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

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.

• How many algebraic numbers are there? Nov 24, 2014 at 18:07
• 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. Nov 24, 2014 at 18:12

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".

• There are infinitely countable many algebraic numbers in the interval $[0,1]$
• There are infinitely uncountable many transcendental numbers in the interval $[0,1]$
• Hence the probability of choosing a transcendental number in the interval $[0,1]$ is $1$
• It is possible to specify some transcendental numbers exactly, e.g. $\pi$ is (among other things) the first positive zero of the $\sin$ function. But if we require a "specification" to be expressible as a finite sequence of, say, ASCII characters, there are only countably many of those, so it's still true that almost all transcendental numbers don't have such a specification. Oct 16, 2022 at 16:13