Though the ability to generate a precise
decimal number by cutting a
notch a set length down a stick may be surprisingly limited, no such limitations would appear to apply if what is denoted by the mark is instead a
fraction or some system indicating a series of fractions. For example, suppose we divide the stick into 22 equally distributed notches, and then placed a mark at the seventh notch. This would indicate that the number to be generated would be 22/7 -- which was for some time the best equivalent available for
pi, and the reason why fraction lovers celebrate
Pi Day on July 22 (using the European tradition of indicating day/month) instead of raising a ruckus on at 1:59 on March 14. Then again, anyone who would be excited about Pi Approximation Day would likely already have been excited by regular Pi Day. But back to fractions, 22/7 is a fraction convertable to the decimal 3.142857..., where that last '...' represents an
infinite repetition of the '142857' sequence.
Such infinite
strings of fractions are in fact far more common than the alternative, there being far more potential divisions of
numerator by
denominator generating like strings than those generating neat resolutions such as 3/4 equaling .75. Concededly, most such strings are simply self-repeating after a point, and so provide no additional information themselves, but at least their infiniteness makes them useful for modifying long strings of non-self repeating numbers. And indeed, in addition to common fractions, there are any number of other numerical functions which generate infinite strings of generally non-reapeating numbers, such as true
pi,
Euler's number, and the vast majority of
square roots. Because pi itself is a non-repeating number, it would appear that a work of a hundred thousand or a million or a billion characters could be encoded into pi if only a fortuitous string of number could be found within it, or within some convenient multiple of it; if for example, having converted an encyclopedia into a string of numbers, we could determine that the entire string existed somewhere within pi (or pi times 3/17 or the like), perhaps starting at the 1,375,712,006th decimal place and ending at the 3,441,255,102nd decimal place. It must be conceded that the chances of finding a single useful string of such length even within an infinite series numbers are slim -- pi is not a random number (like the type we might expect to find
if an infinite number of monkeys pounded out nothing but numbers on an infinite number of keypads), and even were the possibility of it existing substantial enough to consider, but the right string may lay hundreds of trillions or billions of trillions of digits in, making it unlikely that the computing power will ever exist for it to actually be so easily found.
But all hope is yet not lost, for we don't actually require that single billion-digit string. We need only find, in pi or some other irrational decimal, or in some multiplication of such terms (as in 3 times pi times the square root of 7), a string of correctly ordered numbers to give us the few hundred characters needed to allow us to provide instructions of how to calculate the next set of decimals and fractions capable of yielding an even longer string, culminating in one being the length we need. This function is
recursive to a certain degree of
simplicity. The calculation used to generate one billion digit string might be set forth in an instruction containing a few hundred thousand characters -- picture a typical book like the first
Harry Potter book (which is just under 77,000 words, so probably around 400,000 characters), but containing nothing but instructions to generate a longer series of numbers -- describing where to pluck the first string of a few thousand-odd digits, then perhaps the next string of twelve hundred and some, and the next string of a few thousand more.
But once we have described the operations to be performed, we can convert
that description itself into a string of a few hundred thousand numbers, and create a string of a few thousand numbers which provides the instruction of how to generate that few hundred thousand. And this sort of condensation may be continued until we reach the shortest easily generated string capable of telling us how to generate a longer string. Which might be something as simple as telling us "3/5•pi, 2,100 digits from 1,035,721st decimal." And if
that instruction can be fractionally encoded on a
toothpick, there is no actual
limit to what can be encoded, provided a computer exists capable of calculating out and converting each successively expanding string of
instructions at the other end, until the final instruction yields the billions of digits into which the entire encyclopedia has been encoded.