From Various Places

The next two operations come from various places.

11. To get the rational value of an expansion, divide down the first column. Period. Do not take the integer part as in method 3F.

    In Some Definitions I needed to use that operation so many times that I gave it a name, the value function.

    What does the value function look like as a whole? The domain is all continued fraction matrices except case L0. The range is the rational numbers. The mapping is exactly two-to-one because every rational number has two forms, proper and improper. For more about that last detail, see One Bit of Information.

    Can you spot the unusual feature of this equation from Some Definitions?

    2 1/3 = [2,3] < value(T(u)) < [2,2] = 2 1/2

    The value function is automatically applied to both [2,3] and [2,2]! In books about continued fractions, that's completely standard, but if we were writing a computer program, it would be quite error prone to define an implicit conversion to a numeric type.

12. To get the irrational value of a repeating expansion with non-repeating prefix X and repeating part G, follow the instructions from Group Elements and Matrices. In short, first compute the discriminant, …

    D = (tr G)² − 4 det G

    … then fill in the blanks to get the positive solution of the relevant quadratic equation.

    z = (1/2c) [ (a − d) ± root D ]

    If the prefix X is empty, that's the answer. If not, think of the prefix as a group element and apply it to z.

    I have more to say about repeating expansions and irrational values and even the value function, but it doesn't really have anything to do with operations on matrices. So, let's say that there are two possible continuations from here: you can read more about irrational values in The Irrational Value, or you can read more about operations on matrices below.