I am reading about a construction of the Stone-Čech compactification of the set of natural numbers. What I don´t understand is the note about dyadic expansion. How to define such expansion and how to understand it?
$\beta \mathbb{N}$ is built as a closure of a countable set $A = > \{a_n(x): n \in \mathbb{N}\}$ of points in the Tychonoff cube $[0, 1]^{(0, 1)}$. The $a_n(x)$ refers to a dyadic expansion of every $x \in (0, 1)$. One can also identify $\beta \mathbb{N}$ with the set of ultrafilters on $\mathbb{N}$, with the topology generated by sets of the form $\{ F: U \in F \}$ where $U$ is a subset of $\mathbb{N}$. We define filters and ultrafilters in the chapter about the space $l^2$.
Source: Tychonoff A. Über einen Funktionenraum // Math. Ann. 1935. 111, 1. 762–766
The dyadic expansions work just like the decimal expansions, but with only zeros and ones.
If $\langle a_n:n\in\mathbb{N}\rangle$ is a sequence of zeros and ones then the series $\sum_{n=1}^\infty a_n\cdot2^{-n}$ converges and its sum is a number in the interval $[0,1]$. Conversely, every number in $[0,1]$ is equal to the sum of such series; some numbers, like $\frac{1}{2}$, $\frac{1}{4}$, $\frac{3}{4}$, and generally $\frac{k}{2^n}$ are the sum of two such series: $1/2$ corresponds to $\langle 1,0,0,0,\dots\rangle$ and $\langle 0,1,1,1,\dots\rangle$.
For every $x\in[0,1]$ choose one of the sequences and call it $\langle a_n(x):n\in\mathbb{N}\rangle$ (most numbers have just one sequence).
You can think of it as a $[0,1]\times\mathbb{N}$-matrix of zeros and ones, and each column $\langle a_n(x):x\in[0,1]\rangle$ is a point in $[0,1]^{[0,1]}$.