of a real number is defined to be a rational approximation in which the numerator is the best choice for the given denominator. For example, the list of all approximants of
starts off this way:
. For each of the denominators, the numerator is chosen so as to approximate
as closely as possible.
The gold list
is formed by keeping only approximants that successively approach the target more closely. In the case of
, many of the approximants listed above must be eliminated:
are equal to
, not closer to
. Similarly, all approximants having denominators between 8 and 56 are eliminated because none of them are closer to
. The gold list for
Clearly, irrational targets have infinite gold lists. None of the approximants are equal to the target, but as we consider larger denominators, closer approximants can always eventually be found. If the target itself is rational, however, its gold list terminates with the fraction itself. For example, the gold list of
. This coincides with the first five elements of the gold list of
, but then terminates since there are no fractions closer to the rational target
than the fraction itself.
Now, in a move that seems unmotivated and arbitrary, we consider the differences between numerators and denominators of successive gold list entries, referred to here as
, respectively. These are shown in the last column of the table.
Surprisingly, the ratios
are almost always themselves earlier members of the gold list, shown in the table using the same color for convenient identification. For the gold list of
, the sequence of these "delta ratios" starts this way:
. In fact, except for occasional intrusions, we recognize all these new ratios. Initially there seems to be no reason for this mysterious behavior.
The delta ratios sequence is formed by pairing successive gold list elements and "subtracting"—actually subtracting numerators and denominators separately. The inverse operation of "adding" previous elements of the gold list provides useful insights. First, it must be noted that this operation is not "incorrect" addition of fractions, but rather forming the mediant, denoted here by the symbol "⊕". The mediant
is defined as
. Now most elements of the gold list of
can be seen to be mediants of previous elements. A further definition allows all but the first pair to be written in terms of earlier approximants. The weighted mediant
with weighting factor
is defined as
. Using this notation, several subsequences of weighted mediants are revealed in the gold list of
. For example,
Subsequences where the same approximants are used repeatedly to form successive gold list entries always approach the target monotonically from one side or the other, as indicated by the error terms. These repeated mediant subsequences have the form
are themselves approximants on opposite sides of the target. The
mediants are all on the
side of the target, approaching it, until the
mediant has crossed over to the
To allow real-time interactivity, the code used here relies on the ultra-rapid iterative bracket method of generating the gold list, rather than the painfully slow
method based on the gold list definition: testing each approximant for all denominators and either keeping or discarding them. Additional material relating to approximants, mediants, weighted or repeated mediants, and methods for generating the gold list can be found in .
Snapshot 1: The beginning of the gold list of
, containing familiar approximations such as
. These also show up later in the column of numerator and denominator differences, meaning that most gold list entries are actually mediants of earlier approximants. Some of the best approximants (colored in the first column) are used in sequences of repeated mediants, approaching the target from one side or the other. The strikingly precise
is used in a sequence of almost 150 repeated mediants approaching
Snapshot 2: The beginning of the gold list of the golden ratio
, in which every gold list entry appears exactly once in the column of numerator and denominator differences, a situation closely related to the continued fraction expansion of the target.
Snapshot 3: The beginning of the gold list of
, itself a good approximation to
, having an identical gold list up to the point where the rational target is reached.