Not necessarily.

The digits in the decimal expansion of a number which is itself uniformly, randomly selected from the interval [math][0,1][/math] (or any other interval) are random.

The digits in the decimal expansion of certain individual uncomputable numbers, such as Chaitin’s constant, are random.

The digits in the decimal expansion of any explicitly computable number, rational or irrational, are not random. They are entirely predictable. For example, the digits in the decimal expansions of the irrational numbers [math]\sqrt{2}[/math], [math]\pi[/math], [math]\log(2)[/math] are not random.

However, people sometimes use “random” in the weaker sense of “[math]k[/math]-uniformly distributed for all [math]k[/math]”, which means all [math]k[/math]-long combinations of digits appear equally often. This is also called “[math]\infty[/math]-distributed” in Knuth (vol. 2) and elsewhere.

We suspect that the decimal expansion of many explicit irrational numbers are like that, but this is not proven. Normal numbers are [math]\infty[/math]-distributed in all bases, and we in fact suspect that many explicit irrational numbers, including the ones I’ve mentioned, are normal. But again, this is not proven.

On the other hand, it’s easy to write down explicit [math]\infty[/math]-distributed sequences which are obviously decimal expansions of an irrational number, such as Champernowne’s constant. This clarifies the distinction between [math]\infty[/math]-distributed and random: the digits of

[math]0.123456789101112131415\ldots[/math]

are, clearly, very far from random, despite being provably [math]\infty[/math]-distributed.