generate pair of uniformly random variables given a single uniform random variable

Question

Say we have $X \sim U(0,1)$, a standard uniform variable. I want a function $\{Y,Z\}=f(X)$ from $R$ to $R^2$ that gives us a pair ${Y,Z}$ such that $Y$ and $Z$ are independent and both are $U(0,1)$. Does this exist? It seems like conjuring two random variables from a single one, which is counterintuitive, but I do know you can map injectively from R to $R^2$ so I'm a bit lost.

Answer 1

To summarize the discussion in the comments:

You can use the binary expansion of $X$ to create two, or even countably many, independent variables, all uniform on $[0,1]$. To do $2$, for instance, just let $Y$ denote the binary number made from the odd digits of $X$, and $Z$ the binary number made from the even digits.

To make countably many, consider the (infinite) list of primes $\{2,3,5,7,11,\cdots\}$ and make one variable out of the digits in places $\{2, 2^2, 2^3, \cdots\}$, the second out of the digits in places $\{3, 3^2, 3^3, \cdots \}$ and so on.