Numbers, Fractions & Introduction to Geometry
1. Knowing Our Numbers
Place Value (Indian System)
Ones → Tens → Hundreds → Thousands → Ten Thousands → LAKHS (1,00,000) → Ten Lakhs → CRORES (1,00,00,000).
Example: 45,67,890 = Forty-five lakh sixty-seven thousand eight hundred ninety.
Comparing Numbers
More digits = LARGER. Same digits → compare from LEFT.
Roman Numerals
I(1). V(5). X(10). L(50). C(100). D(500). M(1000).
2. Whole Numbers
Properties of Addition and Multiplication
Closure. Commutative (a+b = b+a). Associative ((a+b)+c = a+(b+c)). Distributive (a×(b+c) = a×b + a×c). Identity (a+0=a. a×1=a). 'Zero is the ADDITIVE identity. One is the MULTIPLICATIVE identity.'
3. Fractions
Types
Proper (<1: 3/4). Improper (≥1: 5/3). Mixed (whole + proper: 1⅔).
Equivalent Fractions
Same value, different form: 1/2 = 2/4 = 3/6 = 50/100.
Simplest Form
Divide numerator and denominator by their HCF (Highest Common Factor).
Operations
Add/Subtract: Find LCM of denominators. Convert. Add/subtract numerators. Multiply: (a/b)×(c/d) = (a×c)/(b×d). Divide: Flip the second. Multiply.
Fractions on the Number Line
'Between any two fractions, there are INFINITELY MANY fractions.'
4. Decimals
Place Value: 34.567 = 3 tens + 4 ones + 5 tenths + 6 hundredths + 7 thousandths.
Operations: Align decimal points for addition/subtraction.
Conversions: Fraction → Decimal = divide numerator by denominator.
5. Introduction to Geometry
Basic Elements
- Point: A location. No size. Line Segment: Part of a line. Two endpoints.
- Ray: One endpoint. Extends infinitely in ONE direction.
- Line: Extends infinitely in BOTH directions.
Angles
| Type | Measure |
|---|---|
| Acute | < 90° |
| Right | = 90° |
| Obtuse | 90° – 180° |
Measuring Angles
Use a PROTRACTOR. Place centre at vertex. Align base with one arm. Read where other arm crosses the scale.
Triangles
Equilateral (3 equal sides, each angle 60°). Isosceles (2 equal sides). Scalene (all sides different).
Quadrilaterals
Square (4 equal sides, 90°). Rectangle (opposite sides equal, 90°). Parallelogram (opposite sides parallel).
Circles
Radius (centre to edge). Diameter (2×radius — through centre). Circumference = 2πr (π ≈ 22/7 ≈ 3.14).
6. Practical Geometry — Constructing With Ruler and Compass
- Perpendicular bisector of a segment.
- Angle bisector.
- Circle of given radius.
- 'Construction teaches PRECISION. In geometry, a millimetre MATTERS.'
