In the paper we justify existence and unity  $B$-representation of numbers of segment $(0;1)$, which uses as a basis a positive number $a$ that satisfies the condition $0<a<\frac{1}{3}$ in particular the positive root $\tau$ of the equation $x^2+x-1=0$, bilateral sequence $(\Theta_n)$: $\Theta_0=\frac{1-3a}{1-a}$, $\Theta_{-n}=\Theta_n=a^{|n|}$ and alphabet $Z=\{0,\pm 1, \pm 2, \pm, \dots \},$\\
namely
$$x=b_{\alpha_1}+\sum\limits_{k=2}^{m}b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i}\equiv
  \Delta^{B}_{\alpha_1\alpha_2...\alpha_m(\emptyset)},$$
$$x=b_{\alpha_1}+\sum\limits_{k=2}^{\infty}b_{\alpha_k}\prod\limits_{i=1}^{k-1}\Theta_{\alpha_i}\equiv
  \Delta^{B}_{\alpha_1\alpha_2...\alpha_n...},$$
where $\alpha_n\in Z$, $\Theta_n>0~\forall n\in Z$, $\sum\limits_{n=-\infty}^{+\infty}\Theta_n=1$,
$b_{n+1}\equiv\sum\limits_{i=-\infty}^{n-1}=b_n+\Theta_n$ $\forall n\in Z$.

 The geometry of $B$-representations of numbers is described (geometric content of numbers, properties of cylinder and tail sets, topological and metric properties of sets with restrictions on the use of numbers). The left and right shift operators of numbers are studied, a group of continuous transformations of the unit interval preserving the tails of the $B$-representation of numbers is described.

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