Loading...
Loading...
Real Numbers
├── Rational Numbers (p/q form, q≠0)
│ ├── Integers (...-2,-1,0,1,2...)
│ │ ├── Negative integers
│ │ ├── Zero
│ │ └── Natural/Positive integers (1,2,3...)
│ └── Fractions (1/2, 3/4, -5/7...)
└── Irrational Numbers (√2, √3, π, e...)
Rational Number: Any number expressible as p/q where p,q are integers and q≠0.
Irrational Number: Non-terminating, non-recurring decimal.
Proof √2 is irrational: Assume √2 = p/q (lowest terms) → 2 = p²/q² → p² = 2q² → p is even → p=2k → 4k²=2q² → q²=2k² → q is even. But both p,q even contradicts "lowest terms". So √2 is irrational.
| Law | Example | |---|---| | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ | | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ | | (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ | | aᵐ × bᵐ = (ab)ᵐ | 2³ × 3³ = 6³ | | a⁰ = 1 | 100⁰ = 1 | | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 | | a^(1/n) = ⁿ√a | 8^(1/3) = 2 |
Polynomial: Expression of form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀
Types by degree: Constant (0), Linear (1), Quadratic (2), Cubic (3)
Zero of polynomial: Value of x that makes p(x) = 0
Remainder Theorem: When p(x) divided by (x-a), remainder = p(a) Factor Theorem: (x-a) is a factor of p(x) iff p(a) = 0
Algebraic Identities:
Cartesian Plane: Two perpendicular number lines (x-axis, y-axis) meeting at origin (0,0).
Quadrants:
Plotting a point: (3, 4) → move 3 right on x-axis, 4 up on y-axis.
Distance from axes: Point (x,y) is |y| from x-axis, |x| from y-axis.
Types of Angles:
Angle Pairs:
Parallel Lines and Transversal: When a transversal cuts parallel lines:
Congruence Criteria:
Properties of Triangles:
Area of triangle with sides a, b, c:
Semi-perimeter: s = (a+b+c)/2
Area = √(s(s-a)(s-b)(s-c))
Example: Triangle with sides 13, 14, 15 s = (13+14+15)/2 = 21 Area = √(21×8×7×6) = √7056 = 84 sq units
Data Types:
Frequency Distribution: Organized table of data grouped in classes.
Measures of Central Tendency:
| Measure | Formula | Use | |---|---|---| | Mean | Sum of observations / Count | Numerical average | | Median | Middle value when sorted | Skewed data | | Mode | Most frequent value | Categorical data |
Bar Graph: Bars of equal width, heights represent frequency. Histogram: Like bar graph but for continuous data, bars touch. Frequency Polygon: Line graph through midpoints of histogram bars.
Q1 (2023): Verify x=2 is a zero of p(x) = x³-2x²-5x+6. p(2) = 8-8-10+6 = -4 ≠ 0. So x=2 is NOT a zero.
Wait: p(2) = 8 - 8 - 10 + 6 = -4. Let's check x=3: p(3) = 27-18-15+6 = 0. So x=3 is a zero.
Q2 (2022): The angles of a triangle are in ratio 1:2:3. Find them. x + 2x + 3x = 180° → 6x = 180° → x = 30° Angles: 30°, 60°, 90° (right triangle!)
Q3 (2024): Find area of triangle with sides 5, 12, 13 using Heron's formula. s = 30/2 = 15 Area = √(15×10×3×2) = √900 = 30 sq units (Also: 5-12-13 is a Pythagorean triple → right triangle → area = ½×5×12 = 30 ✓)
Complete Class 9 Mathematics notes — number systems, polynomials, coordinate geometry, lines and angles, triangles, Heron's formula, statistics with CBSE solved questions.
56 pages · 2.8 MB · Updated 2026-03-11
Number Systems (8 marks), Polynomials (10 marks), Triangles (10 marks), and Statistics (10 marks) are highest priority chapters for CBSE.
Complete NCERT exercises, focus on theorems with proofs for triangles and circles, practice constructions, and solve previous year sample papers.
Science Class 9 — Complete Notes NCERT
Science
Science Class 10 — Complete Notes NCERT
Science
Science Class 9 — Complete Notes NCERT
Science
Class 12 Physics — Complete Notes and CBSE PYQs
Class 12 Physics
Class 12 Chemistry — Complete Notes and CBSE PYQs
Class 12 Chemistry
Your feedback helps us improve notes and tutorials.