Mathematical Foundations of Positional Number Systems
A numeral system is a writing system for expressing numbers using mathematical symbols. Positional number systems use a base (or radix), where each digit's value depends on its position relative to the radix point.
Decimal (Base 10)
The standard counting system used by human civilization, utilizing ten digits from 0 to 9. Each digit represents a power of 10 (units, tens, hundreds, thousands).
Binary (Base 2)
The core computational language of modern microprocessors. Uses two voltage states represented by bits: 0 (Off) and 1 (On), representing increasing powers of 2.
Hexadecimal (Base 16)
Uses 16 distinct symbols: digits 0 through 9 followed by letters A through F (where A=10, B=11, C=12, D=13, E=14, F=15). Extensively used for memory dump addresses and CSS colors.
Roman Numerals
The additive and subtractive numeric system of ancient Rome. Widely used for clock faces, super bowl editions, regal names, and book volume chapter titles.
Roman Numeral Conversion Table & Subtractive Rules
The Romans utilized seven uppercase letters to represent quantities. When a symbol of lower value appears before a symbol of higher value, subtraction is performed:
| Decimal (Base 10) | Binary (Base 2) | Octal (Base 8) | Hexadecimal (Base 16) | Roman Numeral |
|---|---|---|---|---|
| 1 | 0001 | 1 | 1 | I |
| 4 | 0100 | 4 | 4 | IV (5 - 1) |
| 9 | 1001 | 11 | 9 | IX (10 - 1) |
| 16 | 10000 | 20 | 10 | XVI |
| 40 | 101000 | 50 | 28 | XL (50 - 10) |
| 90 | 1011010 | 132 | 5A | XC (100 - 10) |
| 255 | 11111111 | 377 | FF | CCLV |
| 400 | 110010000 | 620 | 190 | CD (500 - 100) |
| 900 | 1110000100 | 1604 | 384 | CM (1000 - 100) |
| 2026 | 11111101010 | 3752 | 7EA | MMXXVI |