Encoding unsigned integers as binary

Assume a bit vector $x = [x_{w - 1}, x_{w - 2}, \dots, x_{0}]$
Then, the unsigned integer encoded by $x$ is given by, where the symbol $B2U_{w}$ means binary to unsigned, $w$ -bit word size

B2U_{w} (x) ≐ i = 0 \sum w - 1 x_{i} 2^{i}

Examples

B 2 U_{4} ([0001] = 0 \cdot 2^{3} + 0 \cdot 2^{2} + 0 \cdot 2^{1} + 1 \cdot 2^{0} = 0 + 0 + 0 + 1 = 1 B 2 U_{4} ([0101]) = 0 \cdot 2^{3} + 1 \cdot 2^{2} + 0 \cdot 2^{1} + 1 \cdot 2^{0} = 0 + 4 + 0 + 1 = 5

If we have a bit vector $x$ of word length, $w$ bits.
The least value that can be represented is given by $x = [00 \dots 00]$ , which represents the integer $0$ .
The maximum value can be represented by $x = [11 \dots 11]$ which represents the integer:

U M a x_{w} = i = 0 \sum w - 1 2^{i} = 2^{w} - 1