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Struggling to find one reliable place with all your Class 9 Math notes? This guide brings together every chapter of the Punjab Textbook Board syllabus — Real Numbers, Logarithms, Sets and Functions, Trigonometry, Coordinate Geometry, and more — with clear definitions, key formulas, and solved examples you can revise from before exams.

These notes are based on the Al-Huda Science Academy Class 9 Mathematics study guide and are organized chapter by chapter so you can jump straight to what you need.

Chapter 1: Real Numbers

The set of real numbers (R) includes all rational and irrational numbers. Every real number has a unique position on the number line.

Classification of numbers:

• Natural numbers (N): 1, 2, 3, 4, …
• Whole numbers (W): 0, 1, 2, 3, …
• Integers (Z): …, -2, -1, 0, 1, 2, …
• Rational numbers (Q): numbers of the form p/q, where p, q are integers and q ≠ 0
• Irrational numbers (Q’): cannot be written as p/q, e.g. √2, √3, π
• Real numbers (R) = Q ∪ Q’

Key properties: closure, commutative, associative, distributive, identity, and inverse properties hold for real numbers under addition and multiplication.

Properties of equality/inequality: reflexive, symmetric, transitive, and trichotomy (for any two reals a, b, exactly one of a < b, a = b, a > b is true).

Radicals: ⁿ√a = a^(1/n), with laws ⁿ√a × ⁿ√b = ⁿ√(ab) and ⁿ√a ÷ ⁿ√b = ⁿ√(a/b).

Solved Example — Rationalization: Rationalize 1/(√5 − √3).

Multiply by the conjugate (√5 + √3): (√5+√3) / (5−3) = (√5 + √3)/2

Chapter 2: Logarithms

If aˣ = y (a > 0, a ≠ 1), then x = logₐy. This means aˣ = y ⇔ logₐy = x.

• Common logarithm: base 10, written log y
• Natural logarithm: base e (≈ 2.71828), written ln y

Laws of Logarithms:

• Product law: logₐ(mn) = logₐm + logₐn
• Quotient law: logₐ(m/n) = logₐm − logₐn
• Power law: logₐ(mⁿ) = n·logₐm
• logₐa = 1, logₐ1 = 0
• Change of base: logₐm = log_b m / log_b a

Characteristic and Mantissa: For numbers ≥ 1, characteristic = digits before decimal − 1. For numbers < 1, characteristic is negative.

Solved Example: Simplify log 8 + log 5 (given log 4 = 0.6021). log 8 + log 5 = log 40 = log 4 + log 10 = 0.6021 + 1 = 1.6021

Chapter 3: Sets and Functions

A set is a well-defined collection of distinct objects (elements), denoted by capital letters.

Types of sets: empty/null set (∅), singleton, finite, infinite, subset (A ⊆ B), equal sets, and universal set (U).

Set operations:

• Union: A ∪ B = {x : x ∈ A or x ∈ B}
• Intersection: A ∩ B = {x : x ∈ A and x ∈ B}
• Difference: A − B = {x : x ∈ A and x ∉ B}
• Complement: A′ = {x ∈ U : x ∉ A}

Ordered pairs and relations: (a, b) ≠ (b, a) unless a = b. The Cartesian product A × B is the set of all ordered pairs (a, b).

Functions: A function maps every element of A to exactly one element of B.

• Domain: set of inputs
• Range: set of outputs used
• One-to-one: distinct inputs → distinct outputs
• Onto: every element of B has a pre-image in A

Solved Example: If A = {1, 2, 3} and B = {4, 5}, find n(A × B). n(A × B) = n(A) × n(B) = 3 × 2 = 6

Chapter 4: Factorization and Algebraic Manipulation

Key algebraic identities (must memorize):

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• a² − b² = (a + b)(a − b)
• (a + b)³ = a³ + 3a²b + 3ab² + b³
• (a − b)³ = a³ − 3a²b + 3ab² − b³
• a³ + b³ = (a + b)(a² − ab + b²)
• a³ − b³ = (a − b)(a² + ab + b²)

Factorization methods: common factor, grouping, algebraic identities, and factorizing quadratic trinomials (ax² + bx + c).

H.C.F. and L.C.M.: H.C.F. × L.C.M. = product of the two expressions.

Solved Examples:

1. Factorize x² − 9 → x² − 3² = (x + 3)(x − 3)
2. Simplify (x² − 4)/(x² + 4x + 4) → (x+2)(x−2)/(x+2)² = (x − 2)/(x + 2)

Chapter 5: Linear Equations and Inequalities

A linear equation in one variable has the form ax + b = 0 (a ≠ 0) and exactly one solution. A linear equation in two variables has the form ax + by + c = 0.

