Knowing Our Numbers

• Comparing Numbers

  • Finding Greatest and Smallest
    • More digits = Larger number
    • Equal digits = Compare from leftmost place
  • Ordering
    • Ascending Order: Smallest to largest
    • Descending Order: Largest to smallest
  • Number Formation
    • Rearranging digits to make largest/smallest values
    • Shifting Digits: Changing digit positions alters value

• Place Value & Expansion

  • Place Value: Position of digit determines its value
  • Face Value: Actual digit itself
  • Number Expansion: Writing numbers in expanded form (e.g., $73 = 7 \times 10 + 3$)

• Systems of Numeration

  • Indian System of Numeration
    • Places: Ones, Tens, Hundreds, Thousands, Lakhs, Crores
    • Commas: First after 3 digits from right, then every 2 digits (e.g., $1,23,45,678$)
  • International System of Numeration
    • Places: Ones, Tens, Hundreds, Thousands, Millions, Billions
    • Commas: Placed after every 3 digits from right (e.g., $12,345,678$)

• Large Numbers in Daily Life

  • Length Units: Millimeter ($mm$), Centimeter ($cm$), Meter ($m$), Kilometer ($km$)
    • $1\text{ km} = 1,000\text{ m} = 1,00,000\text{ cm}$
  • Mass Units: Milligram ($mg$), Gram ($g$), Kilogram ($kg$)
    • $1\text{ kg} = 1,000\text{ g} = 10,00,000\text{ mg}$
  • Capacity Units: Milliliter ($ml$), Liter ($l$)
    • $1\text{ l} = 1,000\text{ ml}$

• Estimation & Rounding Off

  • Rounding Off
    • Nearest 10s: Look at ones digit ($<5 \rightarrow$ down, $\ge 5 \rightarrow$ up)
    • Nearest 100s: Look at tens digit
    • Nearest 1000s: Look at hundreds digit
  • Estimating Outcomes
    • Quick approximation of Sum, Difference, and Product

• Using Brackets

  • Avoids confusion in expressions
  • Enables easy multiplication by expanding numbers (e.g., $7 \times 109 = 7 \times (100 + 9)$)

• Roman Numerals

  • Basic Symbols
    • I = 1, V = 5, X = 10, L = 50
    • C = 100, D = 500, M = 1000
  • Key Rules
    • Repetition means addition (max 3 times)
    • V, L, D are never repeated or subtracted
    • Smaller symbol on right = Add (e.g., $\text{VI} = 5 + 1 = 6$)
    • Smaller symbol on left = Subtract (e.g., $\text{IX} = 10 - 1 = 9$)