A definition of Conway base-$13$ function
You simply require to switch over to and fro from the lexicographic definition of the base - 13 development of the number (consider having ABC as opposed to. - npls) and also the crammed definition you provide to the well - created string as a base - 10 number.
An instance of a number for which Conway base - 13 function is 0 is
0.12 - 34+npls 1+2 - 34..11111111111 ...
where the leftmost. is the threedecimal factor (that is, it has a semantic definition), the 3 rightmost. suggest that the base - 13 depiction has a boundless variety of 1 (that is, they have a metameaning), and also the various other 2. are "figures" of the number (that is, they have a syntactic definition).
An instance of a number for which Conway base - 13 function is not 0 is
0.12 - 34+npls 1+2 - 34.11111111111 ...
At that number, the function has value - 34.11111111111 ... (in base 10)
I recognize why the Wikipedia write-up makes use of the symbols it does, yet I locate it aggravating. Below is a transliteration, with some explanation.
Expand x ∈ (0,1) in base 13, making use of figures 0, 1, ..., 9, d , m , p - - - making use of the convention d = 10, m = 11, p = 12. N.B. for sensible numbers whose most lowered expression a/b is such that b is a power of 13, there are 2 such developments : an ending development, and also a non - ending one finishing in duplicated p figures. In such an instance, make use of the ending development.
Allow S ⊂ (0,1) be the set of reals whose development entails finitely several p , m , and also d figures, such that the last d figure takes place after the last p figure and also the last m figure. (We might call for that there go to the very least one figure 0 - - 9 in between the last p or m figure and also the last d figure, yet this does not appear to be essential.) After that, every x ∈ S has a base 13 development of the kind
0. x 1 x 2 ... x n [ p or m ] a 1 a 2 ... a k [ d ] b 1 b 2 ...
for some figures x j ∈ 0, ..., p and also where the figures a j and also b j are restricted to 0, ..., 9 for all j. The square brackets over are just planned for focus ; and also specifically the n+1 st base - 13 figure of x is the last occurance of either p or m in the development of x .
For x ∈ S , we specify f ( x ) by translating the string layout over. We overlook the figures x 1 via x n , translate the p or m as a plus - indicator or minus - indicator, and also the d as a decimal factor. This generates a decimal development for an actual number, either
+a 1 a 2 ... a k . b 1 b 2 ...
− a 1 a 2 ... a k . b 1 b 2 ...
according to whether the n+1 st base - 13 figure of x is a p or an m specifically. For x ∈ S , we set f ( x ) to this number ; for x ∉ S , we set f ( x ) = 0.
Keep in mind : this function is not determinable, as there is no other way that you can establish beforehand whether the base - 13 development of x ∈ (0,1) has just finitely several occurances of any one of the figures p , m , or d ; also if you are given with a number which is assured to have just finitely several, as a whole you can not recognize when you have actually located the last one. Nonetheless, if you are given with a number x ∈ (0,1) for which you recognize the area of the last p , m , and also d figures, you can calculate f ( x ) really straight.
The suggestion of the Conway base - 13 number is to locate a function that is not continual, yet if
f(a)<x<f(b), after that there is some
c in between
a and also
f(c)=x. This a counterexample to the reverse of the intermediate value theorem.
The function is specified by inscribing base - 10 values in the tail (the figures left after missing a limited number). We make use of
. and also the figures to stand for an inscribed number in the tail and also call for the inscribed number to begin with a
-. In base 10, every number finishing in a boundless variety of
9s can be revised to end in a boundless variety of
0s rather (ie. 0.999 ... = 1.0). In a similar way, we determine we will certainly revise each Conway number finishing in a boundless variety of
+, to make certain that each actual number has an one-of-a-kind decimal depiction.
Each number can have up to one base - 10 inscribed value, which is the outcome of using Conway's Base 13 function if it exists. If no such inscribing exists for x (ie.
+ taking place boundless times in the development), after that we specify f (x) = 0.
We after that show that for each and every
a and also
b that we can locate a
c in between with an approximate inscribed value. We first make certain the number being created is in between
a and also
b by replicating adequate figures from
a and also incrementing a figure that will not matter. This is less complicated due to the fact that we have actually forbidden finishing in a boundless variety of
+. We after that concatenate the figures of the authorized base - 10 depiction of the value we desire the function to need to those figures we have actually currently dealt with.