About Positional Notations & Radix Conversions

Numeral systems provide the symbolic foundation for quantifying, calculating, and representing values across human history and digital architecture. Every positional system relies on a radix or base ($b$), which determines the number of unique digits, including zero, used to represent numbers.

While standard arithmetic relies heavily on the decimal system, computing architecture is built upon binary, octal, and hexadecimal notations. Specialized domains also utilize ancient vigesimal structures, sexagesimal timekeeping systems, and non-positional notations like Roman numerals.

Key Numeral System Frameworks ($b$)

When converting numerical quantities between systems such as $\text{Decimal}$ and $\text{Hexadecimal}$, specific mathematical rules govern value translation:

• Positional Notation: Values are represented as sums of powers of the base multiplied by coefficients ($d_n b^n + \dots + d_1 b^1 + d_0 b^0$).
• Computing Radices: Binary ($\text{base} = 2$), octal ($\text{base} = 8$), and hexadecimal ($\text{base} = 16$) provide compact bit-grouping representations for machine code and memory addresses.
• Historical & High-Base Systems: Systems like ancient Sumerian and Babylonian sexagesimal ($\text{base} = 60$) and Mayan vigesimal ($\text{base} = 20$) reflect ancient cultural counting methods and astronomical cycles.