Complete Reference Guide for Class 9 Mathematics
Prepared by Muhammad Tayyab (SSE Math)
Govt. Christian High School Daska
Rational & Irrational Numbers · Surds & Radicals · Properties of Real Numbers
Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025
The number of the form \( \frac{p}{q} \) where \( p,q \) are integers and \( q \neq 0 \) are called rational numbers.
Examples: \( \frac{2}{9}, \quad 7, \quad \sqrt{\frac{25}{16}} \)
\( Q = \left\{ x \,|\, x = \frac{p}{q}, \quad p,q \in \mathbb{Z} \land q \neq 0 \right\} \)
The number which cannot be expressed in the form \( \frac{p}{q} \) where \( p,q \) are integers and \( q \neq 0 \) are called irrational numbers.
Examples: \( e, \quad \pi, \quad \sqrt{2} \)
\( Q' = \left\{ x \,|\, x \neq \frac{p}{q}, \quad p,q \in \mathbb{Z} \land q \neq 0 \right\} \)
The union of the set of rational numbers and irrational numbers is known as the set of real numbers. It is denoted by \( \mathbb{R} \).
\( \mathbb{R} = Q \cup Q' \)
The decimal representations of rational numbers are of two types:
The decimal number with a finite number of digits after the decimal point is called a terminating decimal number.
Examples:
\( \frac{2}{5} = 0.4 \) and \( \frac{3}{8} = 0.375 \)
The decimal numbers with an infinitely repeating pattern of digits after the decimal point are called non-terminating and recurring decimal numbers.
Examples:
\( \frac{1}{3} = 0.333\ldots = 0.\overline{3} \) (3 repeats infinitely)
\( \frac{1}{6} = 0.166\ldots = 0.1\overline{6} \) (6 repeats infinitely)
\( \frac{22}{7} = 3.142857142857\ldots = 3.\overline{142857} \) (the pattern 142857 repeats infinitely)
Decimal numbers that do not repeat a pattern of digits after the decimal point continue indefinitely without terminating.
Non-terminating and non-recurring decimal numbers are known as irrational numbers.
Examples:
Note:
If \( n \) is a positive integer greater than 1 and \( a \) is a real number, then any real number \( x \) such that \( x^{n} = a \) is called (radical) the \( n^{th} \) root of \( a \), and in symbols is written as:
\( x^{n} = a \)
\( \left[ x^{n} \right]^{\frac{1}{n}} = \left[ a \right]^{\frac{1}{n}} \)
\( x = a^{\frac{1}{n}} \)
\( x = \sqrt[n]{a} \)
In the radical \( \sqrt[n]{a} \), the symbol \( \sqrt{} \) is called the radical sign, \( n \) is called the index of the radical and the real number \( a \) under the radical sign is called the radicand or base.
An irrational radical with rational radicand is called a surd.
Examples of surds:
Not surds:
Note: Every surd is an irrational number, but not every irrational number is a surd (e.g., \( \sqrt{\pi} \) is irrational but not a surd) and the product of two conjugate surds is a rational number.
A surd which contains a single term.
Examples: \( \sqrt{2}, \quad \sqrt{3} \)
A surd which contains sum of two monomial surds or sum of a monomial surd and a rational number.
Examples: \( \sqrt{2} + \sqrt{7}, \quad \sqrt{2} + 5 \)
Conjugate surd of \( \sqrt{a} + \sqrt{b} \) is defined as \( \sqrt{a} - \sqrt{b} \).
\( \sqrt{a} + \sqrt{b} \quad \leftrightarrow \quad \sqrt{a} - \sqrt{b} \)
| Name of the Property | ∀ a, b, c ∈ ℝ | Examples |
|---|---|---|
| Closure | \( a + b \in \mathbb{R} \) | \( 2 + 3 = 5 \in \mathbb{R} \) |
| Commutative | \( a + b = b + a \) | \( 2 + 5 = 5 + 2 \) \( 7 = 7 \) |
| Associative | \( a + (b + c) = (a + b) + c \) | \( 2 + (3 + 5) = (2 + 3) + 5 \) \( 2 + 8 = 5 + 5 \) \( 10 = 10 \) |
| Identity | \( a + 0 = a = 0 + a \) | \( 5 + 0 = 5 = 0 + 5 \) |
| Inverse | \( a + (-a) = (-a) + a = 0 \) | \( 6 + (-6) = (-6) + 6 = 0 \) |
| Name of the Property | ∀ a, b, c ∈ ℝ | Examples |
|---|---|---|
| Closure | \( ab \in \mathbb{R} \) | \( 2 \times 5 = 10 \in \mathbb{R} \) |
| Commutative | \( ab = ba \) | \( 2 \times 5 = 5 \times 2 \) \( 10 = 10 \) |
| Associative | \( a(bc) = (ab)c \) | \( 2 \times (3 \times 5) = (2 \times 3) \times 5 \) \( 2 \times 15 = 6 \times 5 \) \( 30 = 30 \) |
| Identity | \( a \times 1 = a = 1 \times a \) | \( 5 \times 1 = 5 = 1 \times 5 \) |
| Inverse | \( a \times \frac{1}{a} = \frac{1}{a} \times a = 1 \) | \( 6 \times \frac{1}{6} = \frac{1}{6} \times 6 = 1 \) |
Note:
| Property | Mathematical Expression |
|---|---|
| Left Distributive over Addition | \( a(b + c) = ab + ac \) |
| Left Distributive over Subtraction | \( a(b - c) = ab - ac \) |
| Right Distributive over Addition | \( (a + b)c = ac + bc \) |
| Right Distributive over Subtraction | \( (a - b)c = ac - bc \) |
| No. | Property Name | Mathematical Expression |
|---|---|---|
| i | Reflexive | \( \forall a \in \mathbb{R}, \quad a = a \) |
| ii | Symmetric | \( \forall a,b \in \mathbb{R}, \; a = b \Rightarrow b = a \) |
| iii | Transitive | \( \forall a,b,c \in \mathbb{R}, \; a = b \land b = c \Rightarrow a = c \) |
| iv | Additive | \( \forall a,b,c \in \mathbb{R}, \; a = b \Rightarrow a + c = b + c \) |
| v | Multiplicative | \( \forall a,b,c \in \mathbb{R}, \; a = b \Rightarrow ac = bc \) |
| vi | Cancellation (Addition) | \( \forall a,b,c \in \mathbb{R}, \; a + c = b + c \Rightarrow a = b \) |
| vii | Cancellation (Multiplication) | \( \forall a,b,c \in \mathbb{R}, \; ac = bc \Rightarrow a = b \) |
| No. | Property Name | Mathematical Expression |
|---|---|---|
| i | Trichotomy | \( \forall a,b \in \mathbb{R}, \) Either \( a = b \) or \( a > b \) or \( a < b \) |
| ii | Transitive | \( a > b \land b > c \Rightarrow a > c \) \( a < b \land b < c \Rightarrow a < c \) |
| iii | Additive | \( a > b \Rightarrow a + c > b + c \) \( a < b \Rightarrow a + c < b + c \) |
| iv | Multiplicative | \( a > b \Rightarrow ac > bc \) if \( c > 0 \) \( a < b \Rightarrow ac < bc \) if \( c > 0 \) \( a > b \Rightarrow ac < bc \) if \( c < 0 \) \( a < b \Rightarrow ac > bc \) if \( c < 0 \) \( a > b \land c > d \Rightarrow ac > bd \) \( a < b \land c < d \Rightarrow ac < bd \) |
| v | Division | \( a > b \Rightarrow \frac{a}{c} > \frac{b}{c} \) if \( c > 0 \) \( a < b \Rightarrow \frac{a}{c} < \frac{b}{c} \) if \( c > 0 \) \( a > b \Rightarrow \frac{a}{c} < \frac{b}{c} \) if \( c < 0 \) \( a < b \Rightarrow \frac{a}{c} > \frac{b}{c} \) if \( c < 0 \) |
| vi | Reciprocal | \( \forall a,b \in \mathbb{R} \) same sign \( a > b \Rightarrow \frac{1}{a} < \frac{1}{b} \) \( a < b \Rightarrow \frac{1}{a} > \frac{1}{b} \) |
| Laws of Radical | Laws of Indices |
|---|---|
| (i) \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \) | (i) \( a^{m} \cdot a^{n} = a^{m + n} \) |
| (ii) \( \sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}} \) | (ii) \( \left( a^{m} \right)^{n} = a^{mn} \) |
| (iii) \( \sqrt[n]{a^{m}} = \left( \sqrt[n]{a} \right)^{m} \) | (iii) \( (ab)^{n} = a^{n}b^{n} \) |
| (iv) \( \left( \sqrt[n]{a} \right)^{n} = a \) | (iv) \( \left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}} \) |
| (v) \( \frac{a^{m}}{a^{n}} = a^{m - n} \) | |
| (vi) \( a^{0} = 1 \) |
Yes, 0 is a rational number because it can be written in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
Examples: \( \frac{0}{1}, \frac{0}{2}, \frac{0}{3} \), etc., are valid rational numbers.
Since 0 divided by any nonzero integer is always 0, it satisfies the definition of a rational number.
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