Positional number representation
This appears to be nothing more than a description of the standard binary notation. List the bits in the string and below them, starting with 1 at the rightmost position and multiplying by 2 at each position to the left:
$$\begin{array}{lccccc}
\mathbf{bits:} &amp; 1 &amp; 1 &amp; 0 &amp; 1 &amp; 0\\
\mathbf{powers\ of\ 2:} &amp; 16 &amp; 8 &amp; 4 &amp; 2 &amp; 1\\
\mathbf{product:} &amp; 16 &amp; 8 &amp; 0 &amp; 2 &amp; 0
\end{array}$$
then add the resulting products: $16 + 8 + 0 + 2 + 0=26$, as it should be, since $\mathtt{11010}_2=26$.

The successive integral powers of 2 are just those that appear in the second line, right to left, and similarly successive bits are those in the first line, again reading from right to left. I must confess to being confused by the phrase except perhaps for the bit in highest position.