Adding to the above, it is worth noting that the transcendental numbers that are commonly known/used in mathematics (e, pi, etc.) belong to the countable subset of transcendental numbers with a finite recursive function generator. The uncountable transcendental numbers, or the random numbers, are those which cannot be generated by a finite recursive function, and they can therefore also not be the solution to a finite analytic equation (unless it contains random numbers). In other words, if you pick a real number at random, the resulting number will be random.

Adding to the above, it is worth noting that the transcendental numbers that are commonly known/used in mathematics (e, pi, etc.) belong to the countable subset of transcendental numbers with a finite recursive function generator. The uncountable transcendental numbers, or the random numbers, are those which cannot be generated by a finite recursive function, and they can therefore also not be the solution to a finite analytic equation (unless it contains random numbers or is solved by a subset of the reals). In other words, if you pick a real number at random, the resulting number will be random.