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To begin with, we have to be careful what we mean by "random." Clearly pi is not "random" in the strict sense, because individual digits are certainly not random but mathematically fixed. Perhaps a better and easier question is whether pi is "normal base 10," which means that each digit, 0 through 9, appears, in the limit, precisely one tenth of the time; every two-digit string appears, in the limit, precisely one one-hundredth of the time; and similarly for every other finite-length string. One can also ask whether pi is "normal base 2," which means that each binary digit (0 or 1) appears half of the time; each two-digit string appears one fourth of the time, etc.
He adds that "at the very least, we have shown why the digits of pi and log(2) appear to be random: because they are closely approximated by a type of generator associated with the field of chaotic dynamics."
Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three things: How much uncertainty we are willing to tolerate in the forecast; how accurately we are able to measure its current state; and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, ~1 millisecond; weather systems, a couple of days (unproven); the solar system, 50 million years. In chaotic systems the uncertainty in a forecast increases exponentially with elapsed time. Hence doubling the forecast time squares the proportional uncertainty in the forecast. This means that in practice a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears to be random.