Methods to solve simultaneous equations:

• Elimination method
• Substitution method
• Graphical method
• Matrix (Cramer’s rule) method

Linear inequalities: Adding/subtracting a number keeps the direction unchanged. Multiplying/dividing by a negative number reverses the inequality sign.

Solved Example: Solve x + y = 7, x − y = 1. Adding: 2x = 8 → x = 4. Substituting: y = 3. Answer: x = 4, y = 3

Chapter 6: Trigonometry

Angle measurement: π radians = 180°, 1 radian ≈ 57.3°

Trigonometric ratios (right triangle, angle θ): sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ

Special angle values:

Fundamental identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ

Applications: angle of elevation/depression used to find heights and distances.

Solved Example: A ladder makes a 60° angle with the ground and reaches 10 m up a wall. Find the ladder’s length. sin 60° = 10/L → L = 20/√3 ≈ 11.55 m

Chapter 7: Coordinate Geometry

Distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)

Slope: m = (y₂ − y₁)/(x₂ − x₁). Parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Equations of a line:

• Slope-intercept: y = mx + c
• Point-slope: y − y₁ = m(x − x₁)
• Two-point form: (y − y₁)/(x − x₁) = (y₂ − y₁)/(x₂ − x₁)

Area of a triangle: Area = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Solved Example: Find the distance between A(2, 3) and B(6, 6). d = √[16 + 9] = √25 = 5 units

Chapter 8: Logic

A statement is a sentence that is either true or false (not both).

Logical connectives:

• Negation (¬p): “not p”
• Conjunction (p ∧ q): true only when both are true
• Disjunction (p ∨ q): true when at least one is true
• Conditional (p → q): “if p then q”
• Biconditional (p ↔ q): “p if and only if q”

Tautology, Contradiction, Contingency: always true / always false / sometimes true.

Converse, Inverse, Contrapositive (for p → q):

• Converse: q → p
• Inverse: ¬p → ¬q
• Contrapositive: ¬q → ¬p (always logically equivalent to the original)

Chapter 9: Similar Figures

A proportion states two ratios are equal: a : b = c : d.

Similar triangles have equal corresponding angles and proportional corresponding sides, tested by:

• AA (Angle-Angle)
• SSS (side-side-side)
• SAS (side-angle-side)

Key theorems:

• Thales’ Theorem: a line parallel to one side of a triangle divides the other two sides proportionally
• The internal angle bisector divides the opposite side in the ratio of the adjacent sides

Areas of similar figures: ratio of areas = (ratio of corresponding sides)²

Solved Example: The sides of two similar triangles are in ratio 2:3. Find the ratio of their areas. (2/3)² = 4/9 → 4 : 9

Chapter 10: Graphs of Functions

Linear function: y = mx + c graphs as a straight line (slope m, y-intercept c).

Quadratic function: y = ax² + bx + c graphs as a parabola — opens upward if a > 0, downward if a < 0. Vertex x-coordinate = −b/2a.

Reading graphs: intercepts (roots), maximum/minimum points, increasing/decreasing intervals.

Solving equations graphically: real solutions of f(x) = 0 are where y = f(x) crosses the x-axis.

Chapter 11: Loci and Construction

A locus is the set of all points satisfying a given condition:

• Points equidistant from two fixed points → perpendicular bisector of the segment
• Points equidistant from two sides of an angle → angle bisector
• Points at a fixed distance from a point → a circle

Basic constructions: bisecting a line segment, bisecting an angle, constructing a perpendicular, and constructing triangles (SSS, SAS, ASA) with compass and straightedge.

Chapter 12: Information Handling (Statistics)

Types of data: raw data, grouped data, frequency distribution tables.

Measures of central tendency:

• Mean (x̄) = Σx / n
• Median: middle value of ordered data
• Mode: most frequent value

Measures of dispersion:

• Range = Highest − Lowest value
• Variance (σ²) = Σ(x − x̄)² / n
• Standard deviation (σ) = √Variance

Graphical representation: bar graph, histogram, frequency polygon, pie chart.

Solved Example: Find the mean of 4, 8, 6, 5, 3, 10. Sum = 36, Mean = 36/6 = 6

Chapter 13: Probability

Basic terms: experiment, sample space (S), event (a subset of S).

Probability formula: P(E) = n(E) / n(S), where 0 ≤ P(E) ≤ 1.

Laws of probability:

• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
• For mutually exclusive events: P(A ∪ B) = P(A) + P(B)
• P(A′) = 1 − P(A)

Independent events: P(A ∩ B) = P(A) × P(B)

Solved Example: A fair die is rolled once. Find P(number > 4). S = {1,2,3,4,5,6}, favorable = {5, 6} → P(E) = 2/6 = 1/3

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