Important formula: A Story of Numbers - Class 8 PDF Download

One-to-One Mapping Rule

• Each object ↔ one counting unit (stick, sound, or symbol).

• Formula:
  Number of Objects = Number of Counting Units

Tally Marks

• 1 object = 1 vertical line ( | ).

• Group of 5 → |||| + / = 5.

• Formula:
  Total = (Number of groups × 5) + Remaining lines

Roman Numerals Rules

Basic symbols:
I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, M = 1000

Repetition Rule:
I, X, C, M can be repeated max 3 times → III = 3, XXX = 30

Subtractive Rule:

• IV = 4 (5 – 1)

• IX = 9 (10 – 1)

• XL = 40 (50 – 10)

• XC = 90 (100 – 10)

• CD = 400 (500 – 100)

• CM = 900 (1000 – 100)

Base-n Number System Formula

• Landmark Numbers = Powers of n

• Formula:
  Base-n landmarks = n⁰, n¹, n², n³ …

• Example:

  • Base-10 (Decimal): 1, 10, 100, 1000 …

  • Base-5: 1, 5, 25, 125 …

• Number Representation Rule:
  A number is expressed as sum of (digit × base power).

  Example in Base-5:
  143 = (1 × 5³) + (0 × 5²) + (3 × 5¹) + (3 × 5⁰)

Place Value Formula

• General Form:
  Number = (dₙ × bⁿ) + (dₙ₋₁ × bⁿ⁻¹) + … + (d₀ × b⁰)
  where b = base, d = digits.

• Example (Decimal):
  3426 = (3×10³) + (4×10²) + (2×10¹) + (6×10⁰)

Mesopotamian Base-60 Rule

• Landmark numbers = 60⁰ = 1, 60¹ = 60, 60² = 3600 …

• Example:
  640 = (10×60) + 40

Mayan Base-20 (Modified) Rule

• Landmark numbers = 20⁰ = 1, 20¹ = 20, then 360 (instead of 400), then 7200 …

• Used: dot = 1, bar = 5, shell = 0 (placeholder).

General Rule of Group Counting

• Humans group objects in 2, 5, 10, or 20 for easier counting.

• Formula (Approx):
  Total = (Number of Groups × Group Size) + Remainder