the computable numbers have density zero. if you choose a real number completely at random, with probability 1, it will be impossible to write a computer program to list it out to arbitrary precision
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(you can't even pick a random natural number uniformly)
You also can't pick uniformly from the entire real set [0,1] because almost all of those numbers are not only not computable, they are not describeable. The argument is similar to non computable: countable infinite descriptions, but uncountable infinite set of numbers.