Overview
This lecture covers the history and features of numeration systems developed by various civilizations, distinguishing between numbers (concepts) and numerals (symbols).
Numbers vs. Numerals
- A number is an abstract concept used to quantify or measure.
- A numeral is a written symbol that represents a number.
- Different cultures can assign different meanings or emotions to the same numeral.
Purpose and Origin of Numeration Systems
- Numeration systems were needed to record, compare, and communicate quantities, not just count.
- Civilizations used familiar objects, simple marks, or religious icons as numerals.
- Early evidence includes tally marks on rocks or bones.
Ancient Numeration Systems
Egyptian Numeration System
- Used picture-like symbols (hieroglyphs) for powers of ten: vertical stroke (1), inverted U (10), coil (100), lotus (1,000), etc.
- Numbers written by repeating symbols; called "simple grouping."
- No place value, only repetition.
Babylonian Numeration System
- Used wedge-shaped ("cuneiform") symbols: one for 1, one for 10.
- Base 60 (sexagesimal); numbers built by repeating and positioning symbols.
- Had beginnings of place value but lacked a zero, causing ambiguity.
Mayan Numeration System
- Used dots (1) and dashes (5), and an open shell for zero.
- Base 20 (vigesimal), read from top to bottom.
- Included a true zero as a placeholder to clarify value.
Chinese Numeration System
- Early rods represented numbers with horizontal and vertical strokes; evolved into unique characters.
- Lacked place value initially; later used a multiplicative grouping system with specific characters for powers of ten and zero.
- System also used in Japan and Korea.
Greek Numeration System
- Used alphabet letters: first nine for 1–9, next nine for tens, next for hundreds.
- Required memorizing many symbols.
- Another system (Attic numerals) used acrophonic symbols and influenced metric prefixes.
Roman Numeration System
- Seven main numerals: I (1), V (5), X (10), L (50), C (100), D (500), M (1,000).
- Incorporated base 5 and base 10.
- Used subtraction notation (e.g., IV for 4) and no zero, making calculations difficult.
The Hindu-Arabic Numeration System
- Uses ten symbols (0–9) including zero.
- Decimal (base 10) and positional, written left to right.
- Zero originated in India; spread through Arab and then European traders and scholars.
- Popularized in Europe by Fibonacci through Liber Abaci.
- Allows efficient calculation and became globally adopted.
Key Terms & Definitions
- Numeration System — a set of symbols and rules for representing numbers.
- Number — a conceptual quantity or value.
- Numeral — a written symbol for a number.
- Place Value — the value given to a digit based on its position.
- Base (Radix) — the number of unique symbols used before "carrying over" (e.g., base 10).
- Placeholder — a symbol (like zero) that indicates empty positions in place value.
Action Items / Next Steps
- Review each ancient numeration system and compare features with the Hindu-Arabic system.
- Reflect on how place value and zero improved calculation efficiency.
- Optional: Read about Fibonacci and the spread of the Hindu-Arabic system in Europe.