The Dinary or the Quasi-binary numbers

Last updated on 14/02/2025

Let’s take a natural number $F_0$ . There are only two possible cases: either $F_0$ is even or odd. Let’s assume it is even. Then it will be divisible by $2$ . Let’s denote with $a_1$ the maximum value that the exponent of $2$ can assume so that $2^{a_1}$ divides $F_0$ . In other words, $a_1$ is the dyadic evaluation of $F_0$ , that is, $a_1 = v_2(F_0)$ .

We can then write:

$F_0 = 2^{a_1}F_1$

where $F_1$ is now an odd number.

Since $F_1$ is odd, it can be written as $F_1 = 1 + F_2$ , where $F_2$ is an even number. So $F_2 = 2^{a_2}F_3$ , with $F_3$ being odd and $a_2 = v_2(F_2)$ .

Repeating the procedure we can conclude that any even natural number $N_e$ can be written as:

(1) $\begin{equation*} N_e = 2^{a_1}\left(1+2^{a_2}\left(1+2^{a_3}\left( \cdots \right)\right)\right)\end{equation*}$

In the case of an odd number $N_o$ , proceeding in the same way, we will have:

(2) $\begin{equation*} N_o = 1+2^{a_1}\left(1+2^{a_2}\left(1+2^{a_3}\left( \cdots \right)\right)\right)\end{equation*}$

In general, any natural number $N$ can be written as

(3) $\begin{equation*} N =2^{\alpha_0}\left(1+2^{\alpha_1}\left(1+2^{\alpha_2}\left( \cdots \right)\right)\right)\end{equation*}$

$\alpha_0$ will be $0$ only if $N$ is odd, otherwise it will be a number not less than $1$ .

The sequence $D(N) = \left(\alpha_0, \alpha_1, \alpha_2, \cdots, \alpha_n \right)$ is the Quasi-binary representation of $N$ .

Developing the product we find:

(4) $\begin{equation*}N=2^{\alpha_0}+2^{\alpha_0+\alpha_1}+2^{\alpha_0+\alpha_1+\alpha_2}+2^{\alpha_0+\alpha_1+\alpha_2+\alpha_3}+\cdots + 2^{\alpha_0+\alpha_1+\alpha_2+\alpha_3+\cdots \alpha_n}\end{equation*}$

Since $\alpha_j \neq 0$ ( $j = 0, \cdots, n$ ), the exponents of the addends are strictly increasing. Representing $N$ in binary, the exponents indicate in which position of the number there is a $1$ , counting from right to left. This explains the name quasi-binary of this representation.

Binary numbers have been known for a long time, but their formalization is attributed to Leibniz, who in 1703 published the work Explication de l’Arithmétique Binaire.

To honor French, the language adopted by Leibniz, the representation proposed here can also be called demi-binaire, anglicized as Dinary.

We will explore soon the properties of the wonderful Dinary numbers!

The Dinary or the Quasi-binary numbers

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