Are all irrational numbers also normal numbers (probability of each decimal is equal)?

Question

When computing an irrational number is there a possibility that NOT ALL INTEGERS appear t the same rate in the constructed irrational number?

I know PI is a normal number but what about other irrational numbers?

Answer 1

First off, $\pi$ is suspected, but not known to be a normal number. In fact, the normal numbers we know with certainty are more or less only the ones explicitly constructed to be normal.

Second, it is easy to construct irrational, non-normal numbers. Take, for instance, the representation of $\pi$ in binary, and interpret it as a number in decimal. There are only $1$ and $0$ in the decimal expansion, so it cannot be normal. However, given a random irrational number, like $\ln 3$ or $\sqrt{\pi/\sin(15)}$, the odds are overwhelming that it is normal.

Answer 2

Liouville number $\lambda = \sum _{n=1}^{\infty} \frac{1}{10^{n!}}$ is not only irrational, but is transcendental, see here, and its digits are only $0$s and $1$s, where the $1$ is at factorial places.

You should take a look at Champernowne constant there