or even irrational).
That's why you can't stop.
That's irrelevant. It's like saying that you can count to infinity if you never stop counting ... but no, every number in the count is finite.
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And don’t die.