Math Notes For Class 9 Chapter 1 Ex 1.1 Solved & Quiz
What are the Core Formulas and Definitions for this Exercise?
In the LATEST 2025–2026 Matric Class 9 Curriculum, Chapter 1: Real Numbers introduces the fundamental structural building blocks of mathematical analysis. The set of Real Numbers, denoted by $\mathbb{R}$, is formed by the union of Rational Numbers ($\mathbb{Q}$) and Irrational Numbers ($\mathbb{Q}'$).
- Rational Numbers ($\mathbb{Q}$): Any number that can be expressed in the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. In decimal representation, rational numbers are either terminating or non-terminating recurring (repeating).
- Irrational Numbers ($\mathbb{Q}'$): Numbers that cannot be written in the form $\frac{p}{q}$ for any integers $p$ and $q$. Their decimal expansion is non-terminating and non-recurring.
- Real Numbers ($\mathbb{R}$): The set union $\mathbb{R} = \mathbb{Q} \cup \mathbb{Q}'$, where $\mathbb{Q} \cap \mathbb{Q}' = \emptyset$.
| Property Name | Addition Rule ($a, b, c \in \mathbb{R}$) | Multiplication Rule ($a, b, c \in \mathbb{R}$) |
|---|---|---|
| Closure Property | $a + b \in \mathbb{R}$ | $a \cdot b \in \mathbb{R}$ |
| Commutative Property | $a + b = b + a$ | $a \cdot b = b \cdot a$ |
| Associative Property | $(a + b) + c = a + (b + c)$ | $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ |
| Identity Element | $a + 0 = a$ ($0$ is Additive Identity) | $a \cdot 1 = a$ ($1$ is Multiplicative Identity) |
| Inverse Element | $a + (-a) = 0$ ($-a$ is Additive Inverse) | $a \cdot \frac{1}{a} = 1$ ($\frac{1}{a}$ is Multiplicative Inverse, $a \neq 0$) |
| Distributive Property | $a \cdot (b + c) = a \cdot b + a \cdot c \quad \text{and} \quad (a + b) \cdot c = a \cdot c + b \cdot c$ | |
How to Solve All Exercise Questions Step-by-Step?
Question 1 (Part i): Identify whether $\sqrt{5}$ is a Rational or an Irrational Number.
Solution:
The integer $5$ is not a perfect square. Taking the square root of a non-perfect square yields a non-terminating, non-recurring decimal value ($\sqrt{5} \approx 2.236067977\dots$). Therefore, it cannot be expressed in the ratio form $\frac{p}{q}$ where $p, q \in \mathbb{Z}$.
Answer: $$\text{Irrational Number } (\mathbb{Q}')$$
Question 1 (Part ii): Identify whether $\frac{22}{7}$ is a Rational or an Irrational Number.
Solution:
The given number is explicitly written in the fractional form $\frac{p}{q}$, where numerator $p = 22 \in \mathbb{Z}$ and denominator $q = 7 \in \mathbb{Z}$ with $q \neq 0$. By definition, any such fraction is rational.
Answer: $$\text{Rational Number } (\mathbb{Q})$$
Question 1 (Part iii): Identify whether $0.333\dots$ is a Rational or an Irrational Number.
Solution:
The decimal fraction $0.333\dots$ is non-terminating but infinitely repeating (recurring) with period $3$. All non-terminating recurring decimal numbers can be represented as ratio of integers (specifically $\frac{1}{3}$). Hence, it belongs to the rational set.
Answer: $$\text{Rational Number } (\mathbb{Q})$$
Question 1 (Part iv): Identify whether $\pi$ is a Rational or an Irrational Number.
Solution:
The mathematical constant $\pi$ represents the ratio of the circumference of a circle to its diameter. Its decimal expansion is $3.1415926535\dots$, which is strictly non-terminating and non-recurring. Thus, it cannot be reduced to exact integer ratio form.
Answer: $$\text{Irrational Number } (\mathbb{Q}')$$
Question 2 (Part i): Convert the fraction $\frac{17}{25}$ into a decimal fraction and state whether it is terminating or repeating.
Solution:
Perform standard long division of $17$ by $25$:
$$17 \div 25 = 0.68$$
Since the division process terminates with a remainder of zero after two decimal places, this is a terminating decimal.
Answer: $$0.68 \quad \text{(Terminating Decimal)}$$
Question 2 (Part ii): Convert the fraction $\frac{2}{9}$ into a decimal fraction and state whether it is terminating or repeating.
Solution:
Perform standard long division of $2$ by $9$:
$$2 \div 9 = 0.2222\dots = 0.\overline{2}$$
The remainder never becomes zero and the digit $2$ repeats infinitely.
Answer: $$0.\overline{2} \quad \text{(Repeating/Recurring Decimal)}$$
Question 2 (Part iii): Convert the fraction $\frac{5}{8}$ into a decimal fraction and state whether it is terminating or repeating.
Solution:
Perform standard long division of $5$ by $8$:
$$5 \div 8 = 0.625$$
The division terminates completely with a zero remainder after three decimal steps.
Answer: $$0.625 \quad \text{(Terminating Decimal)}$$
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