# Positional number representation

There is a definition of integer representation:

A positional representation for integers that uses the binary digits 0 and 1, in which the values represented by successive bits are additive, begin with 1, and are multiplied by successive integral power of 2, except perhaps for the bit with the highest position. (Adapted from the American National Dictionary for Information Processing Systems.)

It is not clear to me. Could you give an example reflects that definition? In general, I don't understand what is the successive integral power of 2 and what is the succesive bits?

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: $$\begin{array}{lccccc} \mathbf{bits:} & 1 & 1 & 0 & 1 & 0\\ \mathbf{powers\ of\ 2:} & 16 & 8 & 4 & 2 & 1\\ \mathbf{product:} & 16 & 8 & 0 & 2 & 0 \end{array}$$ 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.

• The bit in highest position might be used for sign. That is the likely reason for why it is except. – InformedA Oct 18 '14 at 17:31