TITLE: If $\pi $ is a normal number, is $\tau $ one?
QUESTION [5 upvotes]: If $\pi$ is a normal number, would that imply that $\tau =2\pi $ is also a normal number? If so, why? Something tells me that it should be, but I have no idea how to prove it. If all digits of $\pi$ were either $0$, $1$, $2$, $3$ or $4$, the proof is self-evident. Obviously, every decimal digit appears in $\pi$, so that's where that self-evident proof will fall apart.

REPLY [4 votes]: Well according to wikipedia this statement:
"If x is normal in base b and q ≠ 0 is a rational number, then x⋅q is normal in base b."
has been proven by Wall.
Wall, D. D. (1949), Normal Numbers, Ph.D. thesis, Berkeley, California: University of California.