3️⃣ A Story of Numbers – Important Formulas

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Need to Count (Stone Age)

Purposes: Food, animals, trade, rituals, calendars.

No number names/symbols → Used one-to-one mapping with:

  • Sticks/Pebbles/Seeds (Method 1)
  • Sounds/Names (Method 2) → Limited by available sounds/letters.
  • Written symbols (Method 3) → e.g., Roman numerals.

Key Concepts

  • One-to-One Mapping: Each object ↔ one counting unit.
  • Numerals = Written symbols in a number system.
  • Landmark Numbers: Special values to build other numbers (e.g., 1, 5, 10…).

Early Number Systems

(A)Tally Marks

  • Simple notches/lines.
  • Ishango Bone (20k–35k yrs old) – possibly calendar.
  • Lebombo Bone (44k yrs old) – lunar calendar.

(B) Counting in Groups

  • Gumulgal (Australia): Count in 2’s → Numbers = combinations of 2’s & 1’s.
  • Also used by Bakairi (S. America) & Bushmen (S. Africa).
  • Common groups in history: 2, 5, 10, 20.

(C) Roman Numerals

  • Symbols: 
  • Rules:
    1. If a smaller numeral is placed before a larger one, subtract it.
      Example: IV = 5 − 1 = 4.
    2. If a smaller numeral is placed after a larger one, add it.
      Example: VI = 5 + 1 = 6.
  • Advantage: Shorter than tally marks.
  • Limitations:
    1. No zero.
    2. Difficult for large numbers.
    3. Cannot perform complex arithmetic easily.

Idea of Base-n

  • Base-n system: Landmark numbers = powers of n
    Example: Base-5 → 1, 5, 25, 125…
  • Advantages: Consistent grouping, easier addition & multiplication.

Egyptian System

  • Base-10, symbols for 1, 10, 100, 1000, 10,000…
  • Build numbers by repeating symbols. For example 324 which equals 100 + 100 + 100 + 10 + 10 + 4 is written as
  • Limit: Needs infinite symbols for very large numbers.

Abacus

  • Decimal-based calculating tool.
  • Each line = power of 10.
  • Counters above line = value of 5× that landmark.

Mesopotamian System

  • Location: Ancient civilisation in present-day Iraq and nearby regions.
  • Time Period: Around 4000 years ago.
  • Base: Base-60 (sexagesimal system).
  • Symbols:
    • Two main wedge-shaped symbols (cuneiform writing) for numbers.
    • Numbers formed by repeating and combining these symbols.
  • Special Use:
    • Still used today in measuring time (60 seconds in a minute, 60 minutes in an hour) and angles (360° circle).

Mayan Number System Basics

  • Base: Modified base-20.
    1. 1st place: 1’s (units)
    2. 2nd place: 20’s
    3. 3rd place: 360’s (not 400, due to calendar reasons)
    4. 4th place: 7200’s, etc.
  • Symbols:
    1. Dot (•) = 1
    2. Bar (—) = 5
    3. Shell = 0 (placeholder)
  • Numbers are written vertically, lowest value at bottom.

Chinese Rod Numeral System – Key Points1. Purpose

Two systems existed:

  • Written system – for recording quantities.
  • Rod numeral system – for performing calculations efficiently.

2. Rod Numerals

  • Base: Decimal (base-10), like our modern system.
  • Digits 1–9: Represented using vertical or horizontal rods (small sticks or lines).
  • Place value:
    1. Vertical rods → used for units and hundreds places.
    2. Horizontal rods → used for tens and thousands places.
      (This alternation prevented confusion between adjacent digits.)

3. Zero Representation

  • Like the Mesopotamians: used a blank space to indicate an empty place value.
  • Advantage: Due to uniform rod sizes, the blank space was easier to identify.
  • Note: If they had an actual symbol for zero, it would have been a fully developed place value system like the Hindu–Arabic numerals.

Spread of Hindu–Arabic Numerals

  • Base: Base-10 (decimal system).
  • Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
  • Place Value System: Value of a digit depends on its position.
    Example: In 375,
    3 → Hundreds place = 3 × 100 = 300
    7 → Tens place = 7 × 10 = 70
    5 → Ones place = 5 × 1 = 5
  • Use of Zero: A major contribution by Indian mathematicians (Aryabhata, Brahmagupta).
  • Spread: Carried to Europe by Arab traders → became the Hindu–Arabic numerals we use today.

Comparing Systems