Important Formulas: A Story of Numbers | Mathematics Class 8- New NCERT (Ganita Prakash) PDF Download

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 Points

1. 

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