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

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?

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.

• Thank you for the explanation. I understood why the representation of PI in binary would be considered non-normal in decimal although it would still be an irrational number because it has a infinite amount of unpredictive decimals, which are chosen between two digits but nonetheless cannot be predicted. Am i correct in my thinking? – yoyo_fun Jul 18 '17 at 8:06
• @yoyo_fun That's exactly it. Allthough the phrase "can't be predicted" is a bit strong, seing how we may calculate the binary representation of $\pi$ exactly to any finite length, at least in principle. – Arthur Jul 18 '17 at 8:07
• I was going to suggest the Liouville constant $\sum_{k=1}^\infty 1/10^{k!}$ myself. But I am not sure I agree with your final statement: While it is true that a random real number, picked by any continuous probability distribution, is normal with probability 1, the numbers we can name only form a countable collection, and we know next to nothing about the normality of such numbers. – Harald Hanche-Olsen Jul 18 '17 at 8:10
• @HaraldHanche-Olsen Which is to say the probability of normality among those numbers, as far as human knowledge is concerned, is inherited form the probabilities on the real number line. In the absence of concrete results hinting differently, I would say the odds are still overwhelming. At least that's my view on the philosophy of probability. – Arthur Jul 18 '17 at 8:13
• Hmm. I think it is conceivable, but maybe not likely, that no algebraic number is normal in the sense that all length $n$ sequences of digits occur equally often. But I won't press the point. And in any case, your chosen examples aren't likely to be algebraic, and I agree that there is a good chance that they are indeed normal, so long as you don't interpret “a good chance” as being a rigorous statement about probability. 8-) – Harald Hanche-Olsen Jul 18 '17 at 8:20

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