Mathematics Class 9 Definitions

Complete collection of key terms, concepts, formulas, and theorems from all 13 chapters

Chapters

1 Real Numbers 2 Logarithms 3 Sets and Functions 4 Factorization 5 Linear Equations 6 Trigonometry 7 Coordinate Geometry 8 Logic 9 Similar Figures 10 Graphs of Functions 11 Loci and Construction 12 Information Handling 13 Probability

Mathematics Class 9 Definitions

This comprehensive collection contains all key definitions, terms, formulas, and concepts from the Class 9 Mathematics syllabus according to the Punjab Textbook Board. Each chapter is organized systematically with clear explanations and examples.

How to Use This Resource:

  • Click on any chapter in the sidebar to jump directly to its definitions
  • Each term is highlighted for easy reference
  • Important formulas are boxed for quick identification
  • For detailed explanations, visit the individual chapter pages

Chapter 1: Real Numbers

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Real Numbers

The set of all rational and irrational numbers. Represented by \(\mathbb{R}\).

Rational Numbers

Numbers that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\). They have terminating or repeating decimal expansions.

Irrational Numbers

Numbers that cannot be expressed as simple fractions. Their decimal expansions are non-terminating and non-repeating (e.g., \(\sqrt{2}\), \(\pi\)).

Radical Properties:
• \(\displaystyle \sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
• \(\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
• \(\displaystyle (\sqrt{a})^2 = a\)

Chapter 2: Logarithms

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Logarithm

The exponent to which a base must be raised to produce a given number. If \(a^x = n\), then \(\log_a n = x\).

Laws of Logarithms:
\(\log_a(mn) = \log_a m + \log_a n\)
\(\log_a\left(\frac{m}{n}\right) = \log_a m - \log_a n\)
\(\log_a(m^n) = n \log_a m\)
\(\log_a a = 1\)
\(\log_a 1 = 0\)

Common Logarithm

A logarithm with base 10, written as \(\log N\) (base 10 is implied).

Chapter 3: Sets and Functions

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Set

A well-defined collection of distinct objects, called elements.

Function

A relation between two sets where each element of the first set (domain) is paired with exactly one element of the second set (codomain).

Set Operations:
• Union (\(A \cup B\)): All elements in \(A\) or \(B\)
• Intersection (\(A \cap B\)): Elements common to \(A\) and \(B\)
• Complement (\(A^\complement\) or \(A'\)): Elements not in \(A\)
• Difference (\(A \setminus B\)): Elements in \(A\) but not in \(B\)

Chapter 4: Factorization

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Factorization

The process of breaking down an expression into a product of simpler expressions (factors).

Factorization Formulas:
• \(a^2 - b^2 = (a - b)(a + b)\) (Difference of squares)
• \(a^2 \pm 2ab + b^2 = (a \pm b)^2\) (Perfect square)
• \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\) (Sum/Difference of cubes)
• \(x^2 + (a+b)x + ab = (x + a)(x + b)\) (Trinomial factorization)

Common Factor

A factor that appears in all terms of an expression.

Chapter 5: Linear Equations

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Linear Equation

An equation of degree 1 that can be written in the form ax + b = 0 where a ≠ 0.

Inequality

A mathematical statement that compares two expressions using \( > \), \( < \), \( \geq \), or \( \leq \).

Solution Rules:
• Add/subtract same value to both sides: \( a > b \implies a \pm c > b \pm c \)
• Multiply/divide both sides by positive number: \( a > b \) and \( c > 0 \implies ac > bc \)
• Reverse inequality when multiplying/dividing by negative: \( a > b \) and \( c < 0 \implies ac < bc \)

Chapter 6: Trigonometry

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Trigonometric Ratios

Ratios of sides in a right-angled triangle relative to an acute angle θ.

Basic Trigonometric Ratios:
\(\sin \theta\) = \(\frac{\text{Opposite}}{\text{Hypotenuse}}\)
\(\cos \theta\) = \(\frac{\text{Adjacent}}{\text{Hypotenuse}}\)
\(\tan \theta\) = \(\frac{\text{Opposite}}{\text{Adjacent}}\)
\(\csc \theta\) = \(\frac{1}{\sin \theta}\)
\(\sec \theta\) = \(\frac{1}{\cos \theta}\)
\(\cot \theta\) = \(\frac{1}{\tan \theta}\)

Pythagorean Identity

\(\sin^2 \theta + \cos^2 \theta = 1\)

Chapter 7: Coordinate Geometry

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Cartesian Plane

A two-dimensional plane with perpendicular \(x\) and \(y\) axes intersecting at the origin \((0,0)\).

Key Formulas:
• Distance: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)
• Midpoint: \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
• Slope: \(m = \frac{y_2-y_1}{x_2-x_1}\)

Linear Equation Forms

Slope-intercept: \(y = mx + c\)
General form: \(ax + by + c = 0\)
Point-slope: \(y - y_1 = m(x - x_1)\)

Chapter 8: Logic

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Proposition

A declarative statement that is either true or false but not both.

Logical Connectives:
Conjunction (AND): \(p \land q\)
Disjunction (OR): \(p \lor q\)
Negation (NOT): \(\neg p\)
Conditional: \(p \to q\)
Biconditional: \(p \leftrightarrow q\)

Truth Table

A table showing all possible truth values of a compound statement.

Chapter 9: Similar Figures

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Similar Figures

Figures with the same shape but not necessarily the same size, having corresponding angles equal and sides proportional.

Similarity Criteria:
• AA (Angle-Angle)
• SAS (Side-Angle-Side)
• SSS (Side-Side-Side)
• Ratio of areas = \(\left(\frac{a}{b}\right)^2\)

Scale Factor

The ratio of corresponding lengths in similar figures.

Chapter 10: Graphs of Functions

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Linear Graph

A straight line graph representing a linear function \(y = mx + c\).

Quadratic Graph

A parabola-shaped graph representing a quadratic function \(y = ax^2 + bx + c\).

Graph Features:
• Slope (\(m\)): Steepness of linear graph
• Vertex: Turning point of parabola at \(\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)\)
• Roots: x-intercepts found using \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)

Chapter 11: Loci and Construction

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Locus

The set of all points that satisfy a given condition or set of conditions.

Common Loci:
• Circle: Locus of points at fixed distance from center
• Perpendicular bisector: Locus equidistant from two points
• Angle bisector: Locus equidistant from two lines

Geometric Construction

Creating geometric figures using only compass and straightedge.

Chapter 12: Information Handling

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Frequency Distribution

A table showing how often each value or range of values occurs in a data set, organized into classes with their frequencies.

Measures of Central Tendency:
• Mean: \(\bar{x} = \frac{\sum{x_i}}{n}\)
• Median: Middle value of ordered data
• Mode: Most frequent value(s)
• Range: \(x_{max} - x_{min}\)

Histogram

A bar graph representing frequency distribution of numerical data, where:

  • Bars touch each other (continuous data)
  • Area represents frequency
  • X-axis shows class intervals

Chapter 13: Probability

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Probability

A measure of the likelihood that an event will occur, calculated as: \( P(E) = \frac{\text{Favorable outcomes}}{\text{Total possible outcomes}} \)

Probability Formulas:
• Probability of Event: \( P(E) = \frac{n(E)}{n(S)} \)
• Probability Range: \( 0 \leq P(E) \leq 1 \)
• Complement Rule: \( P(E') = 1 - P(E) \)
• Independent Events: \( P(A \cap B) = P(A) \times P(B) \)

Sample Space

The set of all possible outcomes of an experiment, denoted by \( S \).

  • For a coin toss: \( S = \{H, T\} \)
  • For a die: \( S = \{1, 2, 3, 4, 5, 6\} \)