# New numeration system, mapping to binary numeration system

Let us consider

$$Z = X_1 + X_1 X_2 + X_1 X_2 X_3 + \cdots.$$

Here the $$X_i$$'s can only take on two different values: either $$X_i=a$$ or $$X_i=b$$, with $$0 < a < b < 1$$ and $$a+b = 1$$. The $$n$$-th digit of $$Z$$ is said to be equal to one if $$X_n = b$$, and to zero if $$X_n =a$$. All real numbers in $$[\frac{a}{1-a}, \frac{b}{1-b}]$$ can be uniquely represented in this form (uniquely is to be understood in the same sense that the traditional binary representation of a number is "unique".)

A simple algorithm can be used to compute the digits of $$Z$$. See below its coded version (in Perl). Note that $$a, b$$ are represented by $alpha,$beta in the program. The variable $sum is used to double check that the computations were successful, and give you some idea of how many digits can be accurately computed: it converges to $$Z$$, and represents the successive approximations of $$Z$$ using an increasing number of terms in the formula at the very beginning of this article. open(OUT,">att.txt"); $$beta=sqrt(2)/2;$$alpha=1-$beta;

$$min=$$alpha/(1-$$alpha);$$max=$$beta/(1-$$beta);

$z=log(2); $$prod=1;$$upper=$$beta/(1-$$alpha); $$lower=$$alpha/(1-$beta);

if ($$z >$$upper) {
$$sum=$$beta;
$$digit=1; } else { sum=alpha; digit=0; }$$prod=$$sum; print OUT "z =$$z (b = $$beta | min =$$min | max = $$max)\n"; print OUT "$$digit\n";

for ($$k=0;$$k<60; $k++) { $$upper =$$sum+ $$prod *$$beta/(1-$$alpha);$$lower = $$sum+$$prod *$$alpha/(1-$$beta); if ($$lower <$$z) { $$prod =$$prod * $$beta;$$digit=1; } else { $$prod=$$prod * $$alpha;$$digit=0; } $$sum+=$$prod; print OUT "$$digit\n"; } print OUT "sum =$$sum\n"; close(OUT);  My Question: Is there a simple expression or recursion that maps the digit of $$Z$$ in base two, to its digits in the system discussed here? Of course there is a one-to-one mapping: if you know the digits in one system, then compute $$Z$$, then compute the digits back into the other system. But I am looking at something smarter than that, also hoping to find patterns between the two representations (digits) of a number $$Z$$ in both systems. Fun facts In the case $$a=-0.5, b=0.5$$ (not covered here) the one-to-one mapping between the two digit systems (mine versus standard binary) is described in section 5 in this article. Also, if you pick up a number at random in $$[\frac{a}{1-a}, \frac{b}{1-b}]$$, then in my system, the proportion of digits equal to 1, is $$b$$. By contrast, in the standard binary system, that proportion is 50%. In both systems, successive digits are not auto-correlated (true for the immense majority of numbers, called normal numbers.) Finally, consider the $$X_i$$'s as independent random variables with distribution $$P(X_i = b) = b, P(X_i = a) =a$$. Remember that $$a+b =1$$ and $$0< a < b < 1$$. Then $$Z$$ has a uniform distribution on $$[\frac{a}{1-a}, \frac{b}{1-b}]$$. If $$a+b < 1$$, then the support domain for $$Z$$ is full of wholes, and $$Z$$'s distribution is nowhere differentiable; it does not have a density. Yet it has a distribution and you can compute all its moments. See here for more on this topic. The case $$P(X_i = b) = p,P(X_i = a) = 1-p$$, with $$p \neq b$$ can also lead to a very wild distribution for $$Z$$. See chart below, picturing the empirical percentile distribution of $$Z$$, if $$a=0.4, b=0.6, p =0.8$$. It corresponds to a no-where differentiable distribution. If instead $$p=0.6$$, the curve becomes a straight line, corresponding to a uniform distribution. Potential application You can encode a message using the new numeration system proposed here. However, if you use (say) $$b=0.6$$, it will be very obvious to the hacker that you used $$0.6$$ since about 60% of the digits will be equal to one. The hacker will then easily decode your message. But you could pepper the encoded message with a number of digits equal to zero, at specific locations, to reduce the proportion of digits equal to one, from 60% to 50%. Then the hacker won't know which $$b$$ you used, and won't be able to easily decrypt your message. You need to make sure that when adding zero's, you do it in such a way that the correlation between successive digits remains equal to zero. I am also curious to see what happens when $$b\rightarrow 0.5$$ or $$b\rightarrow 1$$. I also plan on investigating the case $$b=3/4$$, or more generally, $$b$$ a rational number, in more details, and I will provide an update if I find something interesting. • I will publish a version of my program based on high precision arithmetic, so that you can correctly compute far more than 60 digits. Dec 16, 2019 at 5:23 • What do you mean when you say that uniquely is to be understood in the same sense that the traditional binary representation of a number is "unique".?$0.11111...=1.00000...=1$in binary Dec 20, 2019 at 8:31 • @Angela: I mean unique up to the fact that (in standard base 2)$0.11111\cdots = 1.0000\cdots\$. We have the same kind of thing going on here in this new system. Of course if you use my algorithm to compute the digits, then digits are uniquely defined. Dec 20, 2019 at 14:12