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Consider the interval $[0,1]$. What is the probability that a number chosen at random in $[0,1]$ is transcendental?

Please give me some points on how to start this problem.

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    $\begingroup$ How many algebraic numbers are there? $\endgroup$ – Kevin Carlson Nov 24 '14 at 18:07
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    $\begingroup$ Usually we'd ask "which distribution?", but if only the distribution is continuous, then a number chosen from it will be transcendental with probablity 1. $\endgroup$ – Henning Makholm Nov 24 '14 at 18:12
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I realize this question is old but I wanted to give my explanation:

  1. The measure of the unit interval $[0,1]$ is $1$.
  2. Cantor proved that the algebraic numbers are countable, and thus the measure of the set of algebraics on $[0,1]$ is the countable union of the measures of the singletons. Since each singleton has measure zero, the set of algebraic numbers on $[0,1]$ has measure zero.
  3. Since the algebraic numbers and the transcendental numbers both partition $[0,1]$, and the set of algebraic numbers has measure zero, the set of transcendental numbers on $[0,1]$ must be of measure $1$.

So the probability of picking a transcendental number on the unit interval is $1$. Measure of a subset of $[0,1]$ more or less translates to "probability".

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Here is a point to start with:

  • There are infinitely countable many algebraic numbers in the interval $[0,1]$
  • There are infinitely uncountable many transcendental numbers in the interval $[0,1]$
  • Add your part here...
  • Hence the probability of choosing a transcendental number in the interval $[0,1]$ is $1$
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