Positional Radix Representation: A number in base $b$ has polynomial value $N = \sum_{i=0}^{k} d_i \cdot b^i$, where each digit $d_i \in \{0, 1, \dots, b-1\}$.
Radix Bases: Binary ($b=2$, bits $0,1$), Octal ($b=8$, digits $0\text{--}7$), Decimal ($b=10$, digits $0\text{--}9$), and Hexadecimal ($b=16$, digits $0\text{--}9, A\text{--}F$).
Bit-Grouping Shortcuts: Every 3 binary bits map directly to 1 octal digit ($2^3 = 8$), and every 4 binary bits (nibble) map to 1 hexadecimal digit ($2^4 = 16$).
Operating Instructions: Enter numerical values into any base box (Decimal, Binary, Hex, Octal) to trigger bidirectional synchronized radix conversions